Functional responses for zooplankton feeding on multiple resources: a review of assumptions and biological dynamics. Deep Sea Research Part II: Topical Studies in Oceanography 50 (22-26): 2847-2875
ARTICLE IN PRESS
Deep-Sea Research II 50 (2003) 2847–2875
Functional responses for zooplankton feeding on multiple
resources: a review of assumptions and biological dynamics
Wendy Gentlemana,*, Andrew Leisingb, Bruce Frostc, Suzanne Stromd,
James Murrayc
a
Engineering Mathematics, Dalhousie University, 1340 Barrington Street, Halifax, NS, Canada B3J 2X4
b
Pacific Fisheries Environmental Laboratory, 1352 Lighthouse Ave., Pacific Grove, CA 93950, USA
c
School of Oceanography, University of Washington, Box 357940, Seattle, WA 98195-7940, USA
d
Shannon Point Marine Center, 1900 Shannon Point, Anacortes, WA 98221, USA
Received 3 May 2002; received in revised form 2 April 2003; accepted 15 July 2003
Abstract
Modelers often need to quantify the rates at which zooplankton consume a variety of species, size classes and trophic
types. Implicit in the equations used to describe the multiple resource functional response (i.e. how nutritional intake
varies with resource densities) are assumptions that are not often stated, let alone tested. This is problematic because
models are sensitive to the details of these formulations. Here, we enable modelers to make more informed decisions by
providing them with a new framework for considering zooplankton feeding on multiple resources. We define a new
classification of multiple resource responses that is based on preference, selection and switching, and we develop a set of
mathematical diagnostics that elucidate model assumptions. We use these tools to evaluate the assumptions and
biological dynamics inherent in published multiple resource responses. These models are shown to simulate different
resource preferences, implied single resource responses, changes in intake with changing resource densities, nutritional
benefits of generalism, and nutritional costs of selection. Certain formulations are further shown to exhibit anomalous
dynamics such as negative switching and sub-optimal feeding. Such varied responses can have vastly different ecological
consequences for both zooplankton and their resources; inappropriate choices may incorrectly quantify biologically-
mediated fluxes and predict spurious dynamics. We discuss how our classes and diagnostics can help constrain
parameters, interpret behaviors, and identify limitations to a formulation’s applicability for both regional (e.g. High-
Nitrate-Low-Chlorophyll regions comprising large areas of the Pacific) and large-scale applications (e.g. global
biogeochemical or climate change models). Strategies for assessing uncertainty and for using the mathematics to guide
future experimental investigations are also discussed.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Plankton dynamics; Functional response; Zooplankton grazing; Preference; Selection; Switching
1. Introduction
Models of planktonic populations and ecosys-
*Corresponding author. Tel.: +1-902-494-6086; fax: +1-902-
tems traditionally consider zooplankton as feeding
423-1801.
upon a single nutritional resource (i.e. only one
E-mail address: wendy.gentleman@dal.ca (W. Gentleman).
0967-0645/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.dsr2.2003.07.001
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2848
input to a ‘‘zooplankton box’’, Fig. 1a) even Zooplankton can exhibit a different functional
though their natural diets are usually comprised of response for each resource when that resource is
a mixture of trophic types, species, size-classes, the only nutrition available (i.e. different single
and detritus. However, models need to explicitly resource responses) due to differences in the
describe the ingestion of multiple resources (i.e. predators’ ability to perceive and capture specific
more than one input, Fig. 1b) in order to assess the prey (Green, 1986; Jonsson and Tiselius, 1990;
importance of omnivory, to estimate secondary DeMott and Watson, 1991). Different single
production, and to predict plankton dynamics in resource responses also arise from differences in
regions where zooplankton are food-limited. the resources’ nutritional content or time-scales for
Quantifying both the total nutritional intake and their handling and assimilation (Fenchel, 1980;
how that intake is derived from the various Jonsson, 1986). The intake rate for any one
resources is complicated because many factors resource may additionally be affected by the
contribute to the functional response (i.e. the way presence of other resources, such as when the time
intake changes with resource density; Solomon, devoted to one is restricted by the time devoted to
1949). others or when behavioral changes occur with
variations in relative resource densities (Donaghay
and Small, 1979; Ambler, 1986; Colton, 1987;
Gifford and Dagg, 1988; Verity, 1991, Kiorboe
et al., 1996; Strom and Loukos, 1998). Responses
may further be influenced by environmental
factors such as temperature and turbulence
(Rothschild and Osborn, 1988; Davis et al., 1991;
Kiorboe et al., 1996; Campbell et al., 2001; Sell
et al., 2001).
The convolution of such factors makes it
virtually impossible to determine the multiple
resource functional response from field data.
Experimental determination requires measurement
of the nutritional intake for ranges of combina-
tions of resource densities (Colton, 1987). Un-
fortunately, few such factorial design experiments
have been performed, leaving us with very
limited knowledge. As a result, most models of
multiple resources are based on explicit assump-
tions about how single resource responses
can be extended (e.g., prescribing additional
parameters or density dependencies). However,
implicit in the resulting equations are other
Fig. 1. Schematics of nutritional resources for zooplankton.
The number of nutritional resources explicitly considered by a assumptions that are not often stated, let alone
given model can easily be determined through examination of
tested. This makes it difficult to choose an
the model’s schematic. (a) Models that consider zooplankton
appropriate equation and to quantify the uncer-
feeding on a single resource (e.g. phytoplankton) have only one
tainty due to ignorance about the actual res-
arrow pointing to a ‘‘zooplankton box’’, and the specific rate of
intake is dictated by the single resource functional response, I Ã : ponse, which is problematic because models are
(b) Models with more than one arrow pointing to a sensitive to the details of these formulations (Jost
‘‘zooplankton box’’ consider zooplankton feeding on multiple
et al., 1973; Oaten and Murdoch, 1975a, b;
resources, such as different trophic types, species, size-classes
Matsuda et al., 1986; Franks et al., 1986;
and/or detritus. In these models, the specific rate of intake of
Gismervik and Andersen, 1997; Leising et al.,
resource i is dictated by the multiple resource responses Ii ; and
2003).
in this example since there are 5 arrows, i ¼ 1; 2; y; 5:
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2849
Our objective here is to enable and encourage theory and is defined by two parameters: handling
time h and successful attack rate a (Fig. 2b, Table 1).
researchers to make more informed decisions,
think critically about their choices, and explore The latter is the combined rate of encounter,
the consequences of alternatives. Beginning with a attack, and capture per capita resource and may
review of the various Types of single resource depend upon factors such as sensory reception,
responses, we then develop a similar classification motility, and turbulence (e.g. Rothschild and
for multiple resource responses, and present a set Osborn, 1988). The Michaelis–Menten equation
of mathematical diagnostics that elucidate model (Michaelis and Menten, 1913), also called the
assumptions. A review of published functional Monod equation (Monod, 1942, 1950), which is
responses for zooplankton feeding on multiple based on enzyme kinetics theory, is mathemati-
resources is presented, and our tools are used to cally equivalent to the Disk model but is char-
evaluate the assumptions and biological dynamics acterized using two different parameters:
maximum rate m and half-saturation constant k:
inherent in those formulations. We consider the
implications of different multiple resource re- The latter is the resource density for which the
intake is exactly half its maximum (i.e. when N ¼
sponses and make recommendations for modelers
k; I Ã ¼ m=2; Fig. 2b). The equivalence of these
who wish to incorporate such ecological structure
into their applications. Strategies for assessing formulations means that the Michaelis–Menten
how sensitive model results are to the assumptions, parameters can be expressed in terms of the Disk
parameters (i.e. m ¼ 1=h; k ¼ 1=ah; Table 1). Type
and how knowledge of the mathematical dynamics
can direct future experimental investigations, also 2 responses also have been described by the Ivlev
are discussed. equation (Ivlev, 1955), which represents the prob-
ability of feeding at the maximal rate m as
exponentially distributed with N according to the
parameter d: The Ivlev model has a different rate
2. Types of single resource functional responses
of change than the Disk/Michaelis–Menten model,
Single resource functional responses relate the even when their half-saturation values are identical
(i.e. even when d ¼ ðln 2=kÞ; Fig. 2b, Table 1).
specific rate (i.e. per capita zooplankton per unit
time) of nutritional intake, I Ã ; to resource density, While there is generally no statistical basis for
N: These models are based on laboratory experi- choosing one Type 2 model over another (Mullin
ments wherein predator populations are acclima- et al., 1975), there is observational evidence
tized to different resource densities, and on supporting the theory underlying the Disk for-
theoretical arguments regarding predator behavior mulation (e.g. Verity, 1991 and references therein).
Type 3 responses exhibit a curved variation of
and physiology. Holling (1959, 1962, 1965) de-
I Ã with N that contains a point of inflection
scribed four ‘‘Types’’ of relationships and alter-
native types have also been observed. Common separating the concave downward portion of the
curve from the portion that is not. Sigmoidal
responses are shown in Fig. 2 and listed in Table 1
along with sample references to where they have models describe moderate or ‘‘S-shaped’’ Type 3
been fit to data. In summary: response (Fig. 2c). The first Sigmoidal model
Type 1 responses exhibit a linear variation of I Ã (‘‘Sigmoidal I’’ in Table 1) assumes the constant
with N according to the constant rate of change r attack rate a of the Type 2 Disk equation now
(Fig. 2a). Type 1 responses may be Non-Satiating, varies linearly with resource density according to
#
the constant c (i.e. Disk’s a is replaced by a ¼ cN).
but are more typically Rectilinear, such that intake
reaches a maximum rate m for resource densities The second Sigmoidal model (‘‘Sigmoidal II’’ in
Table 1) assumes intake occurs in s steps ðs > 1Þ;
above a critical value v (Table 1).
Type 2 responses exhibit a curved variation of where each step s is described by Type 2
I Ã with N that is concave downward. They have Michaelis–Menten kinetics with half-saturation
constant ks and maximum rate m (Jost et al.,
been described by the Disk equation (Holling,
1973). When s ¼ 2; the second Sigmoidal model is
1959, 1965), which is based on predator–prey
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Fig. 2. Single resource functional responses. Plots of I Ã ; the nutritional intake associated with a single resource, versus resource
density, N (see text and Table 1 for model descriptions). (a) Type 1: Rectilinear ðm ¼ 1; v ¼ 2Þ; (b) Type 2: solid line is Disk/Michaelis–
Menten (a ¼ 1; h ¼ 1; equivalent to m ¼ 1; k ¼ 1), dashed line is Ivlev (m ¼ 1; d ¼ ln 2); (c) Type 3: solid line is Sigmoidal I
(c ¼ 1; h ¼ 1; equivalent to m ¼ 1; k ¼ 1); dashed line is Sigmoidal II (m ¼ 1; k1 ¼ k2 ¼ 0:4; s ¼ 2); (d) Type 3: Threshold (m ¼ 1;
k ¼ 1; t ¼ 0:5); (e) Type 4: Toxicity (m ¼ 1; k ¼ 0:1; b ¼ 0:25); (f) Alternative Type: Modified-Ivlev (e ¼ d ¼ ln 2).
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Table 1
Single resource functional responses.
Single Nutritional Intake Parameter Sample Empirical
Resource Models Dimensions References
I Ã ¼ rN ½r ¼ 1=ð½NTÞ
(A) Type 1: Non-satiating N/A
rN ¼ N m for Npv
(B) Type 1: Rectilinear Frost (1972), Hansen
½r ¼ 1=ð½NTÞ
IÃ ¼ v
and Nielsen (1997),
½m ¼ 1=T
m for N > v
½v ¼ ½N Mayzaud et al. (1998),
Hansen et al. (1999)
I Ã ¼ 1þahN ;
aN
(C) Type 2: Disk Mullin et al. (1975),
½a ¼ 1=ð½NTÞ
a.ka Michaelis–Menten Ohman (1984), Jonsson (1986),
½h ¼ T
½m ¼ 1=T
N
(a.k.a. Monod) Mayzaud et al. (1998),
¼ kþN m
½k ¼ ½N Verity (1991)
where m ¼ 1=h and k ¼ 1=ah
I à ¼ ð1 À expðÀdNÞÞm
(D) Type 2: Ivlev Deason (1980), Barthel (1983),
½m ¼ 1=T
Houde and Roman (1987)
½d ¼ 1=½N
I Ã ¼ kþN m
N
(E) Type 3: Threshold Mullin et al. (1975),
½m ¼ 1=T
eff
eff
Reeve (1977),
½k ¼ ½N
for Not
0
½t ¼ ½N
where Neff ¼ Goldman et al. (1989),
N À t for NXt
Strom (1991), Lessard and
Murrell (1998)
I Ã ¼ 1þ#ahN ; where a ¼ cN;
#
aN
(F) Type 3: Sigmoidal I Frost (1975), Ohman (1984),
½h ¼ T
#
½c ¼ 1=ð½N2 TÞ
(from Disk) Wickham (1995), Gismervik and
N2
¼ m;
(from Michaelis–Menten) Andersen (1997)
k2 þN 2 ½m ¼ 1=T
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
#
where m ¼ 1=h and ¼ N=ah ¼ 1= ch ½k ¼ ½N
I Ã ¼ Qs Ns
(G) Type 3: Sigmoidal II (Theoretical Reference)
½s ¼ ND
m; where s > 1
ðks þNÞ
Jost et al. (1973)
½m ¼ 1=T
s¼1
½ks ¼ ½N
When s ¼ 2 :
½k ¼ ½N
I Ã ¼ ðk1 þNÞðk2 þNÞ m ¼ k2 þN 2 þaN m
2 2
N N
½a ¼ ½N
pffiffiffiffiffiffiffiffiffi
ffi
where k ¼ k1 k2 and a ¼ k1 þ k2
I Ã ¼ kþNþbN 2 m
N
(H) Type 4: Prey Toxicity or (Bacteria References)
½m ¼ 1=T
Predator Confusion Veldkamp and Jannasch (1972),
½k ¼ ½N
½b ¼ 1=½N Van Gemerden (1974)
I Ã ¼ ð1 À eÀdN Þm;
# #
(I) Alternate Types: Mayzaud and Poulet (1978)
½d ¼ 1=½N
where m ¼ eN
Modified-Ivlev ½e ¼ 1=ðT½NÞ
similar to the first, but with an extra term ðaNÞ in resource density Neff ¼ N À t: Thus, the Thresh-
the denominator that results in a different rate of old model is a Michaelis–Menten response that is
shifted to the right such that N ¼ k þ t when I Ã ¼
change (Fig. 2c). An extreme Type 3 response
m=2; which makes it inappropriate to refer to
is described by the Threshold model (Fig. 2d,
the Threshold model’s k as the half-saturation
Table 1), where no intake occurs for resource
densities below a feeding threshold t: This thresh- constant.
Type 4 responses are the only ones that do not
old may be biologically justified or may be a proxy
for other processes (Strom et al., 2000). For N > t; increase monotonically with increasing resource
density. Instead, I Ã reaches a maximum rate m at
the Threshold equation is identical to a Michaelis–
an intermediate density Nmax ; and decreases for
Menten equation expressed in terms of an effective
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higher N (Fig. 2e). The decrease may occur Thus, the way Itot changes with the density of any
because of resource toxicity or predator confusion, one resource depends on the net effect of the
and/or may result from use of higher resource associated changes in every resource’s functional
densities in vitro than predators would encounter response.
in situ (i.e. higher than those for which predators Many different definitions have been used for
preference (e.g. Chesson, 1983, and references
have adapted or evolved). Type 4 responses have
been described by an equation similar to the therein; Strom and Loukos, 1998). Here we follow
Type 2 Michaelis–Menten model, but with addi- Chesson (1978, 1983), where the relative contribu-
tional term in the denominator ðbN 2 Þ that results tion of resource i to the total nutritional intake is
in m and the half-saturation value depending on equated to the relative contribution of Ni to a
complicated functions of the model parameters weighted measure of total resource density,
m; k; and b (Table 1). Ii f Ni
¼ni ð2Þ
;
Alternative types include a response that is P
Itot
fr Nr
similar to a Type 3 Sigmoidal model at low N;
r¼1
but that never exhibits satiation (Fig. 2f). This has
where the non-dimensional weights fi are defined
been described by the Modified-Ivlev model
as the preferences. The composition of the diet,
(Mayzaud and Poulet, 1978), in which the Ivlev’s
#
constant m is replaced by m ¼ eN (Table 1). Since therefore, can be thought of as a random sampling
from preferentially-biased resource densities fi Ni :
this formulation has no maximum rate, there is no
Preferences are typically normalized such that
relationship between the Modified-Ivlev d and the
any one fi o1; and Sfi ¼ 1: As Chesson (1983)
half-saturation value of other models.
observed, when timescales considered are small
enough that resource densities are essentially
3. Classification of multiple resource responses constant, the normalized preference for resource i
can be estimated by
The literature discusses multiple resource re-
Ii =Ni
sponses using terms such as preference, switching, fi ¼ ð3Þ
:
P
n
passive and active selection, optimal feeding, and Ir =Nr
generalism. Here we review the definitions of such r¼1
concepts, and develops a new classification of Recognizing that Fi ; the clearance rate of resource
multiple resource responses that is akin to the i; equals Ii =Ni (Frost, 1972), one can define fi in
various Types of single source responses. terms of the relative contribution of Fi to the total
When predators consume n different kinds of of all n resources’ clearance rates, i.e.
resources, the total intake of a particular nutrient
Fi
(e.g. nitrogen) depends on the nutritional intake fi ¼ n ð4Þ
:
P
derived from each resource. We denote Ii as the Fr
specific rate (i.e. per capita zooplankton per unit r¼1
time) of nutritional intake associated with resource It follows that the relative preference for resource i
i; and consider all resource densities, Ni ði ¼ over resource j is
1; 2; y; nÞ; to be expressed in a common currency
fi Fi
(e.g., nitrogen content). Therefore, Itot ; the specific ¼ ðjaiÞ; ð5Þ
fj Fj
rate of total nutritional intake from multiple
resources, is defined by which is equivalent to the relative contribution
X n those two resources makes to the diet as compared
Itot ¼ ð1Þ
Ii ; to their relative densities in the environment.
i¼1
Preferential intake of resource i over resource j
where Ii depends on Ni and may additionally occurs when Fi =Fj > 1; whereas the converse is true
when Fi =Fj o1:
depend on the density of other resources, Nj ðjaiÞ:
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The relative preference of any two resources density-independent (constant) and therefore no
may be constant or density-dependent because fi switching occurs.
Class 2 (Passive switching): Responses for which
are constant or density-dependent. The term
switching describes scenarios where Fi =Fj increases switching arises from passive selection due to
with an increase in relative resource density Ni =Nj density-dependent behaviors associated with the
(Murdoch, 1969); negative switching occurs when single resource responses.
Fi =Fj decreases when Ni =Nj increases (Chesson, Class 3 (Active switching): Responses for which
1983; Hutson, 1984). Switching means intake rates switching arises from active selection due to be-
change disproportionately with changes in re- haviors that depend on the relative densities of two
source densities in a way that can have a stabilizing resources in a manner that may not be predicted
influence on ecological stability (i.e. how robust from knowledge of the single resource responses.
the ecosystem is to environmental perturbations), Classification of a multiple resource response
and can promote biodiversity through predation depends on factors affecting feeding behavior,
which includes total nutritional intake Itot : Most
refuges for low-density resources (Oaten and
Murdoch, 1975a, b; May, 1977; Holt, 1983). foraging theories assume predators behaviorally
In contrast, negative switching can have a de- adapt in ways that maximize their nutritional gain,
stabilizing influence and can lead to resource as this enhances their ability to compete and would
extinction. be favored evolutionarily (e.g., Stephens and
The term selection refers to mechanisms causing Krebs, 1986). One way nutritional intake can be
maximized is for Itot to increase whenever resource
predators to choose among available resources.
Passive selection relates to factors such as differ- densities increase. Following Holt (1983), we
define optimal feeding as responses which exhibit
ential resource vulnerability (including prey moti-
such a positive dynamic and sub-optimal responses
lity and size), predator perceptual biases,
as those for which Itot decreases when available
nutritional or toxic content of the resources, and
time-scales for resource handling and assimilation nutrition increases. Foraging theory argues that
there is a selective advantage to generalism (i.e.
(Strom and Loukos, 1998, and references therein).
consuming ng different resources) over specialism
Thus, passive selection among multiple resources
(i.e. consuming only a subset ns ong ) when intake
arises from factors causing different single re-
source responses. In contrast, active selection of a wider variety of resources increases Itot (Holt,
relates to behaviors that depend on the relative 1983). Similarly, preferential selection of high-
densities of multiple resources, such as alternating quality resources is advantageous when their
improved nutritional content outweighs any cost
between ambush and suspension feeding, rejecting
of selection, such as that due to time lost
less abundant prey, or concentrating search activ-
ity on high-density patches (Landry, 1981; Holt, distinguishing among resources.
1983; Strom and Loukos, 1998, and references
therein). Passive and active selection are commonly
distinguished by the no-switching versus switching 4. Diagnostics for determining the assumed
nature of the response (Chesson, 1983; Strom and biological dynamics
Loukos, 1998). However, this is not a good metric
for making this distinction because passive selec- We have developed seven simple diagnostics
tion may be density-dependent (Landry, 1981; that can assess the biological dynamics inherent in
Holt, 1983), and theoretically active selection modeled multiple resource functional responses.
could result in constant preferences if the beha-
vioral density-dependence canceled in Eq. (5). 4.1. Diagnostic I: Effective preference
Based on the discussion above, we now define
three classes of multiple resource responses: A model’s assumed preferences are diagnosed by
Class 1 (No switching): Responses for which dividing each equation for Ii by Ni to solve for the
assumed clearance rates Fi and substituting these
the relative preference of any two resources are
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2854
constant but those of Ii are density-dependent.
into Eq. (4). Because the influence of any term
appearing in all modeled Fi is canceled in this When both parameters and behaviors are consis-
equation, modeled preferences often can be tent, passive selection is assumed, and preferences
assessed using terms that are mathematically can be predicted from single resource responses.
simpler than Fi : For example, the preference
formula reduces to a relative measure of attack 4.3. Diagnostic III: Change in intake of one
rates for certain responses (Chesson, 1983). We resource as its density increases
define effective preference Ei as the simplest
quantity that can be used in place of Fi in Eq. (4) A model’s assumed rate of change of intake of
to yield the preference fi : That is resource i for small increases in its density is
diagnosed by examining the partial derivative
Ei
fi ¼ n ð6Þ
;
P ð8Þ
@Ii =@Ni :
Er
r¼1
Eq. (8) is equivalent to the slope of the contours of Ii
where Ei may equal Fi or may be something versus Ni when all other resource densities,
that is mathematically simpler (e.g., attack rates). Nj ðjaiÞ; are invariant. The intake of resource i
It follows that relative preference fi =fj ¼ always increases with increasing Ni when the slope is
Ei =Ej ðjaiÞ: Therefore, a multiple resource model always positive. Where the slope is negative, a Type
assumes no switching occurs between resource i 4 kind of toxicity or confusion response is assumed
and j when Ei =Ej is constant (i.e. Class 1), whereas for resource i: Where the slope is zero, the density of
switching is assumed when the ratio depends on resource i is assumed to have no effect on its intake.
the density of at least one of the two resources.
Switching is assumed to be active (i.e. Class 3) 4.4. Diagnostic IV: Change in intake of one
when Ei depends on Nj ðjaiÞ; whereas switching resource as the density of another increases
may be passive (i.e. Class 2) or active when Ei only
depends on Ni ; determination requires investiga- A model’s assumed rate of change of intake of
tion of Diagnostic II. resource i for small increases in the density of
another resource is diagnosed by examining the
4.2. Diagnostic II: Implied single resource response partial derivative
ð9Þ
@Ii =@Nj ; jai:
The implied single resource response, IiÃimp ; is
Eq. (9) is equivalent to the slope of the contours of
the functional response assumed by a multiple
Ii versus Nj ðjaiÞ when the densities of all other
resource model when resource i is the only
available nutrition. IiÃimp is diagnosed by examin- resources—including Ni —are invariant. Where the
slope is zero, the density of resource j has no effect
ing the modeled intake when all other resource
on Ii : Where the slope is negative, resource j is
densities are zero, i.e.
assumed to interfere with the intake of resource i;
I Ãimp ¼ I ðN ¼ 0Þ ¼ I ðN ¼ 0Þ; jai: ð7Þ
j i j
tot
i as when time spent feeding on j reduces time
All parameters of IiÃimp are prescribed by the devoted to i: Where the slope is positive, a
actual single resource response, Iià ; provided the synergistic effect is assumed, as when behavior or
energy gain associated with j increases the ability
assumed type is correct. Any parameters of Ii that
do not appear in IiÃimp cannot be predicted from to detect or capture i:
the single resource responses (i.e. active selection),
4.5. Diagnostic V: Change in total nutritional
and multiple resource experiments are required
intake as resource density increases
to determine parameter values. Active selection
also is assumed when behaviors are inconsis-
tent between the single and multiple resource A model’s assumed rate of change of total
responses, as when attack rates in IiÃimp are nutritional intake for small increases in the density
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of a resource i is diagnosed by examining the as distinct, i.e.
partial derivative
Csel ¼ I Ãimp ðNtot Þ À Itot ;
X n
ð10Þ
@Itot =@Ni :
where Ntot ¼ ð12Þ
Nr :
r¼1
Eq. (10) is equivalent to the slope of the contours
Where Csel is positive, predators that do not
of Itot versus Ni when the densities of all other
resources Nj ðjaiÞ are invariant. Total nutritional distinguish among functionally equivalent resources
are assumed to be more successful. Where Csel is
intake is independent of Ni where the slope is zero.
negative, a model assumes a nutritional benefit to
Where the slope is positive, feeding is assumed to
selection. Where Csel equals zero, there is neither
be optimal. Where the slope is negative, such that
nutritional advantage nor disadvantage to selection.
total nutritional intake decreases when available
nutrition increases, sub-optimal feeding is as-
sumed.
5. Published multiple resource functional responses
and their assumed dynamics
4.6. Diagnostic VI: Nutritional benefit (or cost) of
generalism
Here, we review functional responses for zoo-
plankton feeding on multiple resources that have
A model’s assumptions about the nutritional
been used in the literature, and use the diagnostics
benefits of generalism, Bgen versus specialism is
presented in Section 4 to elucidate their assumed
diagnosed by calculating the difference in the
biological dynamics. Examples from each of the
modeled total nutritional intake for the two
three Classes outlined in Section 3 are considered.
cases, i.e.
ng
X X
ns 5.1. Examples of Class 1: No Switching models
Bgen ¼ Ig À ð11Þ
Is ; ng > ns :
g¼1 s¼1
Examples of Class 1 models and their associated
references are listed in Table 2a. The multiple
When specialists consume only one resource,
resource Disk model (Table 2a) is derived by
ns ¼ 1; and the second term on the right-hand
extending the single resource Disk model (Table 1)
side of Eq. (11) is equivalent to IiÃimp (Diagnostic
assuming: (i) predators attack and handle only one
II). Where Bgen is positive, generalism is assumed
resource at a time, and (ii) density-independence of
to be nutritionally advantageous, whereas special-
resource-dependent handling times hi and success-
ism is the better strategy where Bgen is negative.
ful attack rates ai (Murdoch, 1973; Bartram,
Where Bgen is zero, the assumption is that
1980). The multiple resource Disk and Michae-
nutritional costs and benefits are balanced.
lis–Menten models are equivalent formula-
tions expressed in terms of different parameters
4.7. Diagnostic VII: Nutritional cost (or benefits) (Table 2a), as was true for their single resource
of selection analogs (Table 1). Unlike the single resource
models, however, these multiple resource equa-
Resources that elicit identical IiÃimp (Diagnostic tions require specification of different numbers of
parameters: 2n (ai and hi ; i ¼ 1; y; n) for Disk
II), with respect to both Type and parameters are
versus 2n þ 1 ðmi ; pi and k) for Michaelis–Men-
functionally equivalent. A model’s assumptions
about the nutritional cost (or benefit) of selecting ten. The extra degree of freedom in the Michaelis–
among such equal quality resources, Csel ; is Menten model is made clear by dividing its
numerator and denominator by k; which results
diagnosed by differencing the modeled nutritional
intake when multiple resources are perceived as a in the identical functional response again defined
by only 2n parameters (i.e. Pi and mi ; Table 2a).
single nutrient pool versus when they are perceived
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2856
Table 2a
Class 1 multiple resource functional responses
Intake of resource i
Class I Formulation Parameter Sample References
Dimensions
ai Ni
(A) Disk Murdoch (1973),
½ai ¼ 1=ð½NTÞ
Ii ¼ P
n
a.k.a. Michaelis–Menten Frost (1987),
½hi ¼ T
ar hr Nr
1þ
r¼1
½mi ¼ 1=T Moloney and
P
n
pi Ni
¼ kþR mi where ¼ pr Nr Field (1991),a
½k ¼ ½N
r¼1
½pi ¼ N:D: Verity (1991),
P
n
#
¼ Pi N#i mi where R ¼ Pr Nr ½Pi ¼ 1=½N Gismervik and
1þR
r¼1
Andersen (1997),
where mi ¼ 1=hi ; Pi ¼ pi =k ¼ ai hi
Strom and
Loukos (1998)
(
(B) Threshold Evans (1988),
½m ¼ 1=T
pi Ni
RÀt Pn
m; for R > t
kþRÀt R
Ii ¼ where ¼ pr Nr Lancelot et al. (2000)
½k ¼ ½N
0; for Rot; r¼1
( # ½pi ¼ N:D:
##
t Pi Ni
RÀ# Pn
# m; for R > t ½Pi ¼ 1=½N
#
#t
¼ where R ¼ Pr Nr
1þRÀ# R
#t ½t ¼ ½N
0; for Ro# r¼1
#
where Pi ¼ pi =k and t ¼ t=k
P
n
(C) Ivlev Hofmann and
½m ¼ 1=T
Ii ¼ ½1 À expðÀdRÞpiR i m; where R ¼
N
pr Nr ;
Ambler (1988)
½k ¼ ½N
r¼1
P
n
½pi ¼ N:D:
#N #
¼ ½1 À expðÀRÞPiR i m; where R ¼ Pr Nr
#
½Pi ¼ 1=½N
r¼1
½d ¼ 1=½N
where Pi ¼ dpi
(
(D) Rectilinear Armstrong (1994)
½m ¼ 1=T
pi Ni
P
n
m; for Rpv
Ii ¼ piv i where R ¼
; pr Nr ½v ¼ ½N
N
R m; for R > v r¼1
( ½pi ¼ N:D:
# P
n
Pi Ni m; for Rp1 # ½Pi ¼ 1=½N
¼ Pi Ni where R ¼
; Pr Nr
#
# m; for R > 1 r¼1
R
where Pi ¼ pi =v
a
Moloney and Field (1991) is included as Class 1: Michaelis–Menten because their model implementation used a single value of the
half-saturation constant, k, for all resources. However, their generalized equation (their Eq. 3), which allows different half-saturation
constants for different resources (i.e. ki) is actually Class 3: Modified-Michaelis–Menten.
The usual justification for such overparameteriza- i.e.
Xn
tion is that parameters controlling the dynamics R
Itot ¼ m where R ¼ ð13Þ
pr N r ;
are defined by ones that are easier to measure or kþR r¼1
comprehend. Nonetheless, overparameterization
where pr are the weights. In this case, k becomes the
hides the real influence parameters have on the
value of R when Itot ¼ m=2; which is why k is called
modeled dynamics.
the half-saturation constant in the literature (Fas-
The Michaelis–Menten equation is one of the
ham et al., 1990; Moloney and Field, 1991; Strom
most commonly used formulations for zooplank-
and Loukos, 1998; Loukos et al., 1997; Pitchford
ton feeding on multiple resources, and all applica-
and Brindley, 1999). The equal mi restriction allows
tions of this model that we cite assume maximum
the Michaelis–Menten Ii (Table 2a) to be viewed as
rates are equal for all resources (i.e. all mi ¼ m).
the fraction of Itot that corresponds to the relative
With this restriction, the Michaelis–Menten Itot
contribution of Ni to R; i.e.
takes the form of the single resource Michaelis–
pi N i
Menten model (Table 1) expressed in terms of a
Ii ¼ Itot ð14Þ
:
weighted measure of the total resource density, R; R
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Table 2b
Diagnostics of Class 1 examples.
I II III IV V VI VII
IiÃimp
Ei @Ii =@Ni @Ii =@Nj @Itot =@Ni Bgen Csel
o0 always
ai Ni equal hi or equal hi or
(A) Disk >0 always =0 always
ai 1þai hi Ni
equal mi : equal mi :
pi mi ðor Pi mi Þ > 0 always X0 hi > hj or
mi omj : o0
hi > hj or
Michaelis– Ni
mi
Ã
ki þNi
mi > mj : o0
Menten at high Nj for
k 1
where kià ¼ ¼ at high Nj resource j
pi Pi
for Ni Xtià :
pi ðor Pi Þ near t > 0
(B) Threshhold >0 always X0 always X0 always =0 always
ðNi ÀtÃ Þ elsewhere: o0
i
m
ki þNi ÀtÃ
Ã
i
where kià ¼ pi ¼ Pi
k 1
and tià ¼ pi ¼ Pi
t t
for N otà : 0
i i
½1 À expðÀdià Ni Þm o0 always
pi ðor Pi Þ
(C) Ivlev >0 always X0 always X0 always =0 always
where dià ¼ dpi ¼ Pi
for Ni pvià : và m
Ni
pi ðor Pi Þ
(D) Rectilinear until X0 always X0 always =0 always
X0
i always satiated: =0
Ã:m
for Ni > vi
once
where và ¼ v ¼ 1
satiated: o0
i pi Pi
parameter k: Only when the maximum rates mi are
The multiple resource Threshold, Ivlev, and
Rectilinear models (Table 2a), which always identical for all resources do the Michaelis–
Menten Ei simplify to its pi parameters. Thus,
assume maximum rates are identical for all
despite pi being referred to as ‘‘preferences’’ in the
resources, are derived in an analogous manner to
the Michaelis–Menten models making the same literature (Fasham et al., 1990; Strom and Loukos,
assumption. That is: (i) Itot is described by each 1998; Loukos et al., 1997; Pitchford and Brindley,
1999), the term is a misnomer when any mi are
model’s respective single resource response (from
Table 1) expressed in terms of a weighted measure different. Had the Threshold, Ivlev, and Recti-
of total resource density R and (ii) Ii is defined by linear models allowed for resource-dependent
maximum rates, their Ei would also equal mi pi ;
Eq. (14). These three models also are overparame-
meaning reference to their pi as ‘‘selectivities’’ and
terized, in that the same functional response can be
described using one less parameter (i.e. Threshold: ‘‘vulnerabilities’’ (Hofmann and Ambler, 1988;
Pi ¼ pi =k; Ivlev: Pi ¼ dpi ; Rectilinear, Pi ¼ pi =v; Armstrong, 1994) is somewhat misleading.
Table 2a). Furthermore, measured clearance rates will only
yield independent estimates of pi in the specific
5.2. Dynamics assumed in Class 1: No Switching case when all maximum rates are equal.
examples
5.2.2. Diagnostic II
5.2.1. Diagnostic I All the Class 1 examples assume every resource
No Class 1 example ever assumes switching elicits the same Type of single resource response
(e.g. all Disk IiÃimp are Type 2 Disk, all Threshold
since all their Ei are constant, which is why they
IiÃimp are Type 3 Threshold, etc., Table 2b). They
are in this class (Table 2b). The Disk’s Ei are
the attack rates ai and the Michaelis–Menten’s also all assume selection is passive, as parameters
Ei ð¼ mi pi Þ are the equivalent term scaled by the and behaviors are consistent between the single
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2858
Fig. 3. Class 1 multiple resource functional responses. Contour plots of nutritional intake from two resources versus resource densities
(N1 and N2 ) for Class 1: No Switching examples (see text and Table 2a for model descriptions). (a) Disk/Michaelis–Menten I1 ; equal
preferences (a1 ¼ a2 ¼ 1; h1 ¼ h2 ¼ 1; equivalent to m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1); (b) Disk/Michaelis–Menten I1 ; unequal
preferences (a1 ¼ 1; a2 ¼ 0:25; h1 ¼ h2 ¼ 1; equivalent to m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ 1; p2 ¼ 0:25); (c) Rectilinear I1 ; equal preferences
(m ¼ 1; v ¼ 2; p1 ¼ p2 ¼ 1); (d) Threshold I1 ; equal preferences (m; k ¼ 1; p1 ¼ p2 ¼ 1; t ¼ 0:5); (e) Disk/Michaelis–Menten Itot ;
parameters as in (a); (f) Disk/Michaelis–Menten Itot ; unequal preferences from unequal handling times/maximum rates (a1 ¼ a2 ¼ 1;
h1 ¼ 4; h2 ¼ 1; equivalent to m1 ¼ 0:25; m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1).
model because kià ; the half-saturation constant of
and multiple resource responses, although this is
IiÃimp ; is generally not equal to k; the so-called
not obvious in the overparameterized versions.
‘‘half-saturation constant’’ of Ii (i.e. kià ¼ k=pi ).
For example, it may incorrectly appear that active
However, k and kià should not be directly
selection is assumed by the Michaelis–Menten
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2859
compared because the two have different biologi- feeding is only assumed to cease when a weighted
cal significance; k is related to weighted, not actual, measure of the total resource density is less than t;
so consumption of resource i can occur when
resource densities (Eq. (14)). The same is true for
the feeding thresholds t and tà in the Threshold Ni ot and even when Ni otià ; the implied single
i
model. resource threshold (Table 2b). Analysis of Diag-
Analysis of IiÃimp for the Michaelis–Menten, nostic IV therefore reveals that t is related to
Threshold, Ivlev, and Rectilinear models further minimal nutritional requirements as opposed to
reveals that Pi of their reduced-parameter versions minimal densities required for detection or attack.
are both measurable and meaningful quantities. In Therefore, the Threshold response could represent
contrast, pi of their overparameterized versions suspension-feeders or foragers that only have the
can only be determined when Pi are known a energy to generate feeding currents or successfully
priori. For example, the Michaelis–Menten Pi are attack resources when there sufficient total nutri-
the reciprocal of kià ; whereas pi are set by the tion available.
actual kià once the modeler chooses a value for k:
When the relationship between kià ; k; and pi is 5.2.5. Diagnostic V
unrecognized, modelers unwittingly assume speci- All Class 1 examples assume feeding is always
optimal when maximum rates mi (handling
fic values for the single resource half-saturation
times hi ) are identical for all resources (Table 2b,
constants. Hence, not only is the overparameter-
ization of these models unjustified, but it can Fig. 3e). However, feeding is sub-optimal when
mi ðhi Þ are resource-dependent and resource den-
obfuscate interpretation of behavior and choice of
sities are high, because Itot decreases for increases
appropriate parameter values.
in the relative density of resources with lower mi
5.2.3. Diagnostic III (longer hi ) (Fig. 3f).
Like their single resource analogs, the multiple
resource Disk/Michaelis–Menten and Ivlev Ii 5.2.6. Diagnostic VI
always increases when Ni increases, regardless of All Class 1 examples assume generalism is the
better strategy in regions where Itot increases with
resource preferences (Table 2b, Fig. 3a–b). While
increasing Ni (Table 2b, Fig. 3e). However, where
the Rectilinear model exhibits the same general
dynamic, the rate of change decreases sharply once feeding is sub-optimal, specialism on resources
intake is maximal, and intake never satiates on any with the largest maximum rates (shortest handling
one resource, which is in contrast to its single times) is more nutritionally advantageous (Fig. 3f).
resource analog (Fig. 3c). The Threshold Ii only
increases with Ni where resource densities are 5.2.7. Diagnostic VII
sufficiently high; variations in Ni are assumed to All the Class 1 examples assume there is neither
have no effect where resource densities are low nutritional cost nor benefit to selecting among
(Fig. 3d). functionally equivalent resources (Table 2b, Fig. 3e).
5.2.4. Diagnostic IV 5.3. Examples of Class 2: Passive Switching models
The Disk/Michaelis–Menten and Ivlev models
always assume interference of other resources, Examples of Class 2 models and their associated
regardless of resource preferences (Table 2b, references are listed in Table 3a.The No-Inter-
Fig. 3a–b). The Rectilinear and Threshold Ii ference model assumes the multiple resource
decrease for increasing Nj ðjaiÞ only when re- functional response for each resource is the same
source densities are high. When resource densities as when it is the only available nutrition (i.e.
Ii ¼ Iià ). The Modified-Threshold model, which
are low, Nj is assumed to have no effect on the
Rectilinear Ii (Fig. 3c), whereas the Threshold Ii we developed as an alternative to the Class 1
increases when Nj increases (Fig. 3d). The syner- Threshold model, allows for resource-dependent
maximum rates mi and feeding thresholds ti
gistic effect in the Threshold model arises because
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
2860
Table 3a
Class 2 multiple resource functional responses
Intake of resource Ni
Class 2 Parameter Sample
Formulation Dimensions References
Ii ¼ Iià dictated by I à ðTable 1Þ
(A) No-Interference Leonard et al.
(1999): I Ã = Alternative
where Iià is the single resource intake from Ni
Type: Modified-Ivlev
P
n
(B) Modified-Threshold This paper
½m ¼ 1=T
Pi Ni;eff
Ii ¼ mi ; where R ¼ Pr Nr;eff and
1þR ½Pi ¼ 1=½N
r¼1
½t ¼ ½N
Ni À ti ; for Ni Xti
Ni;eff ¼
for Ni oti
0;
#
½ci ¼ 1=ð½N2 TÞ
#
ai Ni
(C) Sigmoidal I Gismervik and
Ii ¼ ; where ai ¼ ci Ni
P
n
#
(from Disk) Andersen (1997),
½hi ¼ ½T
ar hr Nr
1þ
r¼1
(from Michaelis– Edwards (2001)
½mi ¼ 1=T
P
n
# Ni
# #
¼ kp2iþR mi ; where R ¼ pr Nr and pi ¼ pi Ni ½k ¼ ½N
Menten)
r¼1
½pi ¼ N:D:
Pn
#
# #
Pi Ni
¼ where R ¼
mi ; Pr Nr ½Pi ¼ 1=½N
#
1þR
r¼1
# # #
Pi ¼ pi =k2 ¼ ai hi
where mi ¼ 1=hi ;
#
½fi ¼ 1=ð½N2 TÞ
#
ai Ni fi Ni
(D) Sigmoidal II Chesson (1983)
Ii ¼ where ai ¼ ð1þgi Ni Þ
P
n
# ½hi ¼ ½T
ar hr Nr
1þ
r¼1
½gi ¼ 1=½N
Pn
(E) Abundance-Based I Strom and Loukos (1998)
½mi ¼ 1=T
#
#
pi Ni
Ii ¼ kþR m; where R ¼ pr Nr
½k ¼ ½N
r¼1
1 À wi Ni ; for Ni oZ ½pi ¼ N:D:
and wi ¼ ð1Àpi Þ
#
pi ¼
½Z ¼ ½N
Z
pi ; for Ni XZ
½wi ¼ 1=½N
(Table 3a). The other Class 2 models are derived Menten equation also use one more parameter
by extending the Class 1 Disk model assuming the than is necessary to describe the functional
attack rate for resource i depends upon its density response.
#
(i.e. the constant ai is replaced by ai that depends
on Ni ), but the handling time hi remains constant. 5.4. Dynamics assumed in Class 2: Passive
Switching examples
This is equivalent to extending the Michaelis–
Menten equation by replacing the constant pi with
#
pi that depends on Ni ; but keeping maximum rates 5.4.1. Diagnostic I
mi constant. These three models are distinguished The No-Interference Ei equal the single resource
clearance rates, Fià ; which results in switching
by their assumed density-dependence: (i) linear in
unless Iià is Type 1 Non-Satiating. The Sigmoidal
the first Sigmoidal model (‘‘Sigmoidal I’’ in Table
and Abundance-Based Ei are the density-depen-
3a); (ii) hyperbolic in the second Sigmoidal model
(‘‘Sigmoidal II’’ in Table 3a); and (iii) rectilinear in dent analogs of Class 1 Disk/Michaelis–Menten
#
models upon which they were based (i.e. Ei ¼ ai ¼
the Abundance-Based model (‘‘Abundance-Based
# #
mi Pi or Ei ¼ mi pi ), and the Modified-Threshold Ei
I’’ in Table 3a). The Abundance-Based model
additionally assumes all mi are equal. All formula- additionally depend on how Ni scales with the
tions based on the overparameterized Michaelis– threshold ti (Table 3b). All these examples assume
Table 3b
Diagnostics of Class 2 examples
I II III IV V VI VII
IiÃimp
Ei @Ii =@Ni @Ii =@Nj @Itot =@Ni Bgen Csel
Fià Iià o0; 0 or > 0
¼ 0 always
(A) No- X0 always X0 always
X0
depends on I Ã
Interference always unless Type 4
unless
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
Type 4
p0 always equal mi X0
(B) Modified- equal >0 always
for Ni Xti : for Ni Xti : X0
mi omj : o0
Pi mi ðNNi i Þ
i Àt ðNi Àti Þ mi X0
Threshold always
m
Ã
ki þNi Àti i
always at high Nj for
for Ni oti : 0 where kà ¼ 1
ARTICLE IN PRESS
mi omj o0
i Pi resource j
for Ni oti at high Nj
# o0 always
# ai Ni equal hi or equal hi or
(C) Sigmoidal I >0 >0 always
ai #
1þai hi Ni
#
# equal mi X0 equal mi :
(from disk) always
pi mi ðor Pi mi Þ Ni2
mi
à always
ðki Þ2 þNi2 X0
hi > hj or hi > hj or
(from
where kià ¼ pi ¼ pffiffiffiffi mi omj : mi omj : o0
k 1
Michaelis–
Pi
o 0 at high Nj
Menten) at high Nj for
resource j
# o0 always
Ni2
ai equal hi : X0
(D) Sigmoidal II >0 always equal >0 always
mÃ
à à i
ðki;1 þNi Þðki;2 þNi Þ hi : X0
always
where mà ¼ hi ;
1
hi > hj : hi > hj :
i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o0 at high Nj o0 at high Nj for
Ã
ki;1 ¼ 1 xi þ x2 À 4yi
i
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
resource j
à ¼ 1 x À x2 À 4y
ki;2 2 i i
i
xi ¼ fghi and yi ¼ fi1 i
i
h
i
# o0 at o0 at o0, 0 or 0 depends on o0, 0 or >0
pi
(E) Abundance- 40 at
for Ni XZ :
Ni
Based I intermediate intermediate intermediate dens. and params. depends on dens.
Ãm
ki þNi
dens. dens. dens. and params.
where kià ¼ pi
k
depends depends depends
for Ni oZ : on params. on params. on params.
Ni Àwi Ni2
m
kþNi Àwi Ni2
2861
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2862
Fig. 4. Class 2 multiple resource functional responses. Contour plots of nutritional intake from two resources versus resource densities
(N1 and N2 ) for Class 2: Passive Switching examples (see text and Table 3a for model descriptions). ‘‘PDD’’ = preference density
dependence. (a) Abundance-Based IiÃimp ; (m ¼ 1; k ¼ 0:5; pi ¼ 0:5; Z ¼ 1); (b) Sigmoidal I I1 ; equal PDD (h1 ¼ h2 ¼ 1; c1 ¼ c2 ¼ 1;
equivalent to m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1); (c) Modified-Threshold I1 ; equal PDD (m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1;
t1 ¼ t2 ¼ 0:5); (d) Abundance-Based I1 ; unequal PDD (m ¼ 1; k ¼ 0:5; p1 ¼ 1; p2 ¼ 0:25; Z ¼ 1); (e) Sigmoidal I Itot ; parameters as
per (d); (f) Sigmoidal I Itot ; unequal PDD from unequal handling times/maximum rates (h1 ¼ 4; h2 ¼ 1; c1 ¼ 0:25; c2 ¼ 1 equivalent to
m1 ¼ 0:25 m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1).
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2863
switching, since their Ei are density-dependent. assumed for Sigmoidal Ii (Fig. 4b). The Mod-
However, most examples also assume (essentially) ified-Threshold model assumes no effect when
Nj ðjaiÞ are low and interference when Nj > tj
no switching when resource densities are high.
Thus, when zooplankton behavior is consistent (Fig. 4c). In contrast, the certain parameter values
result in the Abundance-Based Ii exhibiting syner-
with these models, measured clearance rates will
not reveal switching unless experiments are con- gism (Table 3b), which results in regions of
ducted over a sufficiently broad range of densities. negative switching.
Furthermore, because these models’ Ei depend
only upon Ni ; determination of the passive nature 5.4.5. Diagnostic V
of this switching (i.e. the reason they are Class 2) Feeding is always optimal in the No-Interfer-
requires examination of Diagnostic II. ence model, provided none of the single resource
responses are Type 4. When all maximum rates mi
5.4.2. Diagnostic II (handling times hi ) are equal, the Modified-
Threshold and both Sigmoidal Itot also always
The No-Interference model allows for resource-
increase with increasing Ni (Table 3b, Fig. 4e).
dependent Types of single resource responses (e.g.
Type 1 Rectilinear for one resource and Type 3 However, these models assume feeding can be sub-
optimal when mi ðhi Þ are resource-dependent and
Sigmoidal for another), and the Modified-Thresh-
old IiÃimp can also be different Types depending on resource densities are high (Fig. 4f). Sub-optimal
whether a feeding threshold is specified (i.e. either feeding can occur at intermediate resource densi-
Type 3 Threshold or Type 2 Michaelis–Menten). ties when certain parameter values are used in the
In contrast, the Sigmoidal and Abundance-Based Abundance-Based model (Table 3b).
IiÃimp are the same Type for all resources.
5.4.6. Diagnostic VI
Furthermore, certain parameter values result in
the Abundance-Based IiÃimp being uncharacteristic All Class 2 examples assume generalism is the
best strategy where Itot increases with increasing
of any known response (Table 3b, Fig. 4a). The
Ntot (Table 3b, Fig. 4e). However, specialism is
behaviors and parameters are consistent between
the single and multiple resource formulations for more nutritionally advantageous where feeding is
all the Class 2 examples (Table 3b). Thus, passive sub-optimal (i.e. high resource densities for the
selection is assumed, and switching can be Modified-Threshold and both Sigmoidal models;
predicted when the actual I Ã are known. However, intermediate densities for Abundance-Based).
this fact is not obvious in the overparameterized
5.4.7. Diagnostic VII
formulations, and their extra degree of freedom is
unjustified. The Modified-Threshold and both Sigmoidal
models assume there is a nutritional cost to
5.4.3. Diagnostic III selecting among functionally equivalent resources
Ii always increases when Ni increases in both (Table 3b, Fig. 4e). In contrast, it can be
Sigmoidal models (Fig. 4b). The same dynamic is nutritionally beneficial to distinguish among such
assumed by the Modified-Threshold model when resources in the No-Interference and Abundance-
Based models, depending on I Ã and parameter
Ni > ti (Fig. 4c), and by the No-Interference model
when none of the single resource responses are values.
Type 4. However, the Abundance Based Ii can
decrease when Ni increases for certain parameter 5.5. Examples of Class 3: Active Switching models
values (Fig. 4d), which results in negative switch-
ing. Examples of Class 3 models and associated
references are listed in Table 4a. The Proportion-
5.4.4. Diagnostic IV Based model was used in Fasham et al. (1990), a
Other resources have no effect on the No- planktonic ecosystem model cited hundreds of
Interference Ii ; whereas interference is always times in the literature, which suggests that this is
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2864
Table 4a
Class 3 multiple resource functional responses
Intake of resource Ni
Class 3 Parameter References
Formulation Dimensions
P
n
(A) Proportion- Fasham et al. (1990),
½mi ¼ 1=T
#
#
pi Ni
Ii ¼ kþR m; where R ¼ pr Nr
Based Fasham et al. (1993),
½k ¼ ½N
r¼1
½pi ¼ N:D: Chai et al. (1996),
#
and pi ¼ P i Ni
p
n
Loukos et al. (1997),
pr Nr
Strom and Loukos (1998),
r¼1
Pitchford and Brindley (1999)
P
n
(B) Abundance- Strom and Loukos (1998)
½mi ¼ 1=T
#
#
pi Ni
Ii ¼ kþR m; where R ¼ pr Nr
Based II ½k ¼ ½N
r¼1
½pi ¼ N:D:
* *
1 À Nwi for NoZ
wi ¼ ð1Àpi Þ
# ½Z ¼ ½N
and pi ¼ ;
* Z
pi for NXZ ½wi ¼ 1=½N
*
and N is resource with largest pi
#
ai Ni
(C) Modified- Colton (1987)
½ai ¼ 1=ð½NTÞ
Ii ¼ ; where
P
n
## ½Aij ¼ 1=ð½N2 TÞ
Disk ar hr Nr
1þ
r¼1
½hi ¼ ½T
P P
n n
#
#
ai ¼ ai þ Aij Nj and hi ¼ hi þ Hij Nj ½Hij ¼ T=½N
j¼1 j¼1
ja1 ja1
Ni
(D) Modified- Moloney and Field (1991)
½mi ¼ 1=T
Ii ¼ m;
P
n
Michaelis–Menten ½ki ¼ ½N
kþ Nr
r¼1
P
Ni
¼k where kr;eff ¼ ki þ
m Nr
i;eff þNi
rai
and at least two ki are different
(otherwise this model reduces
to Class 1 Michaelis–Menten with
equal preferences)
bi Nil
(E) Switching Tansky (1978),
½ri ¼ 1=ð½NTÞ
Ii ¼ ri;eff Ni ; where ri;eff ¼ ri
R
Matsuda et al. (1986)
½l ¼ N:D:
P
n
½bi ¼ N:D:
br Nrl
and R ¼
r¼1
Ii ¼ Iià zi ;
#
(F) Weighted- Pace et al. (1984):
½zi ¼ N:D:
I Ã ¼ Type 1
intake
where Iià ¼ is the single resource response
Rectilinear with
z IÃ
#
for resource i; and zi ¼ Pi i additional lower
n
à zr Ir feeding threshold
r¼1
densities of other resources Nj ðjaiÞ: The Propor-
perhaps the most frequently used multiple resource
#
tion-Based pi depend on the relative contribution
functional response for zooplankton. Both the
of Ni to a weighted measure of total resource
Proportion-Based and the Abundance-Based
#
density, R: The Class 3 Abundance-Based pi all
(‘‘Abundance-Based II’’ in Table 4a) models
*
vary with N; the density of one particular resource
extend the Class 1 Michaelis–Menten equation
by replacing the constant pi with density-depen- assigned the highest pi ; according to the rectilinear
#
dent pi ; and assuming maximum rates are equal for relationship described for the Class 2 Abundance-
all resources. The difference from the Class 2 Based model. The Modified-Disk model extends
examples derived in a similar manner is that these the Class 1 Disk equation by assuming both attack
#
two Class 3 models assume pi vary with the rates ai and handling times hi vary linearly with
Table 4b
Diagnostics of Class 3 examples
I II III IV V VI VII
IiÃimp
Ei @Ii =@Ni @Ii =@Nj @Itot =@Ni Bgen Csel
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
p0 always o0 for low dens.
Ni
pi Ni > 0 always only > 0 for > 0 always
(A) Proportion- m
kþNi
Based high dens.
* * * o0 at o0 at o0 at o0, 0 or >0
(B) Abundance- >0 at
for i ¼same
for NoZ ¼ 1 À wi N
*
Based II intermediate intermediate intermediate intermediate depends on dens.
as Class 2
for NXZ ¼ pi
dens. dens. dens. dens. and params.
Ab.-Based 1
ARTICLE IN PRESS
* depends depends depends on depends
Ni
for i ai ¼ 1þNi m
on params. on params. params. on params.
# o0 at low dens. o0 at high and low
ai Ni
ai
(C) Modified- >0 at low dens. only >0 at >0 always
1 þ ai hi Ni
Disk depends on depends on dens. low dens.
params. params. depends on
params.
p0 always o0 at high dens. o0 at high Nj
Ni
1
(D) Modified- >0 always N/A
m
P
n ki þNi
ki þ
Michaelis– depends on params. (with
Nr for resource j
r¼1
Menten func. equiv.
resources,
model is
Class 1 M.M.)
p0 always o0 at high and o0 at low dens.
ðbi Ni Þl ri ri Ni
(E) Switching >0 always >0 always
low dens.
depends on
params.
Iià Fià zi Iià p0 always o0 at high and o0 at low dens.
(F) Weighted- X0 always X0 always
Intake unless Type 4 unless Type 4 low dens. unless Type 4
depends on
params.
2865
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2866
Nj ðjaiÞ; according to the constant rates of change for resource-dependent single resource half-satura-
Aij and Hij (Table 3a). tion constants, and the Modified-Disk allows for
The Modified-Michaelis–Menten, Switching and resource-dependent attack rates and handling
Weighted-Intake models are all based on assump- times. In the Class 3 Abundance-Based model,
tions about how other resources affect the single one resource solicits a non-standard response (i.e.
resource response, without linking the changes to Fig. 4a), while the others are assumed to be
any specific behavioral mechanism. The Modified- functionally equivalent with half-saturation con-
Michaelis–Menten model assumes other resources stants always equal to 1 in whatever units the
act to increase the effective half-saturation con- resource densities are measured. In contrast, the
stant ki;eff of a Type 2 Michaelis–Menten Iià (Table Weighted-Intake model allows each single re-
3a). The Switching model assumes other resources source response to be any type.
Examination of IiÃimp further reveals that certain
act to reduce the effective rate of change ri;eff of a
Type 1 Non-Satiating, according to a weighted parameters of the Proportion-Based, Weighted-
exponential measure of the total resource density Intake and Switching models cannot be predicted
(Table 3a). In the Weighted-Intake model, Iià is from knowledge of the single resource responses
(i.e. pi ; bi ; l and zi ; Table 4b) Therefore, behavior
reduced according to its relative contribution to a
weighted measure of the total of all Iià ; where zi in these three models is assumed to depend on the
are the weights (Table 3a). relative resource densities (i.e. active selection—
the reason they are considered Class 3), and these
5.6. Dynamics assumed in Class 3: Active parameters would have to be determined through
multiple resource experiments. IiÃimp also demon-
Switching examples
strates that the overparameterization of the
5.6.1. Diagnostic I Proportion-Based and Switching models is justi-
The Class 3 Modified-Disk, Proportion-Based fied, because their extra degree of freedom relates
and Abundance-Based Ei equal the density-depen- to a measurable quantity. That is the multiple
resource k is the half-saturation constant of the
dent analogs of the Class 1 Disk/Michaelis–
Menten models on which they were based (i.e. single resource response in the Proportion-Based
# #
Ei ¼ ai or Ei ¼ pi ), but the Modified-Michaelis– model, and ri is the single resource clearance rate
Menten Ei take a radically different form in the Switching model.
(Table 4b). The Switching and Weighted-Intake
Ei depend on both the single resource responses, 5.6.3. Diagnostic III
Iià ; and the parameters related to the assumed The Proportion-Based Ii always increases when
Ni increases (Table 4b), and the Weighted-Intake
influence of other resources. The Abundance-
Based, Modified-Disk and Modified-Michaelis– model makes the same assumption when none of
Menten Ei each depend on Nj ðjaiÞ; and therefore the single resource responses are Type 4. Of the
obviously assume active switching (i.e. Class 3). other Class 3 examples, only the Modified-Disk
and Abundance-Based models ever assume Ii can
Classification as active switching for the three
decrease when Ni increases, which results in regions
other examples requires examination of Diag-
nostic II. of negative switching (Fig. 5a). This Type 4 kind of
dynamic is due to the non-standard Abundance-
Based IiÃimp ; whereas in the Modified-Disk model it
5.6.2. Diagnostic II
results from the assumed behavioural changes.
Most Class 3 examples assume the single
5.6.4. Diagnostic IV
resource responses are all the same Type (Table
4b). The Proportion-Based model further assumes The Proportion-Based model always assumes
all resources are functionally equivalent, in that interference (Table 4b). Of the other Class 3
the parameters of IiÃimp are identical for all examples, only the Modified-Disk and Abun-
dance-Based Ii ever increase when Nj increases
resources. The Modified-Michaelis–Menten allows
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2867
Fig. 5. Class 3 multiple resource functional responses. Contour plots of nutritional intake from two resources versus resource densities
(N1 and N2 ) for Class 3: Active Switching examples (see text and Table 4a for model descriptions). ‘‘PDD’’ = preference density
dependence. (a) Modified-Disk I1 ; unequal PDD with equal handling times (a1 ¼ a2 ¼ 1; A12 ¼ 0; A21 ¼ 1; h1 ¼ h2 ¼ 0:25; H12 ¼ 0;
H21 ¼ 2); (b) Modified-Disk I1 ; another unequal PDD with equal handling times (a1 ¼ a2 ¼ 0:5; A12 ¼ 1; A21 ¼ 0; h1 ¼ h2 ¼ 0:5;
H12 ¼ H21 ¼ 0); (c) Modified-Michaelis–Menten Itot ; unequal PDD with equal maximum rates (m1 ¼ m2 ¼ 1; k1 ¼ 5; k2 ¼ 0:5;
p1 ¼ p2 ¼ 1); (d) Modified-Disk Itot ; with parameters as in (a); (e) Proportion-Based Itot ; equal PDD (m1 ¼ m2 ¼ 1;
k ¼ 1; p1 ¼ p2 ¼ 0:5); (f) Weighted-Intake Itot ; equal PDD (z1 ¼ z2 ¼ 1), where Iià is a rectilinear model (Table 1) with a lower
feeding threshold, as in Pace et al. (1984).
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2868
and results in regions of negative switching responses: no, passive, and active switching,
(Fig. 5b). This synergistic effect is again due to although such distinctions were rarely made in
the non-standard Abundance-Based IiÃimp ; and the the literature. The greatest differences among
Class 1 Ii occurred when resource i was rare, and
Modified-Disk’s assumed behavioural changes.
parameter values had little influence on contour
5.6.5. Diagnostic V shapes. In contrast, examples in both switching
In all the Class 3 examples, Itot can decrease classes also exhibited different dynamics for Ii
when Ni increases, and this assumption occurs for when resource i was common, and contour shapes
wider ranges of resource densities and parameter were sensitive to parameter values. The variations
values than the passive selection models. For of Ii versus Ni were generally analogous to the
example, even when all maximum rates mi (hand- different types of single resource responses. Most
ling times hi ) are equal, sub-optimal feeding occurs multiple resource models never assumed Ii de-
at high resource densities in the Modified-Michae- creased when Ni increased; however, this Type 4
lis–Menten and Modified-Disk models (Figs. 5c–d), kind of dynamic arose in some active switching
at intermediate resource densities in the Abun- examples. The assumed variation of Ii versus
dance-Based model, and at low resource densities in Nj ðjaiÞ covered the spectrum of possible re-
the Switching and Proportion-Based models (Fig. sponses (i.e. no effect, interference, and syner-
5e). When formulated as in Pace et al. (1984), the gism), even just among the Class 1 examples.
Weighted-Intake model also assumes sub-optimal Switching models generally assumed interference,
feeding at low resource densities (Fig. 5f). but varying parameter values and/or resource
densities led to negative switching in some cases.
5.6.6. Diagnostic VI We also found a wide diversity of modeled
All the Class 3 examples assume there are regions dynamics for Itot ; especially with respect to the
where specialism is more nutritionally advanta- optimality of feeding. Examples in all three classes
geous than generalism (Table 4b). However, these assumed there were regions where Itot decreased
regions are not restricted to where feeding is sub- when total available nutrition increased, even
optimal. In several models, specialism is assumed to when none of the single resource responses were
be a better strategy even when resources are Type 4. In the Class 1 and 2 passive selection
functionally equivalent and/or resource densities models, such sub-optimal feeding occurred only
low (e.g., Proportional-Based: Fig. 5e). when more nutritious resources became relatively
more rare. In such circumstances, these models
assumed specialization on high quality resources
5.6.7. Diagnostic VII
was a better strategy than generalism. The Active-
There is a nutritional cost to selection in most
Switching models demonstrated sub-optimal feed-
Class 3 examples (Table 4b), including those
ing over wider ranges of resource densities,
assuming resources are functionally equivalent,
including when resources were of equal quality.
such as the Proportion-Based model. The Abun-
Many Class 3 models additionally assumed
dance-Based model additionally assumes there is
specialism was more nutritionally advantageous
sometimes a benefit to distinguishing among
than generalism when resources were rare and
resources.
zooplankton were highly food-limited, although
some assumed the reverse. The No-Switching
models never assumed a cost or benefit to selecting
6. Discussion
among resources of equal quality, whereas switch-
ing models generally assumed zooplankton that
6.1. Assumed dynamics and their ecological
consequences perceived functionally equivalent resources as a
single nutrient pool would be more successful.
Our review identifies published models in all However, some models assumed it was occasion-
three classes of multiple resource functional ally beneficial to distinguish among such resources.
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The different dynamics assumed for Ii and Itot predators are tightly coupled, poor model choices
can have vastly different ecological consequences. also can result in spurious dynamics such as
For example, responses for which Ii increases with erroneously creating (or suppressing) blooms or
Ni tend to have a stabilizing influence on resource erroneously affording resource refuges (or causing
dynamics, in that predation may suppress resource extinction).
blooms, especially when the effective preference In order for a model to be useful, its math needs
for resource i depends on Ni : In contrast, blooms to be consistent with the biology it is trying to
are more likely to arise where predation pressure is represent. Recognition of the assumed dynamics
reduced as Ni increases. The stability of resource i can help modelers make appropriate choices for
is moderated by a model’s assumptions regarding their application. The implied single resource
response, IiÃimp ; is perhaps the most important
the effect of other resources Nj ðjaiÞ: Models
assuming interference decrease predation on i diagnostic for this assessment. It dominates the
when j becomes relatively more abundant, espe- modeled response when other resource densities
cially when predators switch. Such responses can are low, dictates the preferences and switching in
promote biodiversity by affording a refuge for passive selection models, and determines the
resources that are relatively more rare. However, nature of the behavioral assumptions in active
models assuming synergism can result in extinction selection responses. It also can reveal problems
of rare resources, even when this dynamic is not with candidate formulations, thereby ruling them
technically ‘‘negative switching’’ (e.g., Class 1 out or indicating where they need to be modified.
Certain models were shown to have IiÃimp that
Threshold). Models assuming sub-optimal feed-
ing may result in starved predators, whereas are uncharacteristic of any known Types (i.e.
predators may be satiated when feeding is assumed Abundance-Based for certain parameters), which
to be optimal. As these different assumptions recommends against their use. Inappropriate
directly affect zooplankton growth, they indirectly Types may also be assumed by other models,
affect both their ability to compete with other since many examples consider single resource
responses to be the same type for all resources
predators and the losses inflicted upon the
resources. when they are generally resource-dependent. When
actual single resource responses are consistent with
the assumed types, IiÃimp reveals the biological
6.2. How to choose the appropriate model
significance of the multiple resource model para-
We have shown how modeling decisions can be meters, identifies which are more precisely known
confused by the overparameterization and/or and how they relate to experimental measure-
misleading nomenclature of some multiple re- ments. Therefore, Diagnostic II determines
source models. Ignorance about the actual re- whether the overparameterization occurring in
sponse also may prompt modelers to use some models is justified (e.g., Proportion-Based)
or not (e.g., Michaelis–Menten), and indicates
previously published formulations, without con-
sidering whether they are appropriate for the new whether an assumption such as ‘‘equal maximum
application. However, as discussed above, even rates’’ is reasonable (e.g., copepod ingestion of
seemingly subtle differences in parameter values, different species or size classes: Frost, 1972;
density-dependence and/or density ranges result in Ambler, 1986; Gismervik and Andersen, 1997;
drastically different dynamics. Poor model choices ciliate growth on algae: Stoecker et al., 1986;
will incorrectly quantify resource preferences, over Verity, 1991; Montagnes, 1996), or not because
(or under)estimate resource consumption and resources have different handling times, nutri-
tional quality, and/or accessibility (e.g., copepod
predator growth, and predict contrasting effects
of changing resource densities. Such misrepresen- nauplii: Ambler, 1986; ciliate growth on nano-
tations can mislead conclusions about the impor- plankton: Verity, 1991).
tance of omnivory or magnitude of secondary Diagnostics III–V are also helpful in determin-
production. In systems where resources and ing whether a model is appropriate for a specific
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2870
application. For example, while one can hypothe- While any individual species may exhibit sub-
size reasons for a Type 4 response (e.g., toxicity or optimal feeding when there are dramatic changes
confusion) or synergistic effects of other resources in environmental conditions, such changes often
(e.g., more efficient searching), there should be lead to shifts in the local community structure (e.g.
actual biological support for such anomalous HNLC regions when iron is added). Zooplankton
dynamics before employing models that exhibit would likely adapt to long-term changes in
them (e.g., Class 1 Threshold is consistent with conditions and/or new dominant species would
Calanus pacificus ingesting phytoplankton; emerge. Certainly, different regions have different
Landry, 1981). One can similarly hypothesize dominant predators, which employ different feed-
reasons for sub-optimal feeding, such as energetic ing strategies and have different functional re-
costs of switching (Fasham et al., 1990) or fitness sponses. Hence, use of any single formulation and/
being unaffected by foraging yield (Holt, 1983). or set of parameters is not recommended for large-
However, theoretical arguments suggest natural scale applications spanning wide ranges of re-
selection would diminish the degree of sub- source densities and planktonic communities (e.g.,
optimality (Holt, 1983), and observational evi- global biogeochemical or climate-change models)
dence indicates that predators do select more as such approaches can systematically bias results.
nutritious resources (Cowles et al., 1988; Verity, This can be particularly problematic for formula-
1991 and references therein; Strom and Loukos, tions assuming different kinds of dynamics for
1998, and references therein; Meyer-Harms et al., different resource densities (e.g. Proportion-
1999). We know of no observations of decreas- Based). Large-scale applications therefore may
ing nutritional intake for increasing available require regionally and/or temporally varying
nutrition when resource densities are those sub-models and parameters in order to repre-
naturally encountered by the zooplankton. sent adequately differences among planktonic
This suggests that modelers should avoid communities.
use of formulations that assume sub-opti-
6.3. Assessing uncertainty due to assumptions
mal feeding for their system’s normal density
ranges.
Except for the Modified-Disk, all Class 3: Active There is often insufficient knowledge to support
Switching models we cite are based on hypothe- the choice of any one equation. Analyses of how
sized—not observed—behaviors. Authors typi- well different models fit observations can suggest
cally claimed the motivation for their assumed the better candidates (e.g., Carpenter et al., 1993),
behavioral density-dependence was that predators but consistency of a model with data does not
would focus on resources yielding greater nutri- validate assumptions because models of natural
tion. However, all these active selection examples systems are insufficiently constrained (Oreskes
exhibit the same kind of sub-optimal feeding as the et al., 1994). When models results hinge on
passive selection models: there are regions where unsupported assumptions they may incorrectly
total nutritional intake decreases for increasing corroborate or nullify hypotheses and mislead
resource density. Unlike any passive selection future research. This is especially important for
responses, this dynamic occurs even when re- predictive models of food-limited regions, since the
sources are of the same nutritional quality, and greatest differences among most models’ dynamics
when resource densities are low. Yet, the latter occur when resource densities are low. Hence,
condition is where selective pressure to feed sensitivity analyses always should be conducted in
optimally would likely be greatest because nutri- order to assess the uncertainty introduced by our
tional yield would be critical for survival. We ignorance.
therefore recommend against use of any unsup- Many sensitivity analyses are conducted by
ported Class 3 examples, especially for regions varying parameter values, usually only one at a
where predators are highly food-limited (e.g. time and often only in one direction (e.g., Evans,
HNLC). 1999). This is done despite the non-linearity of
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2871
modeled processes, or the fact that different shapes including passive selection, and how feeding
of the functional response can introduce variation behaviors may not optimize nutritional intake
into model results that is at least the magnitude of nor have stabilizing influences on resources.
variation due to uncertainty in parameter values. Therefore, measurements of constant or density-
We have shown how changing parameters can dependent resource preferences are insufficient to
radically change the assumed dynamics (e.g., from determine the nature of a functional response,
interference to synergism, or optimal to sub- especially when experiments cover only a limited
optimal feeding), meaning sensitivity to parameter range of resource densities. Our Classes and
values may actually indicate sensitivity to unsup- Diagnostics can aid experimental design, clarify
ported behavioral assumptions. The literature is parameters’ biological significance and help inter-
also rich with examples of how different dynamics pret zooplankton behavior.
arise from basic choices modelers make, such as In the same way that Diagnostics I and II
explicitly including omnivory or aggregating dif- together determine the class of a mathematical
ferent resources (e.g., May 1972, 1973; Holling, model, classification of an actual response requires
1973; Armstrong, 1994, 1999; Polis and Strong, measurements of clearance rates over ranges of
combinations of resource densities and knowledge
1996; Pahl-Wostl, 1997).
We recommend that assumptions related to the of the single resource responses. Empirical fits of
functional response be tested by varying both the latter indicate single resource behaviors (e.g.,
parameter values and model structure. Our Diag- constant attack rates like Type 1 and 2, or density-
nostics can identify formulations that assume dependent ones like Type 3), and hint at candidate
contrasting dynamics for the range of resource multiple resource models. Comparison of mea-
densities being considered, and thereby indicate sured and modeled preferences reveals whether
which models have the greatest potential to affect behaviors depend on the availability of other
resources (e.g. Iià are Disk, but measured prefer-
results. For example, models assuming optimal
feeding could approximate upper bounds on ences are not attack rates). When active selection
predator growth and resource consumption. These does occur, recognition of factors affecting the
results could be compared with the lowered composition of the diet (Diagnostic I, e.g. max-
growth and consumption resulting from responses imum rates), and the optimality of selection
for which there is a nutritional cost to selection or (Diagnostics V–VII) can suggest nutritionally
resource refuges (e.g. Class 2 Sigmoidal). Further advantageous behaviors (e.g., specialism, prefer-
comparisons could be made between models that ential for high quality resources, etc.), which might
assume resources are perceived as distinct (e.g., explain the data.
multiple resource food webs) versus those wherein Once an empirical model is developed, our
resources are perceived as a single nutrient pool Diagnostics can elucidate the biological dynamics
(e.g., single resource food chains). Confidence in resulting from that response. Recognition of these
conclusions is increased when results are relatively assumptions helps direct future research, especially
robust to the details of the functional response. when the model’s implied dynamics are incon-
However, when the formulation is crucial (as it sistent with what was expected. When the observed
usually is), then the inability to make estimates behavior implies Type 4, synergism, or sub-
with narrow ranges is an important conclusion and optimal feeding (Diagnostic III–V) at unmeasured
aids direction of future research. resource densities, experiments should be per-
formed to confirm whether such anomalous
6.4. Assumed dynamics help experimentalists dynamics are actually exhibited or if behavioral
adaptations occur. Alternatively, when unexpected
Our review found switching (and negative dynamics occur at measured densities, the math-
switching) responses that are no-switching at high ematical model suggests the conceptual model
and low resource densities. We also illustrated how should be revised. Experimental investigation of
switching could arise from a host of mechanisms, selection can be further aided by determining what
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
2872
factors affect preferences (Diagnostic I), and why determination by constraining parameters, inter-
certain resources may be preferred even when they preting behaviors, and recognizing limitations to a
are less abundant than others (Diagnostic V). model’s utility for both regional (e.g., HNLC) and
Diagnostics VI and VII can help formulate large-scale applications (e.g., global biogeochem-
hypotheses, as they suggest why the feeding ical or climate change). We identified published
strategies of dominant predators vary regionally. models with contrasting assumptions that can be
used in sensitivity studies to quantify the un-
6.5. Conclusions certainty introduced due to ignorance about the
actual response. Clarification of model dynamics
The Classes and Diagnostics we defined provide also helps direct future experimental research,
a framework for considering the varied behaviors especially when the math is not consistent with
the concept. We recommend researchers employ
and ecological implications of multiple resource
functional responses. They elucidate a models’ our framework when making decisions about
assumptions regarding resource preferences, im- multiple resource models, and thereby maximize
plied single resource responses, changes in intake the utility of such tools for advancing our ecologi-
with changing resource densities, nutritional ben- cal understanding and predictive capabilities.
efits of generalism, and nutritional costs of
selection. They reveal whether or not switching
Acknowledgements
can occur, the origin of switching when it does,
and where responses result in anomalous dynamics
The authors would like to thank Rob Arm-
such as negative switching or sub-optimal feeding.
strong for his comments on an early version of the
Our review of published multiple resource
paper, which greatly improved the generality and
models was by no means exhaustive; however, it
utility of this work. We would also like to
has still emphasized how model choice can be
acknowledge the helpful editorial feedback pro-
critical. The examples we cited exhibit dramati-
vided by Donald DeAngelis, Michio Kishi, and
cally different dynamics, even for seemingly subtle
anonymous reviewer, as well as Dan Kelley,
differences among formulations. We identified
George Jackson, and Mark Kot. This work was
equations that generally should be avoided, such
supported by National Science Foundation US
as the Abundance-Based models that are unchar-
JGOFS Grant OCE-9818770.
acteristic of any known response, and demon-
strated how there is no good reason to use any
overparameterized Class 1 formulation including
References
Michaelis–Menten. We revealed how passive
selection leads to sub-optimal intake when re-
Ambler, J.W., 1986. Formulations of an ingestion function for
sources are of different quality (e.g. Disk and
a populations of Paracalanus feeding on mixtures of
Sigmoidal models), yet all hypothesized behavioral phytoplankton. Journal of Plankton Research 8 (5),
adaptations in the Class 3 examples, including the 957–972.
popular Proportion-Based model, result in wider Armstrong, R.A., 1994. Grazing limitation and nutrient
limitation in marine ecosystems: steady state solutions of
regions of anomalous dynamics. This suggests use
an ecosystem model with multiple food chains. Limnology
of existing active selection models is hard to justify
and Oceanography 39 (3), 597–608.
for many applications, and points to the need for Armstrong, R.A., 1999. Stable model structures for represent-
theoreticians and experimentalists to develop more ing biogeochemical diversity and size spectra in plankton
realistic formulations. communities. Journal of Plankton Research 21 (3), 445–464.
Barthel, K.G., 1983. Food uptake and growth efficiency of
Modeling the nutritional intake for multiple
Eurytemora affinis (Copepoda: Calanoida). Marine Biology
resources is more complicated than it might seem.
74 (3), 269–274.
Choosing a formulation is not a straightforward; it Bartram, W.C., 1980. Experimental development of a model for
depends on the specific zooplankton and resources the feeding of neritic copepods on phytoplankton. Journal
being considered. Our diagnostics can assist in this of Plankton Research 3 (1), 25–51.
ARTICLE IN PRESS
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2873
Franks, P.J.S., Wroblewski, J.S., Flierl, G.R., 1986. Behavior of
Campbell, R.G., Wagner, M.W., Teegarden, G.J.,
a simple plankton model with food-level acclimation by
Boudreau, C.A., Durbin, E.G., 2001. Growth and
development rates of the copepod Calanus finmarchicus in herbivores. Marine Biology 91, 121–129.
the laboratory. Marine Ecology Progress Series 221, Frost, B.W., 1972. Effects of size and concentration of food
161–183. particles on the feeding behavior of the marine planktonic
Carpenter, S.R., Lathrop, R.C., Munoz-del-Rio, A., 1993. copepod Calanus pacificus. Limnology and Oceanography
Comparison of dynamic models for edible phytoplankton. 17 (6), 805–815.
Canadian Journal of Aquatic Fisheries and Sciences 50, Frost, B.W., 1975. A threshold feeding behavior in
1757–1767. Calanus pacificus. Limnology and Oceanography 20 (2),
Chai, F., Lindey, S.T., Barber, R.T., 1996. Origin and 263–266.
maintenance of a high nitrate condition in the equatorial Frost, B.W., 1987. Grazing control of phytoplankton stock in
Pacific. Deep-Sea Research II 43 (4–6), 1031–1064. the open subarctic Pacific Ocean: a model assessing the role
Chesson, J., 1978. Measuring preference in selective predation. of mesozooplankton, particularly the large calanoid cope-
Ecology 59 (2), 211–215. pods Neocalanus spp. Marine Ecology Progress Series 39,
Chesson, J., 1983. The estimation and analysis of preference 49–68.
and its relationship to foraging models. Ecology 64 (5), Gifford, D.J., Dagg, M.J., 1988. Feeding of the estuarine
1297–1304. copepod Acartia tonsa Dana: carnivory versus herbivory in
Colton, T.F., 1987. Extending functional response models to natural microplankton assemblages. Bulletin of Marine
include a second prey type: an experimental test. Ecology 68 Science 43 (3), 458–468.
(4), 900–912. Gismervik, I., Andersen, T., 1997. Prey switching by Acartia
Cowles, T.J., Olson, R.J., Chisholm, S.W., 1988. Food selection clausi: experimental evidence and implications of intraguild
by copepods: discrimination on the basis of food quality. predation assessed by a model. Marine Ecology Progress
Marine Biology 100, 41–49. Series 157, 247–259.
Davis, C.S., Flierl, G.R., Wiebe, P.H., Franks, P.J.S., 1991. Goldman, J.C., Dennet, M.R., Gordin, H., 1989. Dynamics of
Micropatchiness, turbulence and recruitment in plankton. herbivorous grazing by heterotrophic dinoflagellate Oxy-
Journal of Marine Research 49, 109–151. rrhis marina. Journal of Plankton Research 11 (2), 391–407.
Deason, E.E., 1980. Grazing of Acartia hudsonica (A. clausi) on Green, C.H., 1986. Patterns of prey selection: implications of
Skeletonema costatum in Narragansett Bay (USA): influence predator foraging tactics. American Naturalist 128 (6),
824–839.
of food concentration and temperature. Marine Biology 60
Hansen, P.J., Nielsen, T.G., 1997. Mixotrophic feeding of
(2/3), 101–113.
DeMott, W.R., Watson, M.D., 1991. Remote detection of algae Fragilidium subglobosum (Dinophyceae) on three species of
by copepods: responses to algal size, odors and motility. Ceratium: effects of prey concentration, prey species
Journal of Plankton Research 15, 1203–1222. and light intensity. Marine Ecology Progress Series 147,
Donaghay, P.L., Small, L.F., 1979. Food selection capabilities 187–196.
of the estuarine copepod Acartia clausi. Marine Biology 52, Hansen, B., Tande, K.S., Bergreen, O.C., 1999. On the trophic
fate of Phaeocystic pouchetii (Hariot). 3. Functional
137–146.
response in grazing demonstrated on juvenile stages of
Edwards, A.M., 2001. Adding detritus to a nutrient–
Calanus finmarchicus (Copepoda) fed diatoms and
phytoplankton–zooplankton model: a dynamical-systems
Phaeocystis. Journal of Plankton Research 12 (6),
approach. Journal of Plankton Research 23 (4), 389–413.
1173–1187.
Evans, G.T., 1988. A framework for discussing seasonal
Hofmann, E.E., Ambler, J.W., 1988. Plankton dynamics on the
succession and coexistence of phytoplankton species.
outer southeastern US continental shelf. Part II: a time-
Limnology and Oceanography 33 (5), 1027–1036.
dependent biological model. Journal of Marine Research 40
Evans, G.T., 1999. The role of local models and data sets in the
(4), 883–917.
Joint Global Ocean Flux Study. Deep-Sea Research I 46,
Holling, C.S., 1959. Some characteristics of simple types
1369–1389.
of predation and parasitism. Canadian Entomologist 91,
Fasham, M.J.R., Ducklow, H.W., McKelvie, S.M., 1990.
824–839.
A nitrogen-based model of plankton dynamics in the
Holling, C.S., 1962. Principles of insect predation. Annual
oceanic mixed layer. Journal of Marine Research 48,
Review of Entomology 6, 163–182.
591–639.
Holling, C.S., 1965. The functional response of predators to
Fasham, M.J.R., Sarmiento, J.L., Slater, R.D., Ducklow,
prey density and its role in mimicry and population
H.W., Williams, R., 1993. Ecosystem behavior at Bermuda
regulation. Memoirs of the Entomological Society of
Station ‘S’ and Ocean Weather Station ‘India’: a general
Canada 45, 3–60.
circulation model and observational analysis. Global
Holling, C.S., 1973. Resilience and stability of ecological
Biogeochemical Cycles 7, 379–415.
systems. Annual Review of Ecological Systems 4, 1–24.
Fenchel, T., 1980. Suspension feeding in ciliated protozoa:
Holt, R.D., 1983. Optimal foraging and the form of the
functional response and particle size selection. Microbial
predator isocline. American Naturalist 122 (4), 521–541.
Ecology 6, 1–11.
ARTICLE IN PRESS
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
2874
Houde, S.E.L., Roman, M.R., 1987. Effects of food quality Mayzaud, P., Poulet, S.A., 1978. The importance of the time
on the functional ingestion response of the copepod factor in the response of zooplankton to varying concentra-
Acartia tonsa. Marine Ecology Progress Series 40 (1–2), tions of naturally occurring particulate matter. Limnology
69–77. and Oceanography 23, 1144–1154.
Hutson, V., 1984. Predator mediated coexistence with a Mayzaud, P., Tirelli, V., Bernard, J.M., Roche-Mayzaud, O.,
switching predator. Mathematical Biosciences 68, 233–246. 1998. The influence of food quality on the nutritional
Ivlev, V.S., 1955. Experimental Ecology of the Feeding of acclimation of the copepod Acartia clausi. Journal of
Fishes. Pischepromizdat, Moscow, 302pp. (Translated from Marine Systems 15 (1–4), 483–493.
Russian by D. Scott, Yale University Press, New Haven, Meyer-Harms, B., Irigoien, X., Head, R., Harris, R., 1999.
1961.) Selective feeding on natural phytoplankton by Calanus
Jonsson, P.R., 1986. Particle size selection, feeding rates and finmarchicus before, during and after the 1997 spring bloom
growth dynamics of marine planktonic oligotrichous ciliates in the Norwegian Sea. Limnology and Oceanography 44 (1),
(Ciliophora: Oligotrichina). Marine Ecology Progress Series 154–165.
33, 265–277. Michaelis, L., Menten, M.L., 1913. Die Kinetik der Invertin-
Jonsson, P.R., Tiselius, P., 1990. Feeding behavior, prey wirkung. Biochemistry Z 49, 333–369.
detection, and capture efficiency of the copepod Acartia Moloney, C.L., Field, J.G., 1991. Modeling carbon and
tonsa feeding on planktonic ciliates. Marine Ecology nitrogen flows in a microbial plankton community. In:
Progress Series 60, 35–44. Reid, P.C., et al. (Ed.), Protozoa and their Role in Marine
Jost, J.L., Drake, J.F., Tsuchiya, H.M., Fredrickson, A.G., Processes, NATO ASI Series, Vol. G 25. Springer, Berlin.
1973. Microbial food chains and food webs. Journal of Monod, J., 1942. Recherches sur la croissance des cultures
Theoretical Biology 41, 461–484. bacteriennes. Hermann et Cie, Paris.
Kiorboe, T., Saiz, E., Viitasalo, M., 1996. Prey switching Monod, J., 1950. Annales de l Institut Pasteur, Paris 79, 390.
behavior in the planktonic copepod Acartia tonsa. Marine Montagnes, D.J.S., 1996. Growth responses of planktonic
Ecology Progress Series 143, 65–75. ciliates in the genera Strobilidium and Strombidium. Marine
Lancelot, C., Hannon, E., Becquevort, S., Veth, C., De Baar, Ecology Progress Series 130 (1–3), 241–254.
H.J.W., 2000. Modeling phytoplankton blooms and carbon Mullin, M.M., Stewart, E.F., Fuglister, F.J., 1975. Ingestion by
export in the Southern Ocean: dominant controls by planktonic grazers as a function of food concentration.
light and iron in the Atlantic sector in Austral spring Limnology and Oceanography 20 (2), 259–262.
1992. Deep-Sea Research I 47, 1621–1662. Murdoch, W.W., 1969. Switching in general predators: experi-
Landry, M.R., 1981. Switching between herbivory and carni- ments on prey specificity and stability of prey populations.
vory by the planktonic marine copepod Calanus pacificus. Ecological Monographs 39, 335–354.
Marine Biology 65, 77–82. Murdoch, W.W., 1973. The functional response of predators.
Leising, A., Gentleman, W.C., Frost, B.W., 2003. The threshold Journal of Applied Ecology 10, 335–342.
feeding response of microzooplankton within Pacific high- Oaten, A., Murdoch, W.W., 1975a. Functional response and
nitrate low-chlorophyll ecosystem model under steady stability in predator–prey systems. American Naturalist 109,
and variable iron input. Deep-Sea Research II, this issue 289–298.
(doi: 10.1016/j.dsr2.2003.07.002). Oaten, A., Murdoch, W.W., 1975b. Switching functional
Leonard, C.L., McClain, C.R., Murtugudde, R., Hofmann, response and stability in predator–prey systems. American
E.E., Harding Jr., L.W., 1999. An iron-based ecosystem Naturalist 109, 299–318.
model of the central equatorial Pacific. Journal of Geophy- Ohman, M.D., 1984. Omnivory by Euphausia pacifica: the
role of copepod prey. Marine Ecology Progress Series 19,
sical Research 104 (C1), 1325–1341.
Lessard, E.J., Murrell, M.C., 1998. Microzooplankton herbi- 125–131.
vory and phytoplankton growth in the northwestern Oreskes, N., Shrader-Frechette, K., Belitz, K., 1994. Verifica-
tion, validation and confirmation of numerical models in the
Sargasso Sea. Aquatic Microbial Ecology 16, 173–188.
Loukos, H., Frost, B., Harrison, D.E., Murray, J.W., 1997. An earth sciences. Science 263, 641–646.
ecosystem model with iron limitation of primary production Pace, M.L., Glasser, J.E., Pomeroy, L.R., 1984. A simulation
in the equatorial Pacific at 140 W: Deep Sea Research II 44 analysis of continental shelf food webs. Marine Biology 82,
47–63.
(9–10), 2221–2249.
Pahl-Wostl, C., 1997. Dynamic structure of a food web model:
Matsuda, H., Kawasaki, K., Shigesada, N., Teramoto, E.,
comparison with a food chain model. Ecological Modeling
Ricciardi, L.M., 1986. Switching effect on the stability of the
prey–predator system with three trophic levels. Journal of 100, 103–123.
Pitchford, J.W., Brindley, J., 1999. Iron limitation, grazing pre-
Theoretical Biology 112, 251–262.
ssure and oceanic high nutrient-low Chlorophyll (HNLC)
May, R.M., 1972. Limit cycles in predator–prey communities.
Science 177, 900–902. regions. Journal of Plankton Research 21 (3), 525–547.
May, R.M., 1973. Stability and Complexity in Model Ecosys- Polis, G.A., Strong, D.R., 1996. Food-web complexity and
tems. Princeton University Press, Princeton, NJ, 265pp. community dynamics. The American Naturalist 147 (5),
May, R.M., 1977. Predators that switch. Nature 269, 103–104. 813–846.
ARTICLE IN PRESS
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2875
Reeve, M.R., Walter, M.A., 1977. Observations on the Strom, S.L., Loukos, H., 1998. Selective feeding by protozoa:
existence of lower threshold and upper critical food concen- model and experimental behaviors and their consequences
tration for the copepod Acartia tonsa Dana. Journal of for population stability. Journal of Plankton Research 20
Experimental Marine Biology and Ecology 29 (3), 211–221. (5), 831–846.
Rothschild, B.J., Osborn, T.R., 1988. Small-scale turbulence Strom, S.L., Miller, C.B., Frost, B.W., 2000. What sets lower
and plankton contact rates. Journal of Plankton Research limits to phytoplankton stocks in high-nitrate, low-Chlor-
10, 465–474. ophyll regions of the open ocean? Marine Ecology Progress
Sell, A.F., Van Keuren, D., Madin, L.P., 2001. Predation by Series 193, 19–31.
omnivorous copepods on early developmental stages of Tansky, M., 1978. Switching effect in a prey-predator system.
Calanus finmarchicus and Pseudocalanus spp. Limnology Journal of Theoretical Biology 70, 263–271.
and Oceanography 46 (4), 953–959. Van Gemerden, H., 1974. Coexistence of organisms competing
Solomon, M.E., 1949. The natural control of animal popula- for the same substrate: an example among the purple sulfur
tions. Journal of Animal Ecology 18, 1–35. bacteria. Microbial Ecology 1, 104–119.
Stephens, D.W., Krebs, J.R., 1986. Foraging Theory. Princeton Veldkamp, H., Jannasch, H.W., 1972. Mixed culture studies
University Press, Princeton. with the chemostat. Journal of Applied Chemistry and
Stoecker, D.K., Cucci, T.L., Hulburt, E.M., Yentsch, C.M., 1986. Biotechnology 22, 105–123.
Selective feeding by Balanion sp. (Ciliata: Balanionidae) on Verity, P.G., 1991. Measurement and simulation of prey uptake
phytoplankton that best support its growth. Journal of by marine planktonic ciliates fed plastidic and aplastidic
Experimental Marine Biology and Ecology 95 (2), 113–130. nanoplankton. Limnology and Oceanography 36 (4),
Strom, S.L., 1991. Growth and grazing rates of herbivorous 729–750.
dinoflagellate Gymnodinium sp. from the open subarctic Wickham, S.A, 1995. Cyclops predation on ciliates: species-
Pacific Ocean. Marine Ecology Progress Series 78 (2), specific differences and functional response. Journal of
103–113. Plankton Research 17 (8), 1633–1646.
Deep-Sea Research II 50 (2003) 2847–2875
Functional responses for zooplankton feeding on multiple
resources: a review of assumptions and biological dynamics
Wendy Gentlemana,*, Andrew Leisingb, Bruce Frostc, Suzanne Stromd,
James Murrayc
a
Engineering Mathematics, Dalhousie University, 1340 Barrington Street, Halifax, NS, Canada B3J 2X4
b
Pacific Fisheries Environmental Laboratory, 1352 Lighthouse Ave., Pacific Grove, CA 93950, USA
c
School of Oceanography, University of Washington, Box 357940, Seattle, WA 98195-7940, USA
d
Shannon Point Marine Center, 1900 Shannon Point, Anacortes, WA 98221, USA
Received 3 May 2002; received in revised form 2 April 2003; accepted 15 July 2003
Abstract
Modelers often need to quantify the rates at which zooplankton consume a variety of species, size classes and trophic
types. Implicit in the equations used to describe the multiple resource functional response (i.e. how nutritional intake
varies with resource densities) are assumptions that are not often stated, let alone tested. This is problematic because
models are sensitive to the details of these formulations. Here, we enable modelers to make more informed decisions by
providing them with a new framework for considering zooplankton feeding on multiple resources. We define a new
classification of multiple resource responses that is based on preference, selection and switching, and we develop a set of
mathematical diagnostics that elucidate model assumptions. We use these tools to evaluate the assumptions and
biological dynamics inherent in published multiple resource responses. These models are shown to simulate different
resource preferences, implied single resource responses, changes in intake with changing resource densities, nutritional
benefits of generalism, and nutritional costs of selection. Certain formulations are further shown to exhibit anomalous
dynamics such as negative switching and sub-optimal feeding. Such varied responses can have vastly different ecological
consequences for both zooplankton and their resources; inappropriate choices may incorrectly quantify biologically-
mediated fluxes and predict spurious dynamics. We discuss how our classes and diagnostics can help constrain
parameters, interpret behaviors, and identify limitations to a formulation’s applicability for both regional (e.g. High-
Nitrate-Low-Chlorophyll regions comprising large areas of the Pacific) and large-scale applications (e.g. global
biogeochemical or climate change models). Strategies for assessing uncertainty and for using the mathematics to guide
future experimental investigations are also discussed.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Plankton dynamics; Functional response; Zooplankton grazing; Preference; Selection; Switching
1. Introduction
Models of planktonic populations and ecosys-
*Corresponding author. Tel.: +1-902-494-6086; fax: +1-902-
tems traditionally consider zooplankton as feeding
423-1801.
upon a single nutritional resource (i.e. only one
E-mail address: wendy.gentleman@dal.ca (W. Gentleman).
0967-0645/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.dsr2.2003.07.001
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input to a ‘‘zooplankton box’’, Fig. 1a) even Zooplankton can exhibit a different functional
though their natural diets are usually comprised of response for each resource when that resource is
a mixture of trophic types, species, size-classes, the only nutrition available (i.e. different single
and detritus. However, models need to explicitly resource responses) due to differences in the
describe the ingestion of multiple resources (i.e. predators’ ability to perceive and capture specific
more than one input, Fig. 1b) in order to assess the prey (Green, 1986; Jonsson and Tiselius, 1990;
importance of omnivory, to estimate secondary DeMott and Watson, 1991). Different single
production, and to predict plankton dynamics in resource responses also arise from differences in
regions where zooplankton are food-limited. the resources’ nutritional content or time-scales for
Quantifying both the total nutritional intake and their handling and assimilation (Fenchel, 1980;
how that intake is derived from the various Jonsson, 1986). The intake rate for any one
resources is complicated because many factors resource may additionally be affected by the
contribute to the functional response (i.e. the way presence of other resources, such as when the time
intake changes with resource density; Solomon, devoted to one is restricted by the time devoted to
1949). others or when behavioral changes occur with
variations in relative resource densities (Donaghay
and Small, 1979; Ambler, 1986; Colton, 1987;
Gifford and Dagg, 1988; Verity, 1991, Kiorboe
et al., 1996; Strom and Loukos, 1998). Responses
may further be influenced by environmental
factors such as temperature and turbulence
(Rothschild and Osborn, 1988; Davis et al., 1991;
Kiorboe et al., 1996; Campbell et al., 2001; Sell
et al., 2001).
The convolution of such factors makes it
virtually impossible to determine the multiple
resource functional response from field data.
Experimental determination requires measurement
of the nutritional intake for ranges of combina-
tions of resource densities (Colton, 1987). Un-
fortunately, few such factorial design experiments
have been performed, leaving us with very
limited knowledge. As a result, most models of
multiple resources are based on explicit assump-
tions about how single resource responses
can be extended (e.g., prescribing additional
parameters or density dependencies). However,
implicit in the resulting equations are other
Fig. 1. Schematics of nutritional resources for zooplankton.
The number of nutritional resources explicitly considered by a assumptions that are not often stated, let alone
given model can easily be determined through examination of
tested. This makes it difficult to choose an
the model’s schematic. (a) Models that consider zooplankton
appropriate equation and to quantify the uncer-
feeding on a single resource (e.g. phytoplankton) have only one
tainty due to ignorance about the actual res-
arrow pointing to a ‘‘zooplankton box’’, and the specific rate of
intake is dictated by the single resource functional response, I Ã : ponse, which is problematic because models are
(b) Models with more than one arrow pointing to a sensitive to the details of these formulations (Jost
‘‘zooplankton box’’ consider zooplankton feeding on multiple
et al., 1973; Oaten and Murdoch, 1975a, b;
resources, such as different trophic types, species, size-classes
Matsuda et al., 1986; Franks et al., 1986;
and/or detritus. In these models, the specific rate of intake of
Gismervik and Andersen, 1997; Leising et al.,
resource i is dictated by the multiple resource responses Ii ; and
2003).
in this example since there are 5 arrows, i ¼ 1; 2; y; 5:
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2849
Our objective here is to enable and encourage theory and is defined by two parameters: handling
time h and successful attack rate a (Fig. 2b, Table 1).
researchers to make more informed decisions,
think critically about their choices, and explore The latter is the combined rate of encounter,
the consequences of alternatives. Beginning with a attack, and capture per capita resource and may
review of the various Types of single resource depend upon factors such as sensory reception,
responses, we then develop a similar classification motility, and turbulence (e.g. Rothschild and
for multiple resource responses, and present a set Osborn, 1988). The Michaelis–Menten equation
of mathematical diagnostics that elucidate model (Michaelis and Menten, 1913), also called the
assumptions. A review of published functional Monod equation (Monod, 1942, 1950), which is
responses for zooplankton feeding on multiple based on enzyme kinetics theory, is mathemati-
resources is presented, and our tools are used to cally equivalent to the Disk model but is char-
evaluate the assumptions and biological dynamics acterized using two different parameters:
maximum rate m and half-saturation constant k:
inherent in those formulations. We consider the
implications of different multiple resource re- The latter is the resource density for which the
intake is exactly half its maximum (i.e. when N ¼
sponses and make recommendations for modelers
k; I Ã ¼ m=2; Fig. 2b). The equivalence of these
who wish to incorporate such ecological structure
into their applications. Strategies for assessing formulations means that the Michaelis–Menten
how sensitive model results are to the assumptions, parameters can be expressed in terms of the Disk
parameters (i.e. m ¼ 1=h; k ¼ 1=ah; Table 1). Type
and how knowledge of the mathematical dynamics
can direct future experimental investigations, also 2 responses also have been described by the Ivlev
are discussed. equation (Ivlev, 1955), which represents the prob-
ability of feeding at the maximal rate m as
exponentially distributed with N according to the
parameter d: The Ivlev model has a different rate
2. Types of single resource functional responses
of change than the Disk/Michaelis–Menten model,
Single resource functional responses relate the even when their half-saturation values are identical
(i.e. even when d ¼ ðln 2=kÞ; Fig. 2b, Table 1).
specific rate (i.e. per capita zooplankton per unit
time) of nutritional intake, I Ã ; to resource density, While there is generally no statistical basis for
N: These models are based on laboratory experi- choosing one Type 2 model over another (Mullin
ments wherein predator populations are acclima- et al., 1975), there is observational evidence
tized to different resource densities, and on supporting the theory underlying the Disk for-
theoretical arguments regarding predator behavior mulation (e.g. Verity, 1991 and references therein).
Type 3 responses exhibit a curved variation of
and physiology. Holling (1959, 1962, 1965) de-
I Ã with N that contains a point of inflection
scribed four ‘‘Types’’ of relationships and alter-
native types have also been observed. Common separating the concave downward portion of the
curve from the portion that is not. Sigmoidal
responses are shown in Fig. 2 and listed in Table 1
along with sample references to where they have models describe moderate or ‘‘S-shaped’’ Type 3
been fit to data. In summary: response (Fig. 2c). The first Sigmoidal model
Type 1 responses exhibit a linear variation of I Ã (‘‘Sigmoidal I’’ in Table 1) assumes the constant
with N according to the constant rate of change r attack rate a of the Type 2 Disk equation now
(Fig. 2a). Type 1 responses may be Non-Satiating, varies linearly with resource density according to
#
the constant c (i.e. Disk’s a is replaced by a ¼ cN).
but are more typically Rectilinear, such that intake
reaches a maximum rate m for resource densities The second Sigmoidal model (‘‘Sigmoidal II’’ in
Table 1) assumes intake occurs in s steps ðs > 1Þ;
above a critical value v (Table 1).
Type 2 responses exhibit a curved variation of where each step s is described by Type 2
I Ã with N that is concave downward. They have Michaelis–Menten kinetics with half-saturation
constant ks and maximum rate m (Jost et al.,
been described by the Disk equation (Holling,
1973). When s ¼ 2; the second Sigmoidal model is
1959, 1965), which is based on predator–prey
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2850
Fig. 2. Single resource functional responses. Plots of I Ã ; the nutritional intake associated with a single resource, versus resource
density, N (see text and Table 1 for model descriptions). (a) Type 1: Rectilinear ðm ¼ 1; v ¼ 2Þ; (b) Type 2: solid line is Disk/Michaelis–
Menten (a ¼ 1; h ¼ 1; equivalent to m ¼ 1; k ¼ 1), dashed line is Ivlev (m ¼ 1; d ¼ ln 2); (c) Type 3: solid line is Sigmoidal I
(c ¼ 1; h ¼ 1; equivalent to m ¼ 1; k ¼ 1); dashed line is Sigmoidal II (m ¼ 1; k1 ¼ k2 ¼ 0:4; s ¼ 2); (d) Type 3: Threshold (m ¼ 1;
k ¼ 1; t ¼ 0:5); (e) Type 4: Toxicity (m ¼ 1; k ¼ 0:1; b ¼ 0:25); (f) Alternative Type: Modified-Ivlev (e ¼ d ¼ ln 2).
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Table 1
Single resource functional responses.
Single Nutritional Intake Parameter Sample Empirical
Resource Models Dimensions References
I Ã ¼ rN ½r ¼ 1=ð½NTÞ
(A) Type 1: Non-satiating N/A
rN ¼ N m for Npv
(B) Type 1: Rectilinear Frost (1972), Hansen
½r ¼ 1=ð½NTÞ
IÃ ¼ v
and Nielsen (1997),
½m ¼ 1=T
m for N > v
½v ¼ ½N Mayzaud et al. (1998),
Hansen et al. (1999)
I Ã ¼ 1þahN ;
aN
(C) Type 2: Disk Mullin et al. (1975),
½a ¼ 1=ð½NTÞ
a.ka Michaelis–Menten Ohman (1984), Jonsson (1986),
½h ¼ T
½m ¼ 1=T
N
(a.k.a. Monod) Mayzaud et al. (1998),
¼ kþN m
½k ¼ ½N Verity (1991)
where m ¼ 1=h and k ¼ 1=ah
I à ¼ ð1 À expðÀdNÞÞm
(D) Type 2: Ivlev Deason (1980), Barthel (1983),
½m ¼ 1=T
Houde and Roman (1987)
½d ¼ 1=½N
I Ã ¼ kþN m
N
(E) Type 3: Threshold Mullin et al. (1975),
½m ¼ 1=T
eff
eff
Reeve (1977),
½k ¼ ½N
for Not
0
½t ¼ ½N
where Neff ¼ Goldman et al. (1989),
N À t for NXt
Strom (1991), Lessard and
Murrell (1998)
I Ã ¼ 1þ#ahN ; where a ¼ cN;
#
aN
(F) Type 3: Sigmoidal I Frost (1975), Ohman (1984),
½h ¼ T
#
½c ¼ 1=ð½N2 TÞ
(from Disk) Wickham (1995), Gismervik and
N2
¼ m;
(from Michaelis–Menten) Andersen (1997)
k2 þN 2 ½m ¼ 1=T
pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
#
where m ¼ 1=h and ¼ N=ah ¼ 1= ch ½k ¼ ½N
I Ã ¼ Qs Ns
(G) Type 3: Sigmoidal II (Theoretical Reference)
½s ¼ ND
m; where s > 1
ðks þNÞ
Jost et al. (1973)
½m ¼ 1=T
s¼1
½ks ¼ ½N
When s ¼ 2 :
½k ¼ ½N
I Ã ¼ ðk1 þNÞðk2 þNÞ m ¼ k2 þN 2 þaN m
2 2
N N
½a ¼ ½N
pffiffiffiffiffiffiffiffiffi
ffi
where k ¼ k1 k2 and a ¼ k1 þ k2
I Ã ¼ kþNþbN 2 m
N
(H) Type 4: Prey Toxicity or (Bacteria References)
½m ¼ 1=T
Predator Confusion Veldkamp and Jannasch (1972),
½k ¼ ½N
½b ¼ 1=½N Van Gemerden (1974)
I Ã ¼ ð1 À eÀdN Þm;
# #
(I) Alternate Types: Mayzaud and Poulet (1978)
½d ¼ 1=½N
where m ¼ eN
Modified-Ivlev ½e ¼ 1=ðT½NÞ
similar to the first, but with an extra term ðaNÞ in resource density Neff ¼ N À t: Thus, the Thresh-
the denominator that results in a different rate of old model is a Michaelis–Menten response that is
shifted to the right such that N ¼ k þ t when I Ã ¼
change (Fig. 2c). An extreme Type 3 response
m=2; which makes it inappropriate to refer to
is described by the Threshold model (Fig. 2d,
the Threshold model’s k as the half-saturation
Table 1), where no intake occurs for resource
densities below a feeding threshold t: This thresh- constant.
Type 4 responses are the only ones that do not
old may be biologically justified or may be a proxy
for other processes (Strom et al., 2000). For N > t; increase monotonically with increasing resource
density. Instead, I Ã reaches a maximum rate m at
the Threshold equation is identical to a Michaelis–
an intermediate density Nmax ; and decreases for
Menten equation expressed in terms of an effective
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2852
higher N (Fig. 2e). The decrease may occur Thus, the way Itot changes with the density of any
because of resource toxicity or predator confusion, one resource depends on the net effect of the
and/or may result from use of higher resource associated changes in every resource’s functional
densities in vitro than predators would encounter response.
in situ (i.e. higher than those for which predators Many different definitions have been used for
preference (e.g. Chesson, 1983, and references
have adapted or evolved). Type 4 responses have
been described by an equation similar to the therein; Strom and Loukos, 1998). Here we follow
Type 2 Michaelis–Menten model, but with addi- Chesson (1978, 1983), where the relative contribu-
tional term in the denominator ðbN 2 Þ that results tion of resource i to the total nutritional intake is
in m and the half-saturation value depending on equated to the relative contribution of Ni to a
complicated functions of the model parameters weighted measure of total resource density,
m; k; and b (Table 1). Ii f Ni
¼ni ð2Þ
;
Alternative types include a response that is P
Itot
fr Nr
similar to a Type 3 Sigmoidal model at low N;
r¼1
but that never exhibits satiation (Fig. 2f). This has
where the non-dimensional weights fi are defined
been described by the Modified-Ivlev model
as the preferences. The composition of the diet,
(Mayzaud and Poulet, 1978), in which the Ivlev’s
#
constant m is replaced by m ¼ eN (Table 1). Since therefore, can be thought of as a random sampling
from preferentially-biased resource densities fi Ni :
this formulation has no maximum rate, there is no
Preferences are typically normalized such that
relationship between the Modified-Ivlev d and the
any one fi o1; and Sfi ¼ 1: As Chesson (1983)
half-saturation value of other models.
observed, when timescales considered are small
enough that resource densities are essentially
3. Classification of multiple resource responses constant, the normalized preference for resource i
can be estimated by
The literature discusses multiple resource re-
Ii =Ni
sponses using terms such as preference, switching, fi ¼ ð3Þ
:
P
n
passive and active selection, optimal feeding, and Ir =Nr
generalism. Here we review the definitions of such r¼1
concepts, and develops a new classification of Recognizing that Fi ; the clearance rate of resource
multiple resource responses that is akin to the i; equals Ii =Ni (Frost, 1972), one can define fi in
various Types of single source responses. terms of the relative contribution of Fi to the total
When predators consume n different kinds of of all n resources’ clearance rates, i.e.
resources, the total intake of a particular nutrient
Fi
(e.g. nitrogen) depends on the nutritional intake fi ¼ n ð4Þ
:
P
derived from each resource. We denote Ii as the Fr
specific rate (i.e. per capita zooplankton per unit r¼1
time) of nutritional intake associated with resource It follows that the relative preference for resource i
i; and consider all resource densities, Ni ði ¼ over resource j is
1; 2; y; nÞ; to be expressed in a common currency
fi Fi
(e.g., nitrogen content). Therefore, Itot ; the specific ¼ ðjaiÞ; ð5Þ
fj Fj
rate of total nutritional intake from multiple
resources, is defined by which is equivalent to the relative contribution
X n those two resources makes to the diet as compared
Itot ¼ ð1Þ
Ii ; to their relative densities in the environment.
i¼1
Preferential intake of resource i over resource j
where Ii depends on Ni and may additionally occurs when Fi =Fj > 1; whereas the converse is true
when Fi =Fj o1:
depend on the density of other resources, Nj ðjaiÞ:
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2853
The relative preference of any two resources density-independent (constant) and therefore no
may be constant or density-dependent because fi switching occurs.
Class 2 (Passive switching): Responses for which
are constant or density-dependent. The term
switching describes scenarios where Fi =Fj increases switching arises from passive selection due to
with an increase in relative resource density Ni =Nj density-dependent behaviors associated with the
(Murdoch, 1969); negative switching occurs when single resource responses.
Fi =Fj decreases when Ni =Nj increases (Chesson, Class 3 (Active switching): Responses for which
1983; Hutson, 1984). Switching means intake rates switching arises from active selection due to be-
change disproportionately with changes in re- haviors that depend on the relative densities of two
source densities in a way that can have a stabilizing resources in a manner that may not be predicted
influence on ecological stability (i.e. how robust from knowledge of the single resource responses.
the ecosystem is to environmental perturbations), Classification of a multiple resource response
and can promote biodiversity through predation depends on factors affecting feeding behavior,
which includes total nutritional intake Itot : Most
refuges for low-density resources (Oaten and
Murdoch, 1975a, b; May, 1977; Holt, 1983). foraging theories assume predators behaviorally
In contrast, negative switching can have a de- adapt in ways that maximize their nutritional gain,
stabilizing influence and can lead to resource as this enhances their ability to compete and would
extinction. be favored evolutionarily (e.g., Stephens and
The term selection refers to mechanisms causing Krebs, 1986). One way nutritional intake can be
maximized is for Itot to increase whenever resource
predators to choose among available resources.
Passive selection relates to factors such as differ- densities increase. Following Holt (1983), we
define optimal feeding as responses which exhibit
ential resource vulnerability (including prey moti-
such a positive dynamic and sub-optimal responses
lity and size), predator perceptual biases,
as those for which Itot decreases when available
nutritional or toxic content of the resources, and
time-scales for resource handling and assimilation nutrition increases. Foraging theory argues that
there is a selective advantage to generalism (i.e.
(Strom and Loukos, 1998, and references therein).
consuming ng different resources) over specialism
Thus, passive selection among multiple resources
(i.e. consuming only a subset ns ong ) when intake
arises from factors causing different single re-
source responses. In contrast, active selection of a wider variety of resources increases Itot (Holt,
relates to behaviors that depend on the relative 1983). Similarly, preferential selection of high-
densities of multiple resources, such as alternating quality resources is advantageous when their
improved nutritional content outweighs any cost
between ambush and suspension feeding, rejecting
of selection, such as that due to time lost
less abundant prey, or concentrating search activ-
ity on high-density patches (Landry, 1981; Holt, distinguishing among resources.
1983; Strom and Loukos, 1998, and references
therein). Passive and active selection are commonly
distinguished by the no-switching versus switching 4. Diagnostics for determining the assumed
nature of the response (Chesson, 1983; Strom and biological dynamics
Loukos, 1998). However, this is not a good metric
for making this distinction because passive selec- We have developed seven simple diagnostics
tion may be density-dependent (Landry, 1981; that can assess the biological dynamics inherent in
Holt, 1983), and theoretically active selection modeled multiple resource functional responses.
could result in constant preferences if the beha-
vioral density-dependence canceled in Eq. (5). 4.1. Diagnostic I: Effective preference
Based on the discussion above, we now define
three classes of multiple resource responses: A model’s assumed preferences are diagnosed by
Class 1 (No switching): Responses for which dividing each equation for Ii by Ni to solve for the
assumed clearance rates Fi and substituting these
the relative preference of any two resources are
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2854
constant but those of Ii are density-dependent.
into Eq. (4). Because the influence of any term
appearing in all modeled Fi is canceled in this When both parameters and behaviors are consis-
equation, modeled preferences often can be tent, passive selection is assumed, and preferences
assessed using terms that are mathematically can be predicted from single resource responses.
simpler than Fi : For example, the preference
formula reduces to a relative measure of attack 4.3. Diagnostic III: Change in intake of one
rates for certain responses (Chesson, 1983). We resource as its density increases
define effective preference Ei as the simplest
quantity that can be used in place of Fi in Eq. (4) A model’s assumed rate of change of intake of
to yield the preference fi : That is resource i for small increases in its density is
diagnosed by examining the partial derivative
Ei
fi ¼ n ð6Þ
;
P ð8Þ
@Ii =@Ni :
Er
r¼1
Eq. (8) is equivalent to the slope of the contours of Ii
where Ei may equal Fi or may be something versus Ni when all other resource densities,
that is mathematically simpler (e.g., attack rates). Nj ðjaiÞ; are invariant. The intake of resource i
It follows that relative preference fi =fj ¼ always increases with increasing Ni when the slope is
Ei =Ej ðjaiÞ: Therefore, a multiple resource model always positive. Where the slope is negative, a Type
assumes no switching occurs between resource i 4 kind of toxicity or confusion response is assumed
and j when Ei =Ej is constant (i.e. Class 1), whereas for resource i: Where the slope is zero, the density of
switching is assumed when the ratio depends on resource i is assumed to have no effect on its intake.
the density of at least one of the two resources.
Switching is assumed to be active (i.e. Class 3) 4.4. Diagnostic IV: Change in intake of one
when Ei depends on Nj ðjaiÞ; whereas switching resource as the density of another increases
may be passive (i.e. Class 2) or active when Ei only
depends on Ni ; determination requires investiga- A model’s assumed rate of change of intake of
tion of Diagnostic II. resource i for small increases in the density of
another resource is diagnosed by examining the
4.2. Diagnostic II: Implied single resource response partial derivative
ð9Þ
@Ii =@Nj ; jai:
The implied single resource response, IiÃimp ; is
Eq. (9) is equivalent to the slope of the contours of
the functional response assumed by a multiple
Ii versus Nj ðjaiÞ when the densities of all other
resource model when resource i is the only
available nutrition. IiÃimp is diagnosed by examin- resources—including Ni —are invariant. Where the
slope is zero, the density of resource j has no effect
ing the modeled intake when all other resource
on Ii : Where the slope is negative, resource j is
densities are zero, i.e.
assumed to interfere with the intake of resource i;
I Ãimp ¼ I ðN ¼ 0Þ ¼ I ðN ¼ 0Þ; jai: ð7Þ
j i j
tot
i as when time spent feeding on j reduces time
All parameters of IiÃimp are prescribed by the devoted to i: Where the slope is positive, a
actual single resource response, Iià ; provided the synergistic effect is assumed, as when behavior or
energy gain associated with j increases the ability
assumed type is correct. Any parameters of Ii that
do not appear in IiÃimp cannot be predicted from to detect or capture i:
the single resource responses (i.e. active selection),
4.5. Diagnostic V: Change in total nutritional
and multiple resource experiments are required
intake as resource density increases
to determine parameter values. Active selection
also is assumed when behaviors are inconsis-
tent between the single and multiple resource A model’s assumed rate of change of total
responses, as when attack rates in IiÃimp are nutritional intake for small increases in the density
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2855
of a resource i is diagnosed by examining the as distinct, i.e.
partial derivative
Csel ¼ I Ãimp ðNtot Þ À Itot ;
X n
ð10Þ
@Itot =@Ni :
where Ntot ¼ ð12Þ
Nr :
r¼1
Eq. (10) is equivalent to the slope of the contours
Where Csel is positive, predators that do not
of Itot versus Ni when the densities of all other
resources Nj ðjaiÞ are invariant. Total nutritional distinguish among functionally equivalent resources
are assumed to be more successful. Where Csel is
intake is independent of Ni where the slope is zero.
negative, a model assumes a nutritional benefit to
Where the slope is positive, feeding is assumed to
selection. Where Csel equals zero, there is neither
be optimal. Where the slope is negative, such that
nutritional advantage nor disadvantage to selection.
total nutritional intake decreases when available
nutrition increases, sub-optimal feeding is as-
sumed.
5. Published multiple resource functional responses
and their assumed dynamics
4.6. Diagnostic VI: Nutritional benefit (or cost) of
generalism
Here, we review functional responses for zoo-
plankton feeding on multiple resources that have
A model’s assumptions about the nutritional
been used in the literature, and use the diagnostics
benefits of generalism, Bgen versus specialism is
presented in Section 4 to elucidate their assumed
diagnosed by calculating the difference in the
biological dynamics. Examples from each of the
modeled total nutritional intake for the two
three Classes outlined in Section 3 are considered.
cases, i.e.
ng
X X
ns 5.1. Examples of Class 1: No Switching models
Bgen ¼ Ig À ð11Þ
Is ; ng > ns :
g¼1 s¼1
Examples of Class 1 models and their associated
references are listed in Table 2a. The multiple
When specialists consume only one resource,
resource Disk model (Table 2a) is derived by
ns ¼ 1; and the second term on the right-hand
extending the single resource Disk model (Table 1)
side of Eq. (11) is equivalent to IiÃimp (Diagnostic
assuming: (i) predators attack and handle only one
II). Where Bgen is positive, generalism is assumed
resource at a time, and (ii) density-independence of
to be nutritionally advantageous, whereas special-
resource-dependent handling times hi and success-
ism is the better strategy where Bgen is negative.
ful attack rates ai (Murdoch, 1973; Bartram,
Where Bgen is zero, the assumption is that
1980). The multiple resource Disk and Michae-
nutritional costs and benefits are balanced.
lis–Menten models are equivalent formula-
tions expressed in terms of different parameters
4.7. Diagnostic VII: Nutritional cost (or benefits) (Table 2a), as was true for their single resource
of selection analogs (Table 1). Unlike the single resource
models, however, these multiple resource equa-
Resources that elicit identical IiÃimp (Diagnostic tions require specification of different numbers of
parameters: 2n (ai and hi ; i ¼ 1; y; n) for Disk
II), with respect to both Type and parameters are
versus 2n þ 1 ðmi ; pi and k) for Michaelis–Men-
functionally equivalent. A model’s assumptions
about the nutritional cost (or benefit) of selecting ten. The extra degree of freedom in the Michaelis–
among such equal quality resources, Csel ; is Menten model is made clear by dividing its
numerator and denominator by k; which results
diagnosed by differencing the modeled nutritional
intake when multiple resources are perceived as a in the identical functional response again defined
by only 2n parameters (i.e. Pi and mi ; Table 2a).
single nutrient pool versus when they are perceived
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Table 2a
Class 1 multiple resource functional responses
Intake of resource i
Class I Formulation Parameter Sample References
Dimensions
ai Ni
(A) Disk Murdoch (1973),
½ai ¼ 1=ð½NTÞ
Ii ¼ P
n
a.k.a. Michaelis–Menten Frost (1987),
½hi ¼ T
ar hr Nr
1þ
r¼1
½mi ¼ 1=T Moloney and
P
n
pi Ni
¼ kþR mi where ¼ pr Nr Field (1991),a
½k ¼ ½N
r¼1
½pi ¼ N:D: Verity (1991),
P
n
#
¼ Pi N#i mi where R ¼ Pr Nr ½Pi ¼ 1=½N Gismervik and
1þR
r¼1
Andersen (1997),
where mi ¼ 1=hi ; Pi ¼ pi =k ¼ ai hi
Strom and
Loukos (1998)
(
(B) Threshold Evans (1988),
½m ¼ 1=T
pi Ni
RÀt Pn
m; for R > t
kþRÀt R
Ii ¼ where ¼ pr Nr Lancelot et al. (2000)
½k ¼ ½N
0; for Rot; r¼1
( # ½pi ¼ N:D:
##
t Pi Ni
RÀ# Pn
# m; for R > t ½Pi ¼ 1=½N
#
#t
¼ where R ¼ Pr Nr
1þRÀ# R
#t ½t ¼ ½N
0; for Ro# r¼1
#
where Pi ¼ pi =k and t ¼ t=k
P
n
(C) Ivlev Hofmann and
½m ¼ 1=T
Ii ¼ ½1 À expðÀdRÞpiR i m; where R ¼
N
pr Nr ;
Ambler (1988)
½k ¼ ½N
r¼1
P
n
½pi ¼ N:D:
#N #
¼ ½1 À expðÀRÞPiR i m; where R ¼ Pr Nr
#
½Pi ¼ 1=½N
r¼1
½d ¼ 1=½N
where Pi ¼ dpi
(
(D) Rectilinear Armstrong (1994)
½m ¼ 1=T
pi Ni
P
n
m; for Rpv
Ii ¼ piv i where R ¼
; pr Nr ½v ¼ ½N
N
R m; for R > v r¼1
( ½pi ¼ N:D:
# P
n
Pi Ni m; for Rp1 # ½Pi ¼ 1=½N
¼ Pi Ni where R ¼
; Pr Nr
#
# m; for R > 1 r¼1
R
where Pi ¼ pi =v
a
Moloney and Field (1991) is included as Class 1: Michaelis–Menten because their model implementation used a single value of the
half-saturation constant, k, for all resources. However, their generalized equation (their Eq. 3), which allows different half-saturation
constants for different resources (i.e. ki) is actually Class 3: Modified-Michaelis–Menten.
The usual justification for such overparameteriza- i.e.
Xn
tion is that parameters controlling the dynamics R
Itot ¼ m where R ¼ ð13Þ
pr N r ;
are defined by ones that are easier to measure or kþR r¼1
comprehend. Nonetheless, overparameterization
where pr are the weights. In this case, k becomes the
hides the real influence parameters have on the
value of R when Itot ¼ m=2; which is why k is called
modeled dynamics.
the half-saturation constant in the literature (Fas-
The Michaelis–Menten equation is one of the
ham et al., 1990; Moloney and Field, 1991; Strom
most commonly used formulations for zooplank-
and Loukos, 1998; Loukos et al., 1997; Pitchford
ton feeding on multiple resources, and all applica-
and Brindley, 1999). The equal mi restriction allows
tions of this model that we cite assume maximum
the Michaelis–Menten Ii (Table 2a) to be viewed as
rates are equal for all resources (i.e. all mi ¼ m).
the fraction of Itot that corresponds to the relative
With this restriction, the Michaelis–Menten Itot
contribution of Ni to R; i.e.
takes the form of the single resource Michaelis–
pi N i
Menten model (Table 1) expressed in terms of a
Ii ¼ Itot ð14Þ
:
weighted measure of the total resource density, R; R
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Table 2b
Diagnostics of Class 1 examples.
I II III IV V VI VII
IiÃimp
Ei @Ii =@Ni @Ii =@Nj @Itot =@Ni Bgen Csel
o0 always
ai Ni equal hi or equal hi or
(A) Disk >0 always =0 always
ai 1þai hi Ni
equal mi : equal mi :
pi mi ðor Pi mi Þ > 0 always X0 hi > hj or
mi omj : o0
hi > hj or
Michaelis– Ni
mi
Ã
ki þNi
mi > mj : o0
Menten at high Nj for
k 1
where kià ¼ ¼ at high Nj resource j
pi Pi
for Ni Xtià :
pi ðor Pi Þ near t > 0
(B) Threshhold >0 always X0 always X0 always =0 always
ðNi ÀtÃ Þ elsewhere: o0
i
m
ki þNi ÀtÃ
Ã
i
where kià ¼ pi ¼ Pi
k 1
and tià ¼ pi ¼ Pi
t t
for N otà : 0
i i
½1 À expðÀdià Ni Þm o0 always
pi ðor Pi Þ
(C) Ivlev >0 always X0 always X0 always =0 always
where dià ¼ dpi ¼ Pi
for Ni pvià : và m
Ni
pi ðor Pi Þ
(D) Rectilinear until X0 always X0 always =0 always
X0
i always satiated: =0
Ã:m
for Ni > vi
once
where và ¼ v ¼ 1
satiated: o0
i pi Pi
parameter k: Only when the maximum rates mi are
The multiple resource Threshold, Ivlev, and
Rectilinear models (Table 2a), which always identical for all resources do the Michaelis–
Menten Ei simplify to its pi parameters. Thus,
assume maximum rates are identical for all
despite pi being referred to as ‘‘preferences’’ in the
resources, are derived in an analogous manner to
the Michaelis–Menten models making the same literature (Fasham et al., 1990; Strom and Loukos,
assumption. That is: (i) Itot is described by each 1998; Loukos et al., 1997; Pitchford and Brindley,
1999), the term is a misnomer when any mi are
model’s respective single resource response (from
Table 1) expressed in terms of a weighted measure different. Had the Threshold, Ivlev, and Recti-
of total resource density R and (ii) Ii is defined by linear models allowed for resource-dependent
maximum rates, their Ei would also equal mi pi ;
Eq. (14). These three models also are overparame-
meaning reference to their pi as ‘‘selectivities’’ and
terized, in that the same functional response can be
described using one less parameter (i.e. Threshold: ‘‘vulnerabilities’’ (Hofmann and Ambler, 1988;
Pi ¼ pi =k; Ivlev: Pi ¼ dpi ; Rectilinear, Pi ¼ pi =v; Armstrong, 1994) is somewhat misleading.
Table 2a). Furthermore, measured clearance rates will only
yield independent estimates of pi in the specific
5.2. Dynamics assumed in Class 1: No Switching case when all maximum rates are equal.
examples
5.2.2. Diagnostic II
5.2.1. Diagnostic I All the Class 1 examples assume every resource
No Class 1 example ever assumes switching elicits the same Type of single resource response
(e.g. all Disk IiÃimp are Type 2 Disk, all Threshold
since all their Ei are constant, which is why they
IiÃimp are Type 3 Threshold, etc., Table 2b). They
are in this class (Table 2b). The Disk’s Ei are
the attack rates ai and the Michaelis–Menten’s also all assume selection is passive, as parameters
Ei ð¼ mi pi Þ are the equivalent term scaled by the and behaviors are consistent between the single
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2858
Fig. 3. Class 1 multiple resource functional responses. Contour plots of nutritional intake from two resources versus resource densities
(N1 and N2 ) for Class 1: No Switching examples (see text and Table 2a for model descriptions). (a) Disk/Michaelis–Menten I1 ; equal
preferences (a1 ¼ a2 ¼ 1; h1 ¼ h2 ¼ 1; equivalent to m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1); (b) Disk/Michaelis–Menten I1 ; unequal
preferences (a1 ¼ 1; a2 ¼ 0:25; h1 ¼ h2 ¼ 1; equivalent to m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ 1; p2 ¼ 0:25); (c) Rectilinear I1 ; equal preferences
(m ¼ 1; v ¼ 2; p1 ¼ p2 ¼ 1); (d) Threshold I1 ; equal preferences (m; k ¼ 1; p1 ¼ p2 ¼ 1; t ¼ 0:5); (e) Disk/Michaelis–Menten Itot ;
parameters as in (a); (f) Disk/Michaelis–Menten Itot ; unequal preferences from unequal handling times/maximum rates (a1 ¼ a2 ¼ 1;
h1 ¼ 4; h2 ¼ 1; equivalent to m1 ¼ 0:25; m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1).
model because kià ; the half-saturation constant of
and multiple resource responses, although this is
IiÃimp ; is generally not equal to k; the so-called
not obvious in the overparameterized versions.
‘‘half-saturation constant’’ of Ii (i.e. kià ¼ k=pi ).
For example, it may incorrectly appear that active
However, k and kià should not be directly
selection is assumed by the Michaelis–Menten
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2859
compared because the two have different biologi- feeding is only assumed to cease when a weighted
cal significance; k is related to weighted, not actual, measure of the total resource density is less than t;
so consumption of resource i can occur when
resource densities (Eq. (14)). The same is true for
the feeding thresholds t and tà in the Threshold Ni ot and even when Ni otià ; the implied single
i
model. resource threshold (Table 2b). Analysis of Diag-
Analysis of IiÃimp for the Michaelis–Menten, nostic IV therefore reveals that t is related to
Threshold, Ivlev, and Rectilinear models further minimal nutritional requirements as opposed to
reveals that Pi of their reduced-parameter versions minimal densities required for detection or attack.
are both measurable and meaningful quantities. In Therefore, the Threshold response could represent
contrast, pi of their overparameterized versions suspension-feeders or foragers that only have the
can only be determined when Pi are known a energy to generate feeding currents or successfully
priori. For example, the Michaelis–Menten Pi are attack resources when there sufficient total nutri-
the reciprocal of kià ; whereas pi are set by the tion available.
actual kià once the modeler chooses a value for k:
When the relationship between kià ; k; and pi is 5.2.5. Diagnostic V
unrecognized, modelers unwittingly assume speci- All Class 1 examples assume feeding is always
optimal when maximum rates mi (handling
fic values for the single resource half-saturation
times hi ) are identical for all resources (Table 2b,
constants. Hence, not only is the overparameter-
ization of these models unjustified, but it can Fig. 3e). However, feeding is sub-optimal when
mi ðhi Þ are resource-dependent and resource den-
obfuscate interpretation of behavior and choice of
sities are high, because Itot decreases for increases
appropriate parameter values.
in the relative density of resources with lower mi
5.2.3. Diagnostic III (longer hi ) (Fig. 3f).
Like their single resource analogs, the multiple
resource Disk/Michaelis–Menten and Ivlev Ii 5.2.6. Diagnostic VI
always increases when Ni increases, regardless of All Class 1 examples assume generalism is the
better strategy in regions where Itot increases with
resource preferences (Table 2b, Fig. 3a–b). While
increasing Ni (Table 2b, Fig. 3e). However, where
the Rectilinear model exhibits the same general
dynamic, the rate of change decreases sharply once feeding is sub-optimal, specialism on resources
intake is maximal, and intake never satiates on any with the largest maximum rates (shortest handling
one resource, which is in contrast to its single times) is more nutritionally advantageous (Fig. 3f).
resource analog (Fig. 3c). The Threshold Ii only
increases with Ni where resource densities are 5.2.7. Diagnostic VII
sufficiently high; variations in Ni are assumed to All the Class 1 examples assume there is neither
have no effect where resource densities are low nutritional cost nor benefit to selecting among
(Fig. 3d). functionally equivalent resources (Table 2b, Fig. 3e).
5.2.4. Diagnostic IV 5.3. Examples of Class 2: Passive Switching models
The Disk/Michaelis–Menten and Ivlev models
always assume interference of other resources, Examples of Class 2 models and their associated
regardless of resource preferences (Table 2b, references are listed in Table 3a.The No-Inter-
Fig. 3a–b). The Rectilinear and Threshold Ii ference model assumes the multiple resource
decrease for increasing Nj ðjaiÞ only when re- functional response for each resource is the same
source densities are high. When resource densities as when it is the only available nutrition (i.e.
Ii ¼ Iià ). The Modified-Threshold model, which
are low, Nj is assumed to have no effect on the
Rectilinear Ii (Fig. 3c), whereas the Threshold Ii we developed as an alternative to the Class 1
increases when Nj increases (Fig. 3d). The syner- Threshold model, allows for resource-dependent
maximum rates mi and feeding thresholds ti
gistic effect in the Threshold model arises because
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2860
Table 3a
Class 2 multiple resource functional responses
Intake of resource Ni
Class 2 Parameter Sample
Formulation Dimensions References
Ii ¼ Iià dictated by I à ðTable 1Þ
(A) No-Interference Leonard et al.
(1999): I Ã = Alternative
where Iià is the single resource intake from Ni
Type: Modified-Ivlev
P
n
(B) Modified-Threshold This paper
½m ¼ 1=T
Pi Ni;eff
Ii ¼ mi ; where R ¼ Pr Nr;eff and
1þR ½Pi ¼ 1=½N
r¼1
½t ¼ ½N
Ni À ti ; for Ni Xti
Ni;eff ¼
for Ni oti
0;
#
½ci ¼ 1=ð½N2 TÞ
#
ai Ni
(C) Sigmoidal I Gismervik and
Ii ¼ ; where ai ¼ ci Ni
P
n
#
(from Disk) Andersen (1997),
½hi ¼ ½T
ar hr Nr
1þ
r¼1
(from Michaelis– Edwards (2001)
½mi ¼ 1=T
P
n
# Ni
# #
¼ kp2iþR mi ; where R ¼ pr Nr and pi ¼ pi Ni ½k ¼ ½N
Menten)
r¼1
½pi ¼ N:D:
Pn
#
# #
Pi Ni
¼ where R ¼
mi ; Pr Nr ½Pi ¼ 1=½N
#
1þR
r¼1
# # #
Pi ¼ pi =k2 ¼ ai hi
where mi ¼ 1=hi ;
#
½fi ¼ 1=ð½N2 TÞ
#
ai Ni fi Ni
(D) Sigmoidal II Chesson (1983)
Ii ¼ where ai ¼ ð1þgi Ni Þ
P
n
# ½hi ¼ ½T
ar hr Nr
1þ
r¼1
½gi ¼ 1=½N
Pn
(E) Abundance-Based I Strom and Loukos (1998)
½mi ¼ 1=T
#
#
pi Ni
Ii ¼ kþR m; where R ¼ pr Nr
½k ¼ ½N
r¼1
1 À wi Ni ; for Ni oZ ½pi ¼ N:D:
and wi ¼ ð1Àpi Þ
#
pi ¼
½Z ¼ ½N
Z
pi ; for Ni XZ
½wi ¼ 1=½N
(Table 3a). The other Class 2 models are derived Menten equation also use one more parameter
by extending the Class 1 Disk model assuming the than is necessary to describe the functional
attack rate for resource i depends upon its density response.
#
(i.e. the constant ai is replaced by ai that depends
on Ni ), but the handling time hi remains constant. 5.4. Dynamics assumed in Class 2: Passive
Switching examples
This is equivalent to extending the Michaelis–
Menten equation by replacing the constant pi with
#
pi that depends on Ni ; but keeping maximum rates 5.4.1. Diagnostic I
mi constant. These three models are distinguished The No-Interference Ei equal the single resource
clearance rates, Fià ; which results in switching
by their assumed density-dependence: (i) linear in
unless Iià is Type 1 Non-Satiating. The Sigmoidal
the first Sigmoidal model (‘‘Sigmoidal I’’ in Table
and Abundance-Based Ei are the density-depen-
3a); (ii) hyperbolic in the second Sigmoidal model
(‘‘Sigmoidal II’’ in Table 3a); and (iii) rectilinear in dent analogs of Class 1 Disk/Michaelis–Menten
#
models upon which they were based (i.e. Ei ¼ ai ¼
the Abundance-Based model (‘‘Abundance-Based
# #
mi Pi or Ei ¼ mi pi ), and the Modified-Threshold Ei
I’’ in Table 3a). The Abundance-Based model
additionally assumes all mi are equal. All formula- additionally depend on how Ni scales with the
tions based on the overparameterized Michaelis– threshold ti (Table 3b). All these examples assume
Table 3b
Diagnostics of Class 2 examples
I II III IV V VI VII
IiÃimp
Ei @Ii =@Ni @Ii =@Nj @Itot =@Ni Bgen Csel
Fià Iià o0; 0 or > 0
¼ 0 always
(A) No- X0 always X0 always
X0
depends on I Ã
Interference always unless Type 4
unless
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
Type 4
p0 always equal mi X0
(B) Modified- equal >0 always
for Ni Xti : for Ni Xti : X0
mi omj : o0
Pi mi ðNNi i Þ
i Àt ðNi Àti Þ mi X0
Threshold always
m
Ã
ki þNi Àti i
always at high Nj for
for Ni oti : 0 where kà ¼ 1
ARTICLE IN PRESS
mi omj o0
i Pi resource j
for Ni oti at high Nj
# o0 always
# ai Ni equal hi or equal hi or
(C) Sigmoidal I >0 >0 always
ai #
1þai hi Ni
#
# equal mi X0 equal mi :
(from disk) always
pi mi ðor Pi mi Þ Ni2
mi
à always
ðki Þ2 þNi2 X0
hi > hj or hi > hj or
(from
where kià ¼ pi ¼ pffiffiffiffi mi omj : mi omj : o0
k 1
Michaelis–
Pi
o 0 at high Nj
Menten) at high Nj for
resource j
# o0 always
Ni2
ai equal hi : X0
(D) Sigmoidal II >0 always equal >0 always
mÃ
à à i
ðki;1 þNi Þðki;2 þNi Þ hi : X0
always
where mà ¼ hi ;
1
hi > hj : hi > hj :
i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o0 at high Nj o0 at high Nj for
Ã
ki;1 ¼ 1 xi þ x2 À 4yi
i
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
resource j
à ¼ 1 x À x2 À 4y
ki;2 2 i i
i
xi ¼ fghi and yi ¼ fi1 i
i
h
i
# o0 at o0 at o0, 0 or 0 depends on o0, 0 or >0
pi
(E) Abundance- 40 at
for Ni XZ :
Ni
Based I intermediate intermediate intermediate dens. and params. depends on dens.
Ãm
ki þNi
dens. dens. dens. and params.
where kià ¼ pi
k
depends depends depends
for Ni oZ : on params. on params. on params.
Ni Àwi Ni2
m
kþNi Àwi Ni2
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2862
Fig. 4. Class 2 multiple resource functional responses. Contour plots of nutritional intake from two resources versus resource densities
(N1 and N2 ) for Class 2: Passive Switching examples (see text and Table 3a for model descriptions). ‘‘PDD’’ = preference density
dependence. (a) Abundance-Based IiÃimp ; (m ¼ 1; k ¼ 0:5; pi ¼ 0:5; Z ¼ 1); (b) Sigmoidal I I1 ; equal PDD (h1 ¼ h2 ¼ 1; c1 ¼ c2 ¼ 1;
equivalent to m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1); (c) Modified-Threshold I1 ; equal PDD (m1 ¼ m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1;
t1 ¼ t2 ¼ 0:5); (d) Abundance-Based I1 ; unequal PDD (m ¼ 1; k ¼ 0:5; p1 ¼ 1; p2 ¼ 0:25; Z ¼ 1); (e) Sigmoidal I Itot ; parameters as
per (d); (f) Sigmoidal I Itot ; unequal PDD from unequal handling times/maximum rates (h1 ¼ 4; h2 ¼ 1; c1 ¼ 0:25; c2 ¼ 1 equivalent to
m1 ¼ 0:25 m2 ¼ 1; k ¼ 1; p1 ¼ p2 ¼ 1).
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2863
switching, since their Ei are density-dependent. assumed for Sigmoidal Ii (Fig. 4b). The Mod-
However, most examples also assume (essentially) ified-Threshold model assumes no effect when
Nj ðjaiÞ are low and interference when Nj > tj
no switching when resource densities are high.
Thus, when zooplankton behavior is consistent (Fig. 4c). In contrast, the certain parameter values
result in the Abundance-Based Ii exhibiting syner-
with these models, measured clearance rates will
not reveal switching unless experiments are con- gism (Table 3b), which results in regions of
ducted over a sufficiently broad range of densities. negative switching.
Furthermore, because these models’ Ei depend
only upon Ni ; determination of the passive nature 5.4.5. Diagnostic V
of this switching (i.e. the reason they are Class 2) Feeding is always optimal in the No-Interfer-
requires examination of Diagnostic II. ence model, provided none of the single resource
responses are Type 4. When all maximum rates mi
5.4.2. Diagnostic II (handling times hi ) are equal, the Modified-
Threshold and both Sigmoidal Itot also always
The No-Interference model allows for resource-
increase with increasing Ni (Table 3b, Fig. 4e).
dependent Types of single resource responses (e.g.
Type 1 Rectilinear for one resource and Type 3 However, these models assume feeding can be sub-
optimal when mi ðhi Þ are resource-dependent and
Sigmoidal for another), and the Modified-Thresh-
old IiÃimp can also be different Types depending on resource densities are high (Fig. 4f). Sub-optimal
whether a feeding threshold is specified (i.e. either feeding can occur at intermediate resource densi-
Type 3 Threshold or Type 2 Michaelis–Menten). ties when certain parameter values are used in the
In contrast, the Sigmoidal and Abundance-Based Abundance-Based model (Table 3b).
IiÃimp are the same Type for all resources.
5.4.6. Diagnostic VI
Furthermore, certain parameter values result in
the Abundance-Based IiÃimp being uncharacteristic All Class 2 examples assume generalism is the
best strategy where Itot increases with increasing
of any known response (Table 3b, Fig. 4a). The
Ntot (Table 3b, Fig. 4e). However, specialism is
behaviors and parameters are consistent between
the single and multiple resource formulations for more nutritionally advantageous where feeding is
all the Class 2 examples (Table 3b). Thus, passive sub-optimal (i.e. high resource densities for the
selection is assumed, and switching can be Modified-Threshold and both Sigmoidal models;
predicted when the actual I Ã are known. However, intermediate densities for Abundance-Based).
this fact is not obvious in the overparameterized
5.4.7. Diagnostic VII
formulations, and their extra degree of freedom is
unjustified. The Modified-Threshold and both Sigmoidal
models assume there is a nutritional cost to
5.4.3. Diagnostic III selecting among functionally equivalent resources
Ii always increases when Ni increases in both (Table 3b, Fig. 4e). In contrast, it can be
Sigmoidal models (Fig. 4b). The same dynamic is nutritionally beneficial to distinguish among such
assumed by the Modified-Threshold model when resources in the No-Interference and Abundance-
Based models, depending on I Ã and parameter
Ni > ti (Fig. 4c), and by the No-Interference model
when none of the single resource responses are values.
Type 4. However, the Abundance Based Ii can
decrease when Ni increases for certain parameter 5.5. Examples of Class 3: Active Switching models
values (Fig. 4d), which results in negative switch-
ing. Examples of Class 3 models and associated
references are listed in Table 4a. The Proportion-
5.4.4. Diagnostic IV Based model was used in Fasham et al. (1990), a
Other resources have no effect on the No- planktonic ecosystem model cited hundreds of
Interference Ii ; whereas interference is always times in the literature, which suggests that this is
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2864
Table 4a
Class 3 multiple resource functional responses
Intake of resource Ni
Class 3 Parameter References
Formulation Dimensions
P
n
(A) Proportion- Fasham et al. (1990),
½mi ¼ 1=T
#
#
pi Ni
Ii ¼ kþR m; where R ¼ pr Nr
Based Fasham et al. (1993),
½k ¼ ½N
r¼1
½pi ¼ N:D: Chai et al. (1996),
#
and pi ¼ P i Ni
p
n
Loukos et al. (1997),
pr Nr
Strom and Loukos (1998),
r¼1
Pitchford and Brindley (1999)
P
n
(B) Abundance- Strom and Loukos (1998)
½mi ¼ 1=T
#
#
pi Ni
Ii ¼ kþR m; where R ¼ pr Nr
Based II ½k ¼ ½N
r¼1
½pi ¼ N:D:
* *
1 À Nwi for NoZ
wi ¼ ð1Àpi Þ
# ½Z ¼ ½N
and pi ¼ ;
* Z
pi for NXZ ½wi ¼ 1=½N
*
and N is resource with largest pi
#
ai Ni
(C) Modified- Colton (1987)
½ai ¼ 1=ð½NTÞ
Ii ¼ ; where
P
n
## ½Aij ¼ 1=ð½N2 TÞ
Disk ar hr Nr
1þ
r¼1
½hi ¼ ½T
P P
n n
#
#
ai ¼ ai þ Aij Nj and hi ¼ hi þ Hij Nj ½Hij ¼ T=½N
j¼1 j¼1
ja1 ja1
Ni
(D) Modified- Moloney and Field (1991)
½mi ¼ 1=T
Ii ¼ m;
P
n
Michaelis–Menten ½ki ¼ ½N
kþ Nr
r¼1
P
Ni
¼k where kr;eff ¼ ki þ
m Nr
i;eff þNi
rai
and at least two ki are different
(otherwise this model reduces
to Class 1 Michaelis–Menten with
equal preferences)
bi Nil
(E) Switching Tansky (1978),
½ri ¼ 1=ð½NTÞ
Ii ¼ ri;eff Ni ; where ri;eff ¼ ri
R
Matsuda et al. (1986)
½l ¼ N:D:
P
n
½bi ¼ N:D:
br Nrl
and R ¼
r¼1
Ii ¼ Iià zi ;
#
(F) Weighted- Pace et al. (1984):
½zi ¼ N:D:
I Ã ¼ Type 1
intake
where Iià ¼ is the single resource response
Rectilinear with
z IÃ
#
for resource i; and zi ¼ Pi i additional lower
n
à zr Ir feeding threshold
r¼1
densities of other resources Nj ðjaiÞ: The Propor-
perhaps the most frequently used multiple resource
#
tion-Based pi depend on the relative contribution
functional response for zooplankton. Both the
of Ni to a weighted measure of total resource
Proportion-Based and the Abundance-Based
#
density, R: The Class 3 Abundance-Based pi all
(‘‘Abundance-Based II’’ in Table 4a) models
*
vary with N; the density of one particular resource
extend the Class 1 Michaelis–Menten equation
by replacing the constant pi with density-depen- assigned the highest pi ; according to the rectilinear
#
dent pi ; and assuming maximum rates are equal for relationship described for the Class 2 Abundance-
all resources. The difference from the Class 2 Based model. The Modified-Disk model extends
examples derived in a similar manner is that these the Class 1 Disk equation by assuming both attack
#
two Class 3 models assume pi vary with the rates ai and handling times hi vary linearly with
Table 4b
Diagnostics of Class 3 examples
I II III IV V VI VII
IiÃimp
Ei @Ii =@Ni @Ii =@Nj @Itot =@Ni Bgen Csel
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
p0 always o0 for low dens.
Ni
pi Ni > 0 always only > 0 for > 0 always
(A) Proportion- m
kþNi
Based high dens.
* * * o0 at o0 at o0 at o0, 0 or >0
(B) Abundance- >0 at
for i ¼same
for NoZ ¼ 1 À wi N
*
Based II intermediate intermediate intermediate intermediate depends on dens.
as Class 2
for NXZ ¼ pi
dens. dens. dens. dens. and params.
Ab.-Based 1
ARTICLE IN PRESS
* depends depends depends on depends
Ni
for i ai ¼ 1þNi m
on params. on params. params. on params.
# o0 at low dens. o0 at high and low
ai Ni
ai
(C) Modified- >0 at low dens. only >0 at >0 always
1 þ ai hi Ni
Disk depends on depends on dens. low dens.
params. params. depends on
params.
p0 always o0 at high dens. o0 at high Nj
Ni
1
(D) Modified- >0 always N/A
m
P
n ki þNi
ki þ
Michaelis– depends on params. (with
Nr for resource j
r¼1
Menten func. equiv.
resources,
model is
Class 1 M.M.)
p0 always o0 at high and o0 at low dens.
ðbi Ni Þl ri ri Ni
(E) Switching >0 always >0 always
low dens.
depends on
params.
Iià Fià zi Iià p0 always o0 at high and o0 at low dens.
(F) Weighted- X0 always X0 always
Intake unless Type 4 unless Type 4 low dens. unless Type 4
depends on
params.
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2866
Nj ðjaiÞ; according to the constant rates of change for resource-dependent single resource half-satura-
Aij and Hij (Table 3a). tion constants, and the Modified-Disk allows for
The Modified-Michaelis–Menten, Switching and resource-dependent attack rates and handling
Weighted-Intake models are all based on assump- times. In the Class 3 Abundance-Based model,
tions about how other resources affect the single one resource solicits a non-standard response (i.e.
resource response, without linking the changes to Fig. 4a), while the others are assumed to be
any specific behavioral mechanism. The Modified- functionally equivalent with half-saturation con-
Michaelis–Menten model assumes other resources stants always equal to 1 in whatever units the
act to increase the effective half-saturation con- resource densities are measured. In contrast, the
stant ki;eff of a Type 2 Michaelis–Menten Iià (Table Weighted-Intake model allows each single re-
3a). The Switching model assumes other resources source response to be any type.
Examination of IiÃimp further reveals that certain
act to reduce the effective rate of change ri;eff of a
Type 1 Non-Satiating, according to a weighted parameters of the Proportion-Based, Weighted-
exponential measure of the total resource density Intake and Switching models cannot be predicted
(Table 3a). In the Weighted-Intake model, Iià is from knowledge of the single resource responses
(i.e. pi ; bi ; l and zi ; Table 4b) Therefore, behavior
reduced according to its relative contribution to a
weighted measure of the total of all Iià ; where zi in these three models is assumed to depend on the
are the weights (Table 3a). relative resource densities (i.e. active selection—
the reason they are considered Class 3), and these
5.6. Dynamics assumed in Class 3: Active parameters would have to be determined through
multiple resource experiments. IiÃimp also demon-
Switching examples
strates that the overparameterization of the
5.6.1. Diagnostic I Proportion-Based and Switching models is justi-
The Class 3 Modified-Disk, Proportion-Based fied, because their extra degree of freedom relates
and Abundance-Based Ei equal the density-depen- to a measurable quantity. That is the multiple
resource k is the half-saturation constant of the
dent analogs of the Class 1 Disk/Michaelis–
Menten models on which they were based (i.e. single resource response in the Proportion-Based
# #
Ei ¼ ai or Ei ¼ pi ), but the Modified-Michaelis– model, and ri is the single resource clearance rate
Menten Ei take a radically different form in the Switching model.
(Table 4b). The Switching and Weighted-Intake
Ei depend on both the single resource responses, 5.6.3. Diagnostic III
Iià ; and the parameters related to the assumed The Proportion-Based Ii always increases when
Ni increases (Table 4b), and the Weighted-Intake
influence of other resources. The Abundance-
Based, Modified-Disk and Modified-Michaelis– model makes the same assumption when none of
Menten Ei each depend on Nj ðjaiÞ; and therefore the single resource responses are Type 4. Of the
obviously assume active switching (i.e. Class 3). other Class 3 examples, only the Modified-Disk
and Abundance-Based models ever assume Ii can
Classification as active switching for the three
decrease when Ni increases, which results in regions
other examples requires examination of Diag-
nostic II. of negative switching (Fig. 5a). This Type 4 kind of
dynamic is due to the non-standard Abundance-
Based IiÃimp ; whereas in the Modified-Disk model it
5.6.2. Diagnostic II
results from the assumed behavioural changes.
Most Class 3 examples assume the single
5.6.4. Diagnostic IV
resource responses are all the same Type (Table
4b). The Proportion-Based model further assumes The Proportion-Based model always assumes
all resources are functionally equivalent, in that interference (Table 4b). Of the other Class 3
the parameters of IiÃimp are identical for all examples, only the Modified-Disk and Abun-
dance-Based Ii ever increase when Nj increases
resources. The Modified-Michaelis–Menten allows
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Fig. 5. Class 3 multiple resource functional responses. Contour plots of nutritional intake from two resources versus resource densities
(N1 and N2 ) for Class 3: Active Switching examples (see text and Table 4a for model descriptions). ‘‘PDD’’ = preference density
dependence. (a) Modified-Disk I1 ; unequal PDD with equal handling times (a1 ¼ a2 ¼ 1; A12 ¼ 0; A21 ¼ 1; h1 ¼ h2 ¼ 0:25; H12 ¼ 0;
H21 ¼ 2); (b) Modified-Disk I1 ; another unequal PDD with equal handling times (a1 ¼ a2 ¼ 0:5; A12 ¼ 1; A21 ¼ 0; h1 ¼ h2 ¼ 0:5;
H12 ¼ H21 ¼ 0); (c) Modified-Michaelis–Menten Itot ; unequal PDD with equal maximum rates (m1 ¼ m2 ¼ 1; k1 ¼ 5; k2 ¼ 0:5;
p1 ¼ p2 ¼ 1); (d) Modified-Disk Itot ; with parameters as in (a); (e) Proportion-Based Itot ; equal PDD (m1 ¼ m2 ¼ 1;
k ¼ 1; p1 ¼ p2 ¼ 0:5); (f) Weighted-Intake Itot ; equal PDD (z1 ¼ z2 ¼ 1), where Iià is a rectilinear model (Table 1) with a lower
feeding threshold, as in Pace et al. (1984).
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2868
and results in regions of negative switching responses: no, passive, and active switching,
(Fig. 5b). This synergistic effect is again due to although such distinctions were rarely made in
the non-standard Abundance-Based IiÃimp ; and the the literature. The greatest differences among
Class 1 Ii occurred when resource i was rare, and
Modified-Disk’s assumed behavioural changes.
parameter values had little influence on contour
5.6.5. Diagnostic V shapes. In contrast, examples in both switching
In all the Class 3 examples, Itot can decrease classes also exhibited different dynamics for Ii
when Ni increases, and this assumption occurs for when resource i was common, and contour shapes
wider ranges of resource densities and parameter were sensitive to parameter values. The variations
values than the passive selection models. For of Ii versus Ni were generally analogous to the
example, even when all maximum rates mi (hand- different types of single resource responses. Most
ling times hi ) are equal, sub-optimal feeding occurs multiple resource models never assumed Ii de-
at high resource densities in the Modified-Michae- creased when Ni increased; however, this Type 4
lis–Menten and Modified-Disk models (Figs. 5c–d), kind of dynamic arose in some active switching
at intermediate resource densities in the Abun- examples. The assumed variation of Ii versus
dance-Based model, and at low resource densities in Nj ðjaiÞ covered the spectrum of possible re-
the Switching and Proportion-Based models (Fig. sponses (i.e. no effect, interference, and syner-
5e). When formulated as in Pace et al. (1984), the gism), even just among the Class 1 examples.
Weighted-Intake model also assumes sub-optimal Switching models generally assumed interference,
feeding at low resource densities (Fig. 5f). but varying parameter values and/or resource
densities led to negative switching in some cases.
5.6.6. Diagnostic VI We also found a wide diversity of modeled
All the Class 3 examples assume there are regions dynamics for Itot ; especially with respect to the
where specialism is more nutritionally advanta- optimality of feeding. Examples in all three classes
geous than generalism (Table 4b). However, these assumed there were regions where Itot decreased
regions are not restricted to where feeding is sub- when total available nutrition increased, even
optimal. In several models, specialism is assumed to when none of the single resource responses were
be a better strategy even when resources are Type 4. In the Class 1 and 2 passive selection
functionally equivalent and/or resource densities models, such sub-optimal feeding occurred only
low (e.g., Proportional-Based: Fig. 5e). when more nutritious resources became relatively
more rare. In such circumstances, these models
assumed specialization on high quality resources
5.6.7. Diagnostic VII
was a better strategy than generalism. The Active-
There is a nutritional cost to selection in most
Switching models demonstrated sub-optimal feed-
Class 3 examples (Table 4b), including those
ing over wider ranges of resource densities,
assuming resources are functionally equivalent,
including when resources were of equal quality.
such as the Proportion-Based model. The Abun-
Many Class 3 models additionally assumed
dance-Based model additionally assumes there is
specialism was more nutritionally advantageous
sometimes a benefit to distinguishing among
than generalism when resources were rare and
resources.
zooplankton were highly food-limited, although
some assumed the reverse. The No-Switching
models never assumed a cost or benefit to selecting
6. Discussion
among resources of equal quality, whereas switch-
ing models generally assumed zooplankton that
6.1. Assumed dynamics and their ecological
consequences perceived functionally equivalent resources as a
single nutrient pool would be more successful.
Our review identifies published models in all However, some models assumed it was occasion-
three classes of multiple resource functional ally beneficial to distinguish among such resources.
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2869
The different dynamics assumed for Ii and Itot predators are tightly coupled, poor model choices
can have vastly different ecological consequences. also can result in spurious dynamics such as
For example, responses for which Ii increases with erroneously creating (or suppressing) blooms or
Ni tend to have a stabilizing influence on resource erroneously affording resource refuges (or causing
dynamics, in that predation may suppress resource extinction).
blooms, especially when the effective preference In order for a model to be useful, its math needs
for resource i depends on Ni : In contrast, blooms to be consistent with the biology it is trying to
are more likely to arise where predation pressure is represent. Recognition of the assumed dynamics
reduced as Ni increases. The stability of resource i can help modelers make appropriate choices for
is moderated by a model’s assumptions regarding their application. The implied single resource
response, IiÃimp ; is perhaps the most important
the effect of other resources Nj ðjaiÞ: Models
assuming interference decrease predation on i diagnostic for this assessment. It dominates the
when j becomes relatively more abundant, espe- modeled response when other resource densities
cially when predators switch. Such responses can are low, dictates the preferences and switching in
promote biodiversity by affording a refuge for passive selection models, and determines the
resources that are relatively more rare. However, nature of the behavioral assumptions in active
models assuming synergism can result in extinction selection responses. It also can reveal problems
of rare resources, even when this dynamic is not with candidate formulations, thereby ruling them
technically ‘‘negative switching’’ (e.g., Class 1 out or indicating where they need to be modified.
Certain models were shown to have IiÃimp that
Threshold). Models assuming sub-optimal feed-
ing may result in starved predators, whereas are uncharacteristic of any known Types (i.e.
predators may be satiated when feeding is assumed Abundance-Based for certain parameters), which
to be optimal. As these different assumptions recommends against their use. Inappropriate
directly affect zooplankton growth, they indirectly Types may also be assumed by other models,
affect both their ability to compete with other since many examples consider single resource
responses to be the same type for all resources
predators and the losses inflicted upon the
resources. when they are generally resource-dependent. When
actual single resource responses are consistent with
the assumed types, IiÃimp reveals the biological
6.2. How to choose the appropriate model
significance of the multiple resource model para-
We have shown how modeling decisions can be meters, identifies which are more precisely known
confused by the overparameterization and/or and how they relate to experimental measure-
misleading nomenclature of some multiple re- ments. Therefore, Diagnostic II determines
source models. Ignorance about the actual re- whether the overparameterization occurring in
sponse also may prompt modelers to use some models is justified (e.g., Proportion-Based)
or not (e.g., Michaelis–Menten), and indicates
previously published formulations, without con-
sidering whether they are appropriate for the new whether an assumption such as ‘‘equal maximum
application. However, as discussed above, even rates’’ is reasonable (e.g., copepod ingestion of
seemingly subtle differences in parameter values, different species or size classes: Frost, 1972;
density-dependence and/or density ranges result in Ambler, 1986; Gismervik and Andersen, 1997;
drastically different dynamics. Poor model choices ciliate growth on algae: Stoecker et al., 1986;
will incorrectly quantify resource preferences, over Verity, 1991; Montagnes, 1996), or not because
(or under)estimate resource consumption and resources have different handling times, nutri-
tional quality, and/or accessibility (e.g., copepod
predator growth, and predict contrasting effects
of changing resource densities. Such misrepresen- nauplii: Ambler, 1986; ciliate growth on nano-
tations can mislead conclusions about the impor- plankton: Verity, 1991).
tance of omnivory or magnitude of secondary Diagnostics III–V are also helpful in determin-
production. In systems where resources and ing whether a model is appropriate for a specific
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2870
application. For example, while one can hypothe- While any individual species may exhibit sub-
size reasons for a Type 4 response (e.g., toxicity or optimal feeding when there are dramatic changes
confusion) or synergistic effects of other resources in environmental conditions, such changes often
(e.g., more efficient searching), there should be lead to shifts in the local community structure (e.g.
actual biological support for such anomalous HNLC regions when iron is added). Zooplankton
dynamics before employing models that exhibit would likely adapt to long-term changes in
them (e.g., Class 1 Threshold is consistent with conditions and/or new dominant species would
Calanus pacificus ingesting phytoplankton; emerge. Certainly, different regions have different
Landry, 1981). One can similarly hypothesize dominant predators, which employ different feed-
reasons for sub-optimal feeding, such as energetic ing strategies and have different functional re-
costs of switching (Fasham et al., 1990) or fitness sponses. Hence, use of any single formulation and/
being unaffected by foraging yield (Holt, 1983). or set of parameters is not recommended for large-
However, theoretical arguments suggest natural scale applications spanning wide ranges of re-
selection would diminish the degree of sub- source densities and planktonic communities (e.g.,
optimality (Holt, 1983), and observational evi- global biogeochemical or climate-change models)
dence indicates that predators do select more as such approaches can systematically bias results.
nutritious resources (Cowles et al., 1988; Verity, This can be particularly problematic for formula-
1991 and references therein; Strom and Loukos, tions assuming different kinds of dynamics for
1998, and references therein; Meyer-Harms et al., different resource densities (e.g. Proportion-
1999). We know of no observations of decreas- Based). Large-scale applications therefore may
ing nutritional intake for increasing available require regionally and/or temporally varying
nutrition when resource densities are those sub-models and parameters in order to repre-
naturally encountered by the zooplankton. sent adequately differences among planktonic
This suggests that modelers should avoid communities.
use of formulations that assume sub-opti-
6.3. Assessing uncertainty due to assumptions
mal feeding for their system’s normal density
ranges.
Except for the Modified-Disk, all Class 3: Active There is often insufficient knowledge to support
Switching models we cite are based on hypothe- the choice of any one equation. Analyses of how
sized—not observed—behaviors. Authors typi- well different models fit observations can suggest
cally claimed the motivation for their assumed the better candidates (e.g., Carpenter et al., 1993),
behavioral density-dependence was that predators but consistency of a model with data does not
would focus on resources yielding greater nutri- validate assumptions because models of natural
tion. However, all these active selection examples systems are insufficiently constrained (Oreskes
exhibit the same kind of sub-optimal feeding as the et al., 1994). When models results hinge on
passive selection models: there are regions where unsupported assumptions they may incorrectly
total nutritional intake decreases for increasing corroborate or nullify hypotheses and mislead
resource density. Unlike any passive selection future research. This is especially important for
responses, this dynamic occurs even when re- predictive models of food-limited regions, since the
sources are of the same nutritional quality, and greatest differences among most models’ dynamics
when resource densities are low. Yet, the latter occur when resource densities are low. Hence,
condition is where selective pressure to feed sensitivity analyses always should be conducted in
optimally would likely be greatest because nutri- order to assess the uncertainty introduced by our
tional yield would be critical for survival. We ignorance.
therefore recommend against use of any unsup- Many sensitivity analyses are conducted by
ported Class 3 examples, especially for regions varying parameter values, usually only one at a
where predators are highly food-limited (e.g. time and often only in one direction (e.g., Evans,
HNLC). 1999). This is done despite the non-linearity of
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2871
modeled processes, or the fact that different shapes including passive selection, and how feeding
of the functional response can introduce variation behaviors may not optimize nutritional intake
into model results that is at least the magnitude of nor have stabilizing influences on resources.
variation due to uncertainty in parameter values. Therefore, measurements of constant or density-
We have shown how changing parameters can dependent resource preferences are insufficient to
radically change the assumed dynamics (e.g., from determine the nature of a functional response,
interference to synergism, or optimal to sub- especially when experiments cover only a limited
optimal feeding), meaning sensitivity to parameter range of resource densities. Our Classes and
values may actually indicate sensitivity to unsup- Diagnostics can aid experimental design, clarify
ported behavioral assumptions. The literature is parameters’ biological significance and help inter-
also rich with examples of how different dynamics pret zooplankton behavior.
arise from basic choices modelers make, such as In the same way that Diagnostics I and II
explicitly including omnivory or aggregating dif- together determine the class of a mathematical
ferent resources (e.g., May 1972, 1973; Holling, model, classification of an actual response requires
1973; Armstrong, 1994, 1999; Polis and Strong, measurements of clearance rates over ranges of
combinations of resource densities and knowledge
1996; Pahl-Wostl, 1997).
We recommend that assumptions related to the of the single resource responses. Empirical fits of
functional response be tested by varying both the latter indicate single resource behaviors (e.g.,
parameter values and model structure. Our Diag- constant attack rates like Type 1 and 2, or density-
nostics can identify formulations that assume dependent ones like Type 3), and hint at candidate
contrasting dynamics for the range of resource multiple resource models. Comparison of mea-
densities being considered, and thereby indicate sured and modeled preferences reveals whether
which models have the greatest potential to affect behaviors depend on the availability of other
resources (e.g. Iià are Disk, but measured prefer-
results. For example, models assuming optimal
feeding could approximate upper bounds on ences are not attack rates). When active selection
predator growth and resource consumption. These does occur, recognition of factors affecting the
results could be compared with the lowered composition of the diet (Diagnostic I, e.g. max-
growth and consumption resulting from responses imum rates), and the optimality of selection
for which there is a nutritional cost to selection or (Diagnostics V–VII) can suggest nutritionally
resource refuges (e.g. Class 2 Sigmoidal). Further advantageous behaviors (e.g., specialism, prefer-
comparisons could be made between models that ential for high quality resources, etc.), which might
assume resources are perceived as distinct (e.g., explain the data.
multiple resource food webs) versus those wherein Once an empirical model is developed, our
resources are perceived as a single nutrient pool Diagnostics can elucidate the biological dynamics
(e.g., single resource food chains). Confidence in resulting from that response. Recognition of these
conclusions is increased when results are relatively assumptions helps direct future research, especially
robust to the details of the functional response. when the model’s implied dynamics are incon-
However, when the formulation is crucial (as it sistent with what was expected. When the observed
usually is), then the inability to make estimates behavior implies Type 4, synergism, or sub-
with narrow ranges is an important conclusion and optimal feeding (Diagnostic III–V) at unmeasured
aids direction of future research. resource densities, experiments should be per-
formed to confirm whether such anomalous
6.4. Assumed dynamics help experimentalists dynamics are actually exhibited or if behavioral
adaptations occur. Alternatively, when unexpected
Our review found switching (and negative dynamics occur at measured densities, the math-
switching) responses that are no-switching at high ematical model suggests the conceptual model
and low resource densities. We also illustrated how should be revised. Experimental investigation of
switching could arise from a host of mechanisms, selection can be further aided by determining what
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W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
2872
factors affect preferences (Diagnostic I), and why determination by constraining parameters, inter-
certain resources may be preferred even when they preting behaviors, and recognizing limitations to a
are less abundant than others (Diagnostic V). model’s utility for both regional (e.g., HNLC) and
Diagnostics VI and VII can help formulate large-scale applications (e.g., global biogeochem-
hypotheses, as they suggest why the feeding ical or climate change). We identified published
strategies of dominant predators vary regionally. models with contrasting assumptions that can be
used in sensitivity studies to quantify the un-
6.5. Conclusions certainty introduced due to ignorance about the
actual response. Clarification of model dynamics
The Classes and Diagnostics we defined provide also helps direct future experimental research,
a framework for considering the varied behaviors especially when the math is not consistent with
the concept. We recommend researchers employ
and ecological implications of multiple resource
functional responses. They elucidate a models’ our framework when making decisions about
assumptions regarding resource preferences, im- multiple resource models, and thereby maximize
plied single resource responses, changes in intake the utility of such tools for advancing our ecologi-
with changing resource densities, nutritional ben- cal understanding and predictive capabilities.
efits of generalism, and nutritional costs of
selection. They reveal whether or not switching
Acknowledgements
can occur, the origin of switching when it does,
and where responses result in anomalous dynamics
The authors would like to thank Rob Arm-
such as negative switching or sub-optimal feeding.
strong for his comments on an early version of the
Our review of published multiple resource
paper, which greatly improved the generality and
models was by no means exhaustive; however, it
utility of this work. We would also like to
has still emphasized how model choice can be
acknowledge the helpful editorial feedback pro-
critical. The examples we cited exhibit dramati-
vided by Donald DeAngelis, Michio Kishi, and
cally different dynamics, even for seemingly subtle
anonymous reviewer, as well as Dan Kelley,
differences among formulations. We identified
George Jackson, and Mark Kot. This work was
equations that generally should be avoided, such
supported by National Science Foundation US
as the Abundance-Based models that are unchar-
JGOFS Grant OCE-9818770.
acteristic of any known response, and demon-
strated how there is no good reason to use any
overparameterized Class 1 formulation including
References
Michaelis–Menten. We revealed how passive
selection leads to sub-optimal intake when re-
Ambler, J.W., 1986. Formulations of an ingestion function for
sources are of different quality (e.g. Disk and
a populations of Paracalanus feeding on mixtures of
Sigmoidal models), yet all hypothesized behavioral phytoplankton. Journal of Plankton Research 8 (5),
adaptations in the Class 3 examples, including the 957–972.
popular Proportion-Based model, result in wider Armstrong, R.A., 1994. Grazing limitation and nutrient
limitation in marine ecosystems: steady state solutions of
regions of anomalous dynamics. This suggests use
an ecosystem model with multiple food chains. Limnology
of existing active selection models is hard to justify
and Oceanography 39 (3), 597–608.
for many applications, and points to the need for Armstrong, R.A., 1999. Stable model structures for represent-
theoreticians and experimentalists to develop more ing biogeochemical diversity and size spectra in plankton
realistic formulations. communities. Journal of Plankton Research 21 (3), 445–464.
Barthel, K.G., 1983. Food uptake and growth efficiency of
Modeling the nutritional intake for multiple
Eurytemora affinis (Copepoda: Calanoida). Marine Biology
resources is more complicated than it might seem.
74 (3), 269–274.
Choosing a formulation is not a straightforward; it Bartram, W.C., 1980. Experimental development of a model for
depends on the specific zooplankton and resources the feeding of neritic copepods on phytoplankton. Journal
being considered. Our diagnostics can assist in this of Plankton Research 3 (1), 25–51.
ARTICLE IN PRESS
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2873
Franks, P.J.S., Wroblewski, J.S., Flierl, G.R., 1986. Behavior of
Campbell, R.G., Wagner, M.W., Teegarden, G.J.,
a simple plankton model with food-level acclimation by
Boudreau, C.A., Durbin, E.G., 2001. Growth and
development rates of the copepod Calanus finmarchicus in herbivores. Marine Biology 91, 121–129.
the laboratory. Marine Ecology Progress Series 221, Frost, B.W., 1972. Effects of size and concentration of food
161–183. particles on the feeding behavior of the marine planktonic
Carpenter, S.R., Lathrop, R.C., Munoz-del-Rio, A., 1993. copepod Calanus pacificus. Limnology and Oceanography
Comparison of dynamic models for edible phytoplankton. 17 (6), 805–815.
Canadian Journal of Aquatic Fisheries and Sciences 50, Frost, B.W., 1975. A threshold feeding behavior in
1757–1767. Calanus pacificus. Limnology and Oceanography 20 (2),
Chai, F., Lindey, S.T., Barber, R.T., 1996. Origin and 263–266.
maintenance of a high nitrate condition in the equatorial Frost, B.W., 1987. Grazing control of phytoplankton stock in
Pacific. Deep-Sea Research II 43 (4–6), 1031–1064. the open subarctic Pacific Ocean: a model assessing the role
Chesson, J., 1978. Measuring preference in selective predation. of mesozooplankton, particularly the large calanoid cope-
Ecology 59 (2), 211–215. pods Neocalanus spp. Marine Ecology Progress Series 39,
Chesson, J., 1983. The estimation and analysis of preference 49–68.
and its relationship to foraging models. Ecology 64 (5), Gifford, D.J., Dagg, M.J., 1988. Feeding of the estuarine
1297–1304. copepod Acartia tonsa Dana: carnivory versus herbivory in
Colton, T.F., 1987. Extending functional response models to natural microplankton assemblages. Bulletin of Marine
include a second prey type: an experimental test. Ecology 68 Science 43 (3), 458–468.
(4), 900–912. Gismervik, I., Andersen, T., 1997. Prey switching by Acartia
Cowles, T.J., Olson, R.J., Chisholm, S.W., 1988. Food selection clausi: experimental evidence and implications of intraguild
by copepods: discrimination on the basis of food quality. predation assessed by a model. Marine Ecology Progress
Marine Biology 100, 41–49. Series 157, 247–259.
Davis, C.S., Flierl, G.R., Wiebe, P.H., Franks, P.J.S., 1991. Goldman, J.C., Dennet, M.R., Gordin, H., 1989. Dynamics of
Micropatchiness, turbulence and recruitment in plankton. herbivorous grazing by heterotrophic dinoflagellate Oxy-
Journal of Marine Research 49, 109–151. rrhis marina. Journal of Plankton Research 11 (2), 391–407.
Deason, E.E., 1980. Grazing of Acartia hudsonica (A. clausi) on Green, C.H., 1986. Patterns of prey selection: implications of
Skeletonema costatum in Narragansett Bay (USA): influence predator foraging tactics. American Naturalist 128 (6),
824–839.
of food concentration and temperature. Marine Biology 60
Hansen, P.J., Nielsen, T.G., 1997. Mixotrophic feeding of
(2/3), 101–113.
DeMott, W.R., Watson, M.D., 1991. Remote detection of algae Fragilidium subglobosum (Dinophyceae) on three species of
by copepods: responses to algal size, odors and motility. Ceratium: effects of prey concentration, prey species
Journal of Plankton Research 15, 1203–1222. and light intensity. Marine Ecology Progress Series 147,
Donaghay, P.L., Small, L.F., 1979. Food selection capabilities 187–196.
of the estuarine copepod Acartia clausi. Marine Biology 52, Hansen, B., Tande, K.S., Bergreen, O.C., 1999. On the trophic
fate of Phaeocystic pouchetii (Hariot). 3. Functional
137–146.
response in grazing demonstrated on juvenile stages of
Edwards, A.M., 2001. Adding detritus to a nutrient–
Calanus finmarchicus (Copepoda) fed diatoms and
phytoplankton–zooplankton model: a dynamical-systems
Phaeocystis. Journal of Plankton Research 12 (6),
approach. Journal of Plankton Research 23 (4), 389–413.
1173–1187.
Evans, G.T., 1988. A framework for discussing seasonal
Hofmann, E.E., Ambler, J.W., 1988. Plankton dynamics on the
succession and coexistence of phytoplankton species.
outer southeastern US continental shelf. Part II: a time-
Limnology and Oceanography 33 (5), 1027–1036.
dependent biological model. Journal of Marine Research 40
Evans, G.T., 1999. The role of local models and data sets in the
(4), 883–917.
Joint Global Ocean Flux Study. Deep-Sea Research I 46,
Holling, C.S., 1959. Some characteristics of simple types
1369–1389.
of predation and parasitism. Canadian Entomologist 91,
Fasham, M.J.R., Ducklow, H.W., McKelvie, S.M., 1990.
824–839.
A nitrogen-based model of plankton dynamics in the
Holling, C.S., 1962. Principles of insect predation. Annual
oceanic mixed layer. Journal of Marine Research 48,
Review of Entomology 6, 163–182.
591–639.
Holling, C.S., 1965. The functional response of predators to
Fasham, M.J.R., Sarmiento, J.L., Slater, R.D., Ducklow,
prey density and its role in mimicry and population
H.W., Williams, R., 1993. Ecosystem behavior at Bermuda
regulation. Memoirs of the Entomological Society of
Station ‘S’ and Ocean Weather Station ‘India’: a general
Canada 45, 3–60.
circulation model and observational analysis. Global
Holling, C.S., 1973. Resilience and stability of ecological
Biogeochemical Cycles 7, 379–415.
systems. Annual Review of Ecological Systems 4, 1–24.
Fenchel, T., 1980. Suspension feeding in ciliated protozoa:
Holt, R.D., 1983. Optimal foraging and the form of the
functional response and particle size selection. Microbial
predator isocline. American Naturalist 122 (4), 521–541.
Ecology 6, 1–11.
ARTICLE IN PRESS
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875
2874
Houde, S.E.L., Roman, M.R., 1987. Effects of food quality Mayzaud, P., Poulet, S.A., 1978. The importance of the time
on the functional ingestion response of the copepod factor in the response of zooplankton to varying concentra-
Acartia tonsa. Marine Ecology Progress Series 40 (1–2), tions of naturally occurring particulate matter. Limnology
69–77. and Oceanography 23, 1144–1154.
Hutson, V., 1984. Predator mediated coexistence with a Mayzaud, P., Tirelli, V., Bernard, J.M., Roche-Mayzaud, O.,
switching predator. Mathematical Biosciences 68, 233–246. 1998. The influence of food quality on the nutritional
Ivlev, V.S., 1955. Experimental Ecology of the Feeding of acclimation of the copepod Acartia clausi. Journal of
Fishes. Pischepromizdat, Moscow, 302pp. (Translated from Marine Systems 15 (1–4), 483–493.
Russian by D. Scott, Yale University Press, New Haven, Meyer-Harms, B., Irigoien, X., Head, R., Harris, R., 1999.
1961.) Selective feeding on natural phytoplankton by Calanus
Jonsson, P.R., 1986. Particle size selection, feeding rates and finmarchicus before, during and after the 1997 spring bloom
growth dynamics of marine planktonic oligotrichous ciliates in the Norwegian Sea. Limnology and Oceanography 44 (1),
(Ciliophora: Oligotrichina). Marine Ecology Progress Series 154–165.
33, 265–277. Michaelis, L., Menten, M.L., 1913. Die Kinetik der Invertin-
Jonsson, P.R., Tiselius, P., 1990. Feeding behavior, prey wirkung. Biochemistry Z 49, 333–369.
detection, and capture efficiency of the copepod Acartia Moloney, C.L., Field, J.G., 1991. Modeling carbon and
tonsa feeding on planktonic ciliates. Marine Ecology nitrogen flows in a microbial plankton community. In:
Progress Series 60, 35–44. Reid, P.C., et al. (Ed.), Protozoa and their Role in Marine
Jost, J.L., Drake, J.F., Tsuchiya, H.M., Fredrickson, A.G., Processes, NATO ASI Series, Vol. G 25. Springer, Berlin.
1973. Microbial food chains and food webs. Journal of Monod, J., 1942. Recherches sur la croissance des cultures
Theoretical Biology 41, 461–484. bacteriennes. Hermann et Cie, Paris.
Kiorboe, T., Saiz, E., Viitasalo, M., 1996. Prey switching Monod, J., 1950. Annales de l Institut Pasteur, Paris 79, 390.
behavior in the planktonic copepod Acartia tonsa. Marine Montagnes, D.J.S., 1996. Growth responses of planktonic
Ecology Progress Series 143, 65–75. ciliates in the genera Strobilidium and Strombidium. Marine
Lancelot, C., Hannon, E., Becquevort, S., Veth, C., De Baar, Ecology Progress Series 130 (1–3), 241–254.
H.J.W., 2000. Modeling phytoplankton blooms and carbon Mullin, M.M., Stewart, E.F., Fuglister, F.J., 1975. Ingestion by
export in the Southern Ocean: dominant controls by planktonic grazers as a function of food concentration.
light and iron in the Atlantic sector in Austral spring Limnology and Oceanography 20 (2), 259–262.
1992. Deep-Sea Research I 47, 1621–1662. Murdoch, W.W., 1969. Switching in general predators: experi-
Landry, M.R., 1981. Switching between herbivory and carni- ments on prey specificity and stability of prey populations.
vory by the planktonic marine copepod Calanus pacificus. Ecological Monographs 39, 335–354.
Marine Biology 65, 77–82. Murdoch, W.W., 1973. The functional response of predators.
Leising, A., Gentleman, W.C., Frost, B.W., 2003. The threshold Journal of Applied Ecology 10, 335–342.
feeding response of microzooplankton within Pacific high- Oaten, A., Murdoch, W.W., 1975a. Functional response and
nitrate low-chlorophyll ecosystem model under steady stability in predator–prey systems. American Naturalist 109,
and variable iron input. Deep-Sea Research II, this issue 289–298.
(doi: 10.1016/j.dsr2.2003.07.002). Oaten, A., Murdoch, W.W., 1975b. Switching functional
Leonard, C.L., McClain, C.R., Murtugudde, R., Hofmann, response and stability in predator–prey systems. American
E.E., Harding Jr., L.W., 1999. An iron-based ecosystem Naturalist 109, 299–318.
model of the central equatorial Pacific. Journal of Geophy- Ohman, M.D., 1984. Omnivory by Euphausia pacifica: the
role of copepod prey. Marine Ecology Progress Series 19,
sical Research 104 (C1), 1325–1341.
Lessard, E.J., Murrell, M.C., 1998. Microzooplankton herbi- 125–131.
vory and phytoplankton growth in the northwestern Oreskes, N., Shrader-Frechette, K., Belitz, K., 1994. Verifica-
tion, validation and confirmation of numerical models in the
Sargasso Sea. Aquatic Microbial Ecology 16, 173–188.
Loukos, H., Frost, B., Harrison, D.E., Murray, J.W., 1997. An earth sciences. Science 263, 641–646.
ecosystem model with iron limitation of primary production Pace, M.L., Glasser, J.E., Pomeroy, L.R., 1984. A simulation
in the equatorial Pacific at 140 W: Deep Sea Research II 44 analysis of continental shelf food webs. Marine Biology 82,
47–63.
(9–10), 2221–2249.
Pahl-Wostl, C., 1997. Dynamic structure of a food web model:
Matsuda, H., Kawasaki, K., Shigesada, N., Teramoto, E.,
comparison with a food chain model. Ecological Modeling
Ricciardi, L.M., 1986. Switching effect on the stability of the
prey–predator system with three trophic levels. Journal of 100, 103–123.
Pitchford, J.W., Brindley, J., 1999. Iron limitation, grazing pre-
Theoretical Biology 112, 251–262.
ssure and oceanic high nutrient-low Chlorophyll (HNLC)
May, R.M., 1972. Limit cycles in predator–prey communities.
Science 177, 900–902. regions. Journal of Plankton Research 21 (3), 525–547.
May, R.M., 1973. Stability and Complexity in Model Ecosys- Polis, G.A., Strong, D.R., 1996. Food-web complexity and
tems. Princeton University Press, Princeton, NJ, 265pp. community dynamics. The American Naturalist 147 (5),
May, R.M., 1977. Predators that switch. Nature 269, 103–104. 813–846.
ARTICLE IN PRESS
W. Gentleman et al. / Deep-Sea Research II 50 (2003) 2847–2875 2875
Reeve, M.R., Walter, M.A., 1977. Observations on the Strom, S.L., Loukos, H., 1998. Selective feeding by protozoa:
existence of lower threshold and upper critical food concen- model and experimental behaviors and their consequences
tration for the copepod Acartia tonsa Dana. Journal of for population stability. Journal of Plankton Research 20
Experimental Marine Biology and Ecology 29 (3), 211–221. (5), 831–846.
Rothschild, B.J., Osborn, T.R., 1988. Small-scale turbulence Strom, S.L., Miller, C.B., Frost, B.W., 2000. What sets lower
and plankton contact rates. Journal of Plankton Research limits to phytoplankton stocks in high-nitrate, low-Chlor-
10, 465–474. ophyll regions of the open ocean? Marine Ecology Progress
Sell, A.F., Van Keuren, D., Madin, L.P., 2001. Predation by Series 193, 19–31.
omnivorous copepods on early developmental stages of Tansky, M., 1978. Switching effect in a prey-predator system.
Calanus finmarchicus and Pseudocalanus spp. Limnology Journal of Theoretical Biology 70, 263–271.
and Oceanography 46 (4), 953–959. Van Gemerden, H., 1974. Coexistence of organisms competing
Solomon, M.E., 1949. The natural control of animal popula- for the same substrate: an example among the purple sulfur
tions. Journal of Animal Ecology 18, 1–35. bacteria. Microbial Ecology 1, 104–119.
Stephens, D.W., Krebs, J.R., 1986. Foraging Theory. Princeton Veldkamp, H., Jannasch, H.W., 1972. Mixed culture studies
University Press, Princeton. with the chemostat. Journal of Applied Chemistry and
Stoecker, D.K., Cucci, T.L., Hulburt, E.M., Yentsch, C.M., 1986. Biotechnology 22, 105–123.
Selective feeding by Balanion sp. (Ciliata: Balanionidae) on Verity, P.G., 1991. Measurement and simulation of prey uptake
phytoplankton that best support its growth. Journal of by marine planktonic ciliates fed plastidic and aplastidic
Experimental Marine Biology and Ecology 95 (2), 113–130. nanoplankton. Limnology and Oceanography 36 (4),
Strom, S.L., 1991. Growth and grazing rates of herbivorous 729–750.
dinoflagellate Gymnodinium sp. from the open subarctic Wickham, S.A, 1995. Cyclops predation on ciliates: species-
Pacific Ocean. Marine Ecology Progress Series 78 (2), specific differences and functional response. Journal of
103–113. Plankton Research 17 (8), 1633–1646.