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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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  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 à    (‘‘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 à ¼ rN                         ½rŠ ¼ 1=ð½NŠTÞ
(A) Type 1: Non-satiating                                           N/A
                    
                     rN ¼ N m for Npv
(B) Type 1: Rectilinear                                            Frost (1972), Hansen
                                              ½rŠ ¼ 1=ð½NŠTÞ
                 IÃ ¼     v
                                                       and Nielsen (1997),
                                              ½mŠ ¼ 1=T
                     m for N > v
                                              ½vŠ ¼ ½NŠ     Mayzaud et al. (1998),
                                                       Hansen et al. (1999)

                 I Ã ¼ 1þahN ;
                     aN
(C) Type 2: Disk                                               Mullin et al. (1975),
                                              ½aŠ ¼ 1=ð½NŠTÞ
a.ka Michaelis–Menten                                             Ohman (1984), Jonsson (1986),
                                              ½hŠ ¼ T
                                              ½mŠ ¼ 1=T
                   N
(a.k.a. Monod)                                                Mayzaud et al. (1998),
                 ¼ kþN m
                                              ½kŠ ¼ ½NŠ     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),
                                              ½mŠ ¼ 1=T
                                                       Houde and Roman (1987)
                                              ½dŠ ¼ 1=½NŠ

                 I Ã ¼ kþN m
                      N
(E) Type 3: Threshold                                             Mullin et al. (1975),
                                              ½mŠ ¼ 1=T
                     eff

                              
                         eff
                                                       Reeve (1977),
                                              ½kŠ ¼ ½NŠ
                                   for Not
                                0
                                              ½tŠ ¼ ½NŠ
                 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),
                                              ½hŠ ¼ T
                      #
                                              ½cŠ ¼ 1=ð½NŠ2 TÞ
(from Disk)                                                  Wickham (1995), Gismervik and
                    N2
                 ¼       m;
(from Michaelis–Menten)                                            Andersen (1997)
                   k2 þN 2                       ½mŠ ¼ 1=T
                                         pffiffiffiffiffi
                                  pffiffiffiffiffiffiffiffiffiffiffiffi
                                      #
                 where m ¼ 1=h and ¼        N=ah ¼ 1= ch     ½kŠ ¼ ½NŠ

                 I Ã ¼ Qs    Ns
(G) Type 3: Sigmoidal II                                           (Theoretical Reference)
                                              ½sŠ ¼ ND
                              m;  where s > 1
                          ðks þNÞ
                                                       Jost et al. (1973)
                                              ½mŠ ¼ 1=T
                        s¼1

                                              ½ks Š ¼ ½NŠ
                 When s ¼ 2 :
                                              ½kŠ ¼ ½NŠ
                 I Ã ¼ ðk1 þNÞðk2 þNÞ m ¼ k2 þN 2 þaN m
                            2        2
                       N        N
                                              ½aŠ ¼ ½NŠ
                        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)
                                              ½mŠ ¼ 1=T
Predator Confusion                                              Veldkamp and Jannasch (1972),
                                              ½kŠ ¼ ½NŠ
                                              ½bŠ ¼ 1=½NŠ    Van Gemerden (1974)

                 I Ã ¼ ð1 À eÀdN Þm;
                         #           #
(I) Alternate Types:                                             Mayzaud and Poulet (1978)
                                              ½dŠ ¼ 1=½NŠ
                                 where m ¼ eN
Modified-Ivlev                                       ½eŠ ¼ 1=ðT½NŠÞ




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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                                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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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 Š ¼ 1=ð½NŠTÞ
                Ii ¼  P
                    n
a.k.a. Michaelis–Menten                                                     Frost (1987),
                                                       ½hi Š ¼ T
                         ar hr Nr
                   1þ
                      r¼1
                                                       ½mi Š ¼ 1=T    Moloney and
                                 P
                                 n
                  pi Ni
                 ¼ kþR mi       where ¼     pr Nr                           Field (1991),a
                                                       ½kŠ ¼ ½NŠ
                                 r¼1
                                                       ½pi Š ¼ N:D:    Verity (1991),
                                    P
                                    n
                               #
                 ¼ Pi N#i mi      where R ¼      Pr Nr               ½Pi Š ¼ 1=½NŠ   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),
                                                       ½mŠ ¼ 1=T
                            pi Ni
                         RÀt          Pn
                         m; for R > t
                         kþRÀt R
                Ii ¼              where ¼   pr Nr                        Lancelot et al. (2000)
                                                       ½kŠ ¼ ½NŠ
                    0; for Rot;           r¼1
                  ( #                                   ½pi Š ¼ N:D:
                            ##
                     t Pi Ni
                    RÀ#                Pn
                        # m; for R > t                         ½Pi Š ¼ 1=½NŠ
                                   #
                     #t
                 ¼               where R ¼   Pr Nr
                   1þRÀ#  R
                       #t                                ½tŠ ¼ ½NŠ
                   0; for Ro#              r¼1
                          #
                where Pi ¼ pi =k and t ¼ t=k

                                          P
                                          n
(C) Ivlev                                                            Hofmann and
                                                       ½mŠ ¼ 1=T
                Ii ¼ ½1 À expðÀdRފpiR i m; where R ¼
                          N
                                              pr Nr ;
                                                                 Ambler (1988)
                                                       ½kŠ ¼ ½NŠ
                                          r¼1
                                          P
                                          n
                                                       ½pi Š ¼ N:D:
                       #N        #
                 ¼ ½1 À expðÀRފPiR i m; where R ¼            Pr Nr
                         #
                                                       ½Pi Š ¼ 1=½NŠ
                                          r¼1
                                                       ½dŠ ¼ 1=½NŠ
                where Pi ¼ dpi
                  (
(D) Rectilinear                                                         Armstrong (1994)
                                                       ½mŠ ¼ 1=T
                   pi Ni
                                            P
                                            n
                      m; for Rpv
                Ii ¼ piv i                 where R ¼
                           ;                   pr Nr      ½vŠ ¼ ½NŠ
                    N
                   R m; for R > v                  r¼1
                  (                                     ½pi Š ¼ N:D:
                          #                  P
                                            n
                   Pi Ni m; for Rp1              #              ½Pi Š ¼ 1=½NŠ
                 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 Àtà ޠ               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 ފm             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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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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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
                                                ½mŠ ¼ 1=T
                 Pi Ni;eff
             Ii ¼      mi ; where R ¼    Pr Nr;eff and
                 1þR                              ½Pi Š ¼ 1=½NŠ
                             r¼1
                                                ½tŠ ¼ ½NŠ
                        Ni À ti ; for Ni Xti
                  Ni;eff ¼
                             for Ni oti
                        0;


                     #
                                                ½ci Š ¼ 1=ð½NŠ2 TÞ
                                #
                     ai Ni
(C) Sigmoidal I                                                       Gismervik and
             Ii ¼             ; where ai ¼ ci Ni
                   P
                   n
                       #
(from Disk)                                                         Andersen (1997),
                                                ½hi Š ¼ ½TŠ
                       ar hr Nr
                 1þ
                   r¼1
(from Michaelis–                                                      Edwards (2001)
                                                ½mi Š ¼ 1=T
                                  P
                                  n
                 # Ni
                                     #     #
               ¼ kp2iþR mi ; where R ¼           pr Nr and pi ¼ pi Ni  ½kŠ ¼ ½NŠ
Menten)
                                  r¼1
                                                ½pi Š ¼ N:D:
                                  Pn
                 #
                             #       #
                 Pi Ni
               ¼           where R ¼
                     mi ;              Pr Nr         ½Pi Š ¼ 1=½NŠ
                   #
                 1þR
                                  r¼1
                               #  #    #
                               Pi ¼ pi =k2 ¼ ai hi
             where mi ¼ 1=hi ;


                    #
                                                ½fi Š ¼ 1=ð½NŠ2 TÞ
                                #
                    ai Ni             fi Ni
(D) Sigmoidal II                                                      Chesson (1983)
             Ii ¼              where ai ¼ ð1þgi Ni Þ
                   P
                   n
                      #                          ½hi Š ¼ ½TŠ
                      ar hr Nr
                 1þ
                   r¼1
                                                ½gi Š ¼ 1=½NŠ


                            Pn
(E) Abundance-Based I                                                    Strom and Loukos (1998)
                                                ½mi Š ¼ 1=T
                #
                              #
                pi Ni
             Ii ¼ kþR m; where R ¼      pr Nr
                                                ½kŠ ¼ ½NŠ
                            r¼1
                  1 À wi Ni ; for Ni oZ                    ½pi Š ¼ N:D:
                               and wi ¼ ð1Àpi Þ
             #
             pi ¼
                                                ½ZŠ ¼ ½NŠ
                                    Z
                  pi ;    for Ni XZ
                                                ½wi Š ¼ 1=½NŠ



(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




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                                                       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
                     Ã                                 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
                     Ã     Ã      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
                     Ã 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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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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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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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 Š ¼ 1=T
               #
                                   #
               pi Ni
             Ii ¼ kþR m;       where R ¼      pr Nr
Based                                                            Fasham et al. (1993),
                                                   ½kŠ ¼ ½NŠ
                                r¼1
                                                   ½pi Š 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 Š ¼ 1=T
                #
                                   #
               pi Ni
             Ii ¼ kþR m;       where R ¼      pr Nr
Based II                                               ½kŠ ¼ ½NŠ
                                r¼1
                                                   ½pi Š ¼ N:D:
                          *      *
                        1 À Nwi   for NoZ
                                          wi ¼ ð1Àpi Þ
               #                                    ½ZŠ ¼ ½NŠ
             and pi ¼                  ;
                                *             Z
                        pi      for NXZ                  ½wi Š ¼ 1=½NŠ
               *
             and N is resource with largest pi
                   #
                   ai Ni
(C) Modified-                                                        Colton (1987)
                                                   ½ai Š ¼ 1=ð½NŠTÞ
             Ii ¼             ; where
                   P
                   n
                      ##                             ½Aij Š ¼ 1=ð½NŠ2 TÞ
Disk                    ar hr Nr
                1þ
                   r¼1
                                                   ½hi Š ¼ ½TŠ
                      P             P
                      n                  n
                              #
             #
             ai ¼ ai þ        Aij Nj and hi ¼ hi þ  Hij Nj            ½Hij Š ¼ T=½NŠ
                      j¼1                 j¼1
                      ja1                 ja1

                   Ni
(D) Modified-                                                        Moloney and Field (1991)
                                                   ½mi Š ¼ 1=T
             Ii ¼          m;
                   P
                   n
Michaelis–Menten                                           ½ki Š ¼ ½NŠ
                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 Š ¼ 1=ð½NŠTÞ
             Ii ¼ ri;eff Ni ;     where ri;eff ¼       ri
                                    R
                                                              Matsuda et al. (1986)
                                                   ½lŠ ¼ N:D:
                      P
                      n
                                                   ½bi Š ¼ N:D:
                         br Nrl
             and R ¼
                      r¼1

             Ii ¼ Iià zi ;
                 #
(F) Weighted-                                                        Pace et al. (1984):
                                                   ½zi Š 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
                             Ã      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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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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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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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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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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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
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