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Predator functional responses: Discriminating between handling and digesting prey. Ecological Monographs 72 (1): 95-112
Keywords: consumer-resource systems; consumption rate; digestion-limited predators; digestion time; functional response models; handling-limited predators; handling time; hunger level; predation rate; predator–prey systems; steady-state satiation (SSS) equation.
Abstract: We present a handy mechanistic functional response model that realistically
incorporates handling (i.e., attacking and eating) and digesting prey. We briefly review
current functional response theory and thereby demonstrate that such a model has been
lacking so far. In our model, we treat digestion as a background process that does not
prevent further foraging activities (i.e., searching and handling). Instead, we let the hunger
level determine the probability that the predator searches for new prey. Additionally, our
model takes into account time wasted through unsuccessful attacks. Since a main assumption
of our model is that the predator’s hunger is in a steady state, we term it the steady-state
satiation (SSS) equation.
The SSS equation yields a new formula for the asymptotic maximum predation rate
(i.e., asymptotic maximum number of prey eaten per unit time, for prey density approaching
infinity). According to this formula, maximum predation rate is determined not by the sum
of the time spent for handling and digesting prey, but solely by the larger of these two
terms. As a consequence, predators can be categorized into two types: handling-limited
predators (where maximum predation rate is limited by handling time) and digestion-limited
predators (where maximum predation rate is limited by digestion time). We give examples
of both predator types. Based on available data, we suggest that most predators are digestion
limited.
The SSS equation is a conceptual mechanistic model. Two possible applications of this
model are that (1) it can be used to calculate the effects of changing predator or prey
characteristics (e.g., defenses) on predation rate and (2) optimal foraging models based on
the SSS equation are testable alternatives to other approaches. This may improve optimal
foraging theory, since one of its major problems has been the lack of alternative models.
Ecological Monographs, 72(1), 2002, pp. 95–112
2002 by the Ecological Society of America
PREDATOR FUNCTIONAL RESPONSES: DISCRIMINATING BETWEEN
HANDLING AND DIGESTING PREY
JONATHAN M. JESCHKE,1,3 MICHAEL KOPP,2 RALPH TOLLRIAN1
AND
1Department of Ecology, Zoological Institute, Ludwig-Maximilians-Universitat Munchen, Karlstrasse 25,
¨ ¨
D-80333 Munchen, Germany
¨
¨
2Max-Planck-Institut fur Limnologie, Abteilung Okophysiologie, Postfach 165, D-24302 Plon, Germany
¨ ¨
Abstract. We present a handy mechanistic functional response model that realistically
incorporates handling (i.e., attacking and eating) and digesting prey. We briefly review
current functional response theory and thereby demonstrate that such a model has been
lacking so far. In our model, we treat digestion as a background process that does not
prevent further foraging activities (i.e., searching and handling). Instead, we let the hunger
level determine the probability that the predator searches for new prey. Additionally, our
model takes into account time wasted through unsuccessful attacks. Since a main assumption
of our model is that the predator’s hunger is in a steady state, we term it the steady-state
satiation (SSS) equation.
The SSS equation yields a new formula for the asymptotic maximum predation rate
(i.e., asymptotic maximum number of prey eaten per unit time, for prey density approaching
infinity). According to this formula, maximum predation rate is determined not by the sum
of the time spent for handling and digesting prey, but solely by the larger of these two
terms. As a consequence, predators can be categorized into two types: handling-limited
predators (where maximum predation rate is limited by handling time) and digestion-limited
predators (where maximum predation rate is limited by digestion time). We give examples
of both predator types. Based on available data, we suggest that most predators are digestion
limited.
The SSS equation is a conceptual mechanistic model. Two possible applications of this
model are that (1) it can be used to calculate the effects of changing predator or prey
characteristics (e.g., defenses) on predation rate and (2) optimal foraging models based on
the SSS equation are testable alternatives to other approaches. This may improve optimal
foraging theory, since one of its major problems has been the lack of alternative models.
Key words: consumer-resource systems; consumption rate; digestion-limited predators; digestion
time; functional response models; handling-limited predators; handling time; hunger level; predation
rate; predator–prey systems; steady-state satiation (SSS) equation.
INTRODUCTION of changing predator or prey characteristics (e.g., de-
fenses) on predation rate.
The relationship between predation rate (i.e., number
of prey eaten per predator per unit time) and prey den- PREVIOUS MODELS: A BRIEF REVIEW
sity is termed the ‘‘functional response’’ (Solomon
Scientists have been modeling functional responses
1949). It is specific for each predator–prey system. The
since the 1920s (reviewed by Holling 1966, Royama
term predator is meant in its broadest sense here, i.e.,
1971), although the term ‘‘functional response’’ was
it includes carnivores, herbivores, parasites, and par-
only introduced in 1949 by Solomon. Since, to our
asitoids. The functional response is an important char-
knowledge, the last review of functional response mod-
acteristic of predator–prey systems and an essential
els dates back to 1971 (Royama), we provide an over-
component of predator–prey models: Multiplying the
view of models published since 1959 together with the
functional response with predator population density
most important factors incorporated in each model (Ta-
and a time factor yields the total number of prey eaten
ble 1). In addition, Fig. 1 shows a ‘‘family tree’’ of
in the period of interest, e.g., one year or one prey
these functional response models. Holling (1959a) has
generation. Given further information, such as actual
categorized functional responses into three main types,
predator density and an energy conversion factor, one
which he called type I, II, and III. Our discussion will
can assess future population densities of both predator
focus on type II functional responses, since these have
and prey. With a mechanistic functional response mod-
been most frequently observed (Hassell et al. 1976,
el, as presented in this study, one can predict the effects
Begon et al. 1996). They are characterized by a hy-
perbolic curve. Starting at low prey densities on the
Manuscript received 8 May 2000; revised 14 December 2000;
abscissa, predation rate first increases almost linearly
accepted 7 February 2001; final version received 12 March 2001.
until it gradually slows down to reach an upper limit.
3 E-mail: jonathan.jeschke@gmx.net
95
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
96
Vol. 72, No. 1
TABLE 1. A selection of functional response models.
Model
A B C D E F G H I J K L
Features (C) (C) (CP) (C) (P) (C) (F) (F) (F) (P) (C) (C)
Components
Success rate1
Probability of attack
Handling time2
Searching and handling overlap-
ping3
Hunger and satiation4
Handling prey digesting prey
Adaptive behavior5
Incomplete consumption6
Nonforaging activities7
Spatial heterogeneity8
Temporal heterogeneity9
Stochasticity10
Environmental conditions11
Predator injury by prey
Inducible defenses12
Dependent on prey density
Prey density
Decreasing prey density13
Alternative prey14
Learning or switching15
Swarming effect16
Dependent on predator density
Predator density
Interference between predators17
Multiple predator effects18
Functional response types
Type I
Type II
Type III
Dome shaped
Other forms19
Notes: Small capital letters in parentheses under models indicate the kind of predator that the model was primarily designed
for: C, carnivores; F, filter feeders; H, herbivores, P, parasites or parasitoids. In the body of the table, ‘‘ ’’ means the model
includes that component, ‘‘ ’’ means the model additionally includes subcomponents, and ‘‘ ’’ means the model does not
include that component. Sources for models are as follows: (A) Gause (1934), Ivlev (1961), Eq. 1; (B) Rashevsky (1959;
no overall model but different equations); (C) Watt (1959); (D) Royama (1971: Eq. 3.12), see also Nakamura (1974: Eq.
15); (E) Royama (1971: Eq. 3.24); (F) Nakamura (1974); (G) Sjoberg (1980); (H) Lam and Frost (1976); (I) Lehman (1976);
¨
(J) Casas et al. (1993); (K) Disc equation (Holling 1959b), Eq. 2; (L) Invertebrate model (Holling 1966; see also Metz and
van Batenburg 1985a,b); (M) Vertebrate model (Holling 1965); (N) Holling and Buckingham (1976); (O) Rao and Kshiragar
(1978); (P) Metz et al. (1988; see also Metz and van Batenburg 1985a, b); (Q) Cushing (1968); (R) Tostowaryk (1972); (S)
Random predator equation (Royama 1971, Rogers 1972); (T) Random parasite equation (Royama 1971, Rogers 1972); (U)
Beddington (1975); (V) Hassell et al. (1977); (W) Longstaff (1980); (X) Mills (1982); (Y) Crowley (1973); (Z) Oaten and
Murdoch (1975); (AA) Real (1977); (BB) McNair (1980); (CC) Abrams (1982); (DD) Dunbrack and Giguere (1987); (EE)
Abrams (1990a); (FF) Descriptive equation (Fujii et al. 1986); (GG) Ungar and Noy-Meir (1988); (HH) Random patch model
˚¨
(Lundberg and Astrom 1990; see also Lundberg and Danell 1990); (II) Juliano (1989); (JJ) Fryxell (1991; see also Wilmshurst
et al. 1995, 1999, 2000); (KK) Spalinger and Hobbs (1992; see also Laca et al. 1994, Shipley et al. 1994); (LL) Farnsworth
and Illius (1996; see also Laca et al. 1994, Shipley et al. 1994); (MM) Hirakawa (1997b; see also Laca et al. 1994, Shipley
et al. 1994); (NN) Farnsworth and Illius (1998; see also Laca et al. 1994, Shipley et al. 1994); (OO) Ruxton and Gurney
(1994); (PP) Cosner et al. (1999) [This model closes a gap between density dependent and ratio dependent functional response
models. Purely ratio dependent models are not included in Table 1, but see Arditi and Ginzburg (1989). However, as Berryman
et al. (1995) have written: ‘‘Note that prey-dependent functional responses can be transformed into ratio-dependent functional
responses by substituting the prey/predator ratio for prey density in the equation.’’]; (QQ) Streams (1994); (RR) Schmitz
(1995; see also Abrams [1990c] and review by Schmitz et al. [1997]); (SS) Abrams and Schmitz (1999); (TT) Berec (2000;
see also Engen and Stenseth 1984); (UU) SSS equation (Eq. 13).
1 Success rate consists of four subcomponents: (1) encounter rate, (2) probability of detection, (3) hunger-independent
probability of attack, and (4) efficiency of attack; empirical values for the attack efficiencies of predators have been reviewed
by Curio (1976), Vermeij (1982), and Packer and Ruttan (1988).
2 Handling time (per prey item) includes attacking time (including evaluating, pursuing, and catching time) and eating
time. See also Anholt et al. (1987), Demment and Greenwood (1988), Laca et al. (1994), Parsons et al. (1994), and Shipley
et al. (1994).
3 Important for queueing predators (Juliano 1989; see also Visser and Reinders 1981, Lucas 1985, Lucas and Grafen 1985)
and vertebrate herbivores (Spalinger and Hobbs 1992, Parsons et al. 1994, Farnsworth and Illius 1996, 1998, Hirakawa
1997b; see also Laca et al. 1994, Shipley et al. 1994).
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 97
TABLE 1. Extended.
Model
M N O P Q R S T U V W X Y Z AA BB CC DD EE
(C) (C) (C) (C) (F) (C) (C) (P) (CP) (CP) (C) (C) (F) (C) (C) (C) (C) (CF) (C)
4 Some models include predator satiation via a maximum predation rate determined by the characteristics of the digestive
system (‘‘ ’’). Other models include the fact that the predator’s gut content is increased by ingestion and decreased by
digestion (‘‘ ’’). See also Campling et al. (1961), Curio (1976), Belovsky (1978, 1984a, b, c, 1986a, b, 1987), Mayzaud and
Poulet (1978), Bernays and Simpson (1982), Murtaugh (1984), Crisp et al. (1985), Demment and Greenwood (1988), Verlinden
and Wiley (1989), Illius and Gordon (1991), Doucet and Fryxell (1993), Forchhammer and Boomsma (1995), Henson and
Hallam (1995), Hirakawa (1997a), and Wilmshurst et al. (2000).
5 See also Belovsky (1978, 1984a, b, c, 1986a, b, 1987), Cook and Cockrell (1978), Sih (1980, 1984), Owen-Smith and
Novellie (1982), McNair (1983), Abrams (1984, 1987, 1989, 1990b, c, 1991, 1992, 1993), Engen and Stenseth (1984),
Formanowicz (1984), Lucas (1985), Wanink and Zwarts (1985), Stephens and Krebs (1986), Anholt et al. (1987), Demment
˚¨
and Greenwood (1988), Belovsky et al. (1989), Verlinden and Wiley (1989), Astrom et al. (1990), Lundberg and Danell
(1990), Mitchell and Brown (1990), Abrams and Matsuda (1993), Doucet and Fryxell (1993), Werner and Anholt (1993),
McNamara and Houston (1994), Forchhammer and Boomsma (1995), Hirakawa (1995, 1997a), Fryxell and Lundberg (1997),
Leonardsson and Johansson (1997), Rothley et al. (1997), Schmitz et al. (1997), and Wilmshurst et al. (2000).
6 See also Buckner (1964), Johnson et al. (1975), Curio (1976), Cook and Cockrell (1978), Sih (1980), Owen-Smith and
Novellie (1982), McNair (1983), Formanowicz (1984), Lucas (1985), Lucas and Grafen (1985), Metz and van Batenburg
˚¨
(1985a, b), Astrom et al. (1990), Lundberg and Danell (1990), and Fryxell and Lundberg (1997).
7 For example, avoidance of top predators, migration, molting, reproductive activities, resting, sleeping, territorial behavior,
thermoregulation, and times of slow rates of metabolism like winter dormancy; see also Belovsky (1978, 1984a, b, c, 1986a,
b, 1987), Caraco (1979), Herbers (1981), Bernays and Simpson (1982), Owen-Smith and Novellie (1982), Abrams (1984,
1991, 1993), Stephens and Krebs (1986), Belovsky et al. (1989), Verlinden and Wiley (1989), Bunnell and Harestad (1990),
Mitchell and Brown (1990), McNamara and Houston (1994), Forchhammer and Boomsma (1995), Hirakawa (1997a), Leon-
ardsson and Johansson (1997), and Rothley et al. (1997).
8 See also Griffiths and Holling (1969), Paloheimo (1971a, b), Oaten (1977), May (1978), Real (1979), McNair (1983),
Belovsky et al. (1989), Blaine and DeAngelis (1997), Fryxell and Lundberg (1997), and Wilmshurst et al. (2000), among others.
9 For example, diel or annual periodicity (Curio 1976, Bernays and Simpson 1982, Belovsky et al. 1989, Forchhammer
and Boomsma 1995).
10 See also Paloheimo (1971a, b), Curry and DeMichele (1977), Curry and Feldman (1979), McNair (1983), Lucas (1985),
and Metz and van Batenburg (1985a, b).
11 For example, precipitation, temperature (Fedorenko 1975, Thompson 1978, Bernays and Simpson 1982), and wind.
12 Behavioral and morphological defenses, that are not permanently present but are induced by the predator (e.g., Fryxell
and Lundberg 1997, Karban and Baldwin 1997, Tollrian and Harvell 1999, Jeschke and Tollrian (2000).
13 See also Curry and DeMichele (1977).
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
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Vol. 72, No. 1
TABLE 1. Extended.
Model
FF GG HH II JJ KK LL MM NN OO PP QQ RR SS TT UU
Features (CP) (H) (H) (C) (H) (H) (H) (H) (H) (C) (C) (C) (CH) (H) (C) (C)
Components
Success rate1
Probability of attack
Handling time2
Searching and handling over-
lapping3
Hunger and satiation4
Handling prey digesting prey
Adaptive behavior5
Incomplete consumption6
Nonforaging activities7
Spatial heterogeneity8
Temporal heterogeneity9
Stochasticity10
Environmental conditions11
Predator injury by prey
Inducible defenses12
Dependent on prey density
Prey density
Decreasing prey density13
Alternative prey14
Learning or switching15
Swarming effect16
Dependent on predator density
Predator density
Interference between predators17
Multiple predator effects18
Functional response types
Type I
Type II
Type III
Dome shaped
Other forms19
14 See also Belovsky (1978, 1984a, b, c, 1986a, b, 1987), Owen-Smith and Novellie (1982), Engen and Stenseth (1984),
Metz and van Batenburg (1985a), Wanink and Zwarts (1985), Abrams (1987, 1989, 1990b, c), Belovsky et al. (1989), Abrams
and Matsuda (1993), Doucet and Fryxell (1993), Parsons et al. (1994), Forchhammer and Boomsma (1995), Fryxell and
Lundberg (1997), Rothley et al. (1997), Schmitz et al. (1997), and Wilmshurst et al. (2000).
15 Learning includes training effects; switching means either switching between prey types (in this case, there is a ‘‘ ’’
at ‘‘alternative prey’’) or behavioral switching, e.g., from sitting and waiting to cruising. Only those models that explicitly
consider learning or switching have ‘‘ ’’ here. Optimal foraging models where switching is a simulation result have ‘‘ ’’
here. For an experimental example of the interaction between learning and spatial distribution see Real (1979); see also
Fryxell and Lundberg (1997) and Kaiser (1998).
16 A swarming effect decreases predation rate with increasing prey density. It can be the result of (1) a better or earlier
detection of the predator by prey, (2) a worse detection of prey by the predator, (3) a better active prey defense, (4) predator
confusion which usually decreases probability or efficiency of attack, (5) clogging of filters (in case of filter feeders), or (6)
accumulation of toxic prey substances. The form of the functional response can be dome shaped in this case. See Miller
(1922), Brock and Riffenburgh (1960), Mori and Chant (1966), Tostowaryk (1972), Halbach and Halbach-Keup (1974), Neill
and Cullen (1974), Nelmes (1974), Milinski and Curio (1975), Bertram (1978), Lazarus (1979), Williamson (1984), Morgan
and Godin (1985), Landeau and Terborgh (1986), and Inman and Krebs (1987).
17 Interference also includes prey exploitation by other predators. Only those models that consider interference inclusively
and prey exploitation explicitly have ‘‘ ’’ here. Models that account for a decreasing prey density through predation and the
number of predators present and include prey exploitation in an implicit way have ‘‘ ’’ here. See also models by Griffiths and
Holling (1969), Hassell and Varley (1969), Royama (1971, model in §4i), DeAngelis et al. (1975), Curry and DeMichele (1977),
Parker and Sutherland (1986), Korona (1989), Ruxton et al. (1992), Holmgren (1995), Fryxell and Lundberg (1997), and
Doncaster (1999); for empirical studies, see Norris and Johnstone (1998), Triplet et al. (1999), or references in Holmgren (1995).
18 Soluk (1993), Sih et al. (1998).
19 Crowley (1973) and Farnsworth and Illius (1996), intermediate type I/II; Nakamura (1974), type II similar; Lam and
Frost (1976), Fujii et al. (1986), type I similar; Lehman (1976), partly type I similar; Metz et al. (1988) and random patch
˚¨
model (Lundberg and Astrom 1990), hyperbolic (type II similar) functional response without an asymptote; Abrams (1982),
Juliano (1989), Fryxell (1991), Schmitz (1995), Hirakawa (1997b), Farnsworth and Illius (1998), Abrams and Schmitz (1999),
and Berec (2000), various forms; see also Parsons et al. (1967, type II with a threshold prey density, corresponding empirical
curves in the same study and in Parsons et al. (1969), Cook and Cockrell (1978; double plateau functional responses), Abrams
(1987, 1989; decreasing functional responses), and Fryxell and Lundberg (1997; various forms).
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 99
FIG. 1. A ‘‘family tree’’ of functional response models.
All functional response models include a factor that Although purely phenomenological, the Gause-Ivlev
determines the curve’s gradient at the origin (‘‘success equation (Gause 1934, Ivlev 1961) has usually been
rate’’ which is a measure of the predator’s hunting ef- viewed as the classical satiation model
ficiency; it has been termed the ‘‘rate of successful
y(x) ymax (1 exp [ a x]) (1)
search’’ by Holling [1959a, b, 1965, 1966]).
where a is hunting success (dimension in SI units: m2
Handling and digestion for a two-dimensional system, e.g., a terrestrial system,
and m3 for a three-dimensional system, e.g., an aquatic
Limitation of predation rate at high prey densities
system), x is prey density (individuals/m2 or individ-
has usually been attributed to either handling time or
uals/m3, respectively), y is predation rate (s 1), and ymax
satiation. However, the exact nature of these two factors
is asymptotic maximum predation rate as x approaches
and their relationship has been modeled in a variety of
infinity (s 1). In the common interpretation, the diges-
different ways and this has led to considerable con-
tive system determines ymax, and the functional response
fusion. The point is that handling prey is an active
curve gradually rises to this value. Rashevsky (1959)
process whereas digestion is a background process. As
has extended the Gause-Ivlev equation by modeling
a consequence, in contrast to handling prey, digestion
satiation more mechanistically: the predator’s gut con-
does not directly prevent the predator from further
tent is increased by ingestion and decreased by diges-
searching or handling. Rather, digestion influences the
tion. Other models including satiation but not handling
predator’s hunger level, which in turn influences the
time have been developed by Watt (1959), Parsons et
probability that the predator searches for new prey. It
al. (1967), Royama (1971), Nakamura (1974), Lam and
is thus necessary to discriminate digestion from han-
Frost (1976), Lehman (1976), Sjoberg (1980), Crisp et
¨
dling in a functional response model. In the following,
al. (1985), Metz and van Batenburg (1985b), Metz et
we briefly review existing models with respect to their
al. (1988), Abrams (1990c), Casas et al. (1993; para-
treatment of these two factors. In our opinion, no com-
pletely satisfying solution to the problem exists to date. sitoid egg load as analogous to hunger level), Henson
Models including satiation but not handling time.— and Hallam (1995), and Abrams and Schmitz (1999).
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
100
Vol. 72, No. 1
Models including handling time but not satiation.— b tatt / teat
In contrast, there are a number of models that include
⇔ asymptotic maximum predation rate
handling time but no predator satiation effects (Holling
1959b, Cushing 1968, Royama 1971, Rogers 1972, teat ) 1.
(tatt / (4)
Tostowaryk 1972, Beddington 1975, Hassell et al.
In using this definition, handling time includes time
1977, Real 1977, Cook and Cockrell 1978, Curry and
wasted through unsuccessful attacks (see also Mills
Feldman 1979, Longstaff 1980, McNair 1980, Visser
1982, Abrams 1990a, Streams 1994).
and Reinders 1981, Abrams 1982, 1987, 1990a, Fujii
Models including both handling time and satia-
et al. 1986, Dunbrack and Giguere 1987, Ungar and
`
tion.—One approach to include both handling time and
˚¨
Noy-Meir 1988, Juliano 1989, Lundberg and A strom
digestion time is to sum them up or to increase handling
1990, Spalinger and Hobbs 1992, Parsons et al. 1994,
time by a ‘‘digestive pause’’ (Crowley 1973, Rao and
Ruxton and Gurney 1994, Streams 1994, Farnsworth
Kshirsagar 1978, Mills 1982, Henson and Hallam
and Illius 1996, Fryxell and Lundberg 1997, Cosner et
1995), i.e., an inactive time period related to digestion
al. 1999, Berec 2000). The most popular functional
(Holling 1965, 1966). When modeled this way, diges-
response model today, Holling’s (1959b) disc equation,
tion is not distinguished from handling. Mills (1982)
belongs to this class:
used this concept to extend the disc equation by inter-
ax preting its parameter b as
y(x) (2)
abx
1
b tatt teat stdig
where a is success rate (dimension in SI units: m2/s or
⇔ asymptotic maximum predation rate
m3/s, respectively; note that the dimensions of a and
a [Gause-Ivlev equation] differ), b is predator han- 1
teat stdig )
(tatt (5)
dling time per prey item (s), x is prey density (indi-
where s is satiation per prey item (dimensionless) and
viduals/m2 or individuals/m3, respectively), and y is
tdig is digestion time per prey item (s; see Table 2).
predation rate (s 1). The curve’s gradient at the origin
A second way to consider both handling and diges-
is equal to a, and the asymptotic maximum for x as x
tion time is to combine the disc equation (which already
approaches infinity is 1/b. The disc equation is math-
includes handling time) with a digestive capacity con-
ematically equivalent to the Michaelis-Menten model
straint (Fryxell 1991, Schmitz 1995, Hirakawa 1997b,
of enzyme kinetics and the Monod formula for bacterial
Farnsworth and Illius 1998). This constraint limits
growth. The Royama-Rogers random predator equation
maximum predation rate but does not otherwise affect
(Royama 1971, Rogers 1972) is a modification of the
the functional response. These models therefore dis-
disc equation that accounts for a decreasing prey den-
criminate between handling and digesting prey. How-
sity in the course of an experiment or between discrete
ever, neither the process of digestion, nor the predator
prey generations. In the original paper (Holling 1959b),
satiation level are considered. The approach to combine
the parameter b of the disc equation denoted the general
handling time with a digestive capacity constraint has
meaning of ‘‘handling time’’ at that time, i.e., the sum
its origins in linear programming models (e.g., Belov-
of attacking time tatt (per prey item; including evalu-
sky 1978, 1984a, b, c, 1986a, b, 1987, Doucet and
ating, pursuing, and catching time) and eating time teat
Fryxell 1993, Forchhammer and Boomsma 1995).
(per prey item):
The only family of models that treats digestion as a
b tatt teat background process, which influences foraging activ-
⇔ asymptotic maximum predation rate ities but does not prevent them, is Holling’s (1966)
invertebrate model and its extensions (Holling 1965,
1
teat )
(tatt (3) Holling and Buckingham 1976, Curry and DeMichele
1977, Metz and van Batenburg 1985a). In the inver-
with handling time b as it was originally defined by
tebrate model, the predation cycle is subdivided into
Holling (1959b).
several stages, and each stage depends on predator hun-
Holling originally developed the disc equation as a
ger level. After a meal, the predator is assumed to
mechanistic model for an artificial predator–prey sys-
undergo a digestive pause and then continues searching
tem: humans ‘‘preying’’ on paper discs (Holling
when it is hungry again. While searching, the predator
1959b). Compared to natural predator–prey systems,
simultaneously continues digestion of its last meal. The
however, it is now clear that the underlying assump-
invertebrate model therefore discriminates between
tions are unrealistic (Hassell et al. 1976). Two points
handling and digesting prey. Here, the length of the
have met the most severe criticism: First, the predator
digestive pause depends on hunger level. Since hunger
does not become satiated, and second, the disc equation
level in turn depends on prey density, the length of the
assumes that every attack by the predator is successful,
digestive pause depends on prey density. This is in
i.e., attack efficiency 100%. Attack efficiencies
100% can be incorporated into the disc equation by contrast to the models mentioned above (Eq. 5), where
defining b as the length of the digestive pause is unrealistically as-
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 101
TABLE 2. The SSS equation parameters.
Parameter Description Dimension† Defined for
m3/s
Encounter rate number of encounters between a searching predator and [0; ]
a single prey item; an encounter is defined as an arrival of a prey item
in the predator’s encounter volume
Probability that the predator detects encountered prey [0; 1]
Efficiency of attack proportion of successful attacks [0; 1]
s Satiation per prey item reciprocal capacity of the hunger-determining [0; ]
part of the gut (mostly stomach or crop); example: if the stomach ca-
pacity of a human is equal to 10 potatoes, then s 0.1
tatt Attacking time per prey item time between prey detection and end of s [0; ]
attack
tdig Digestion time per prey item food transit time ( 50% emptying time) s [0; ]
for the hunger-determining part of the gut, e.g., stomach transit time
for humans
teat Eating time per prey item time between capture and finished ingestion s [0; ]
Note: The parameters can be summarized by a (success rate [m3/s]), b (corrected handling time [s]), and c (corrected
digestion time [s]), see Eq. 13.
† In SI units and given for a three-dimensional system, e.g., an aquatic system; in the case of a two-dimensional system,
e.g., a terrestrial system, m3 must be replaced by m2.
sumed to be constant. The term ‘‘digestive pause’’ re- is no mechanistic linkage to the processes of ingestion
lates to foraging activities only: predators may well use and digestion.
the digestive pause for nonforaging activities, for ex- Their mathematical simplicity renders the disc, the
ample, for looking out for top predators or for sleeping. random predator, and the Gause-Ivlev equation as func-
However, because of its 22 parameters, the invertebrate tional response submodels in predator–prey population
model is extremely unwieldy, and its extensions are models. However, for a deeper understanding of the
even more elaborate. functional response, mechanistic models are necessary.
The parameters of mechanistic models can all be mech-
Phenomenological vs. mechanistic models anistically explained. These models can thus, for ex-
ample, be used to calculate the effects of changing
Probably because of their mathematical simplicity,
predator or prey characteristics (e.g., defenses) on pre-
the Holling (1959b) disc equation (Eq. 2), the Royama-
dation rate.
Rogers random predator equation (Royama 1971, Rog-
ers 1972), and the Gause-Ivlev equation (Gause 1934,
THE STEADY-STATE SATIATION (SSS) EQUATION
Ivlev 1961; Eq. 1) have been the most popular func-
We have shown that a handy mechanistic functional
tional response models. However, they must be con-
response model that realistically incorporates handling
sidered phenomenological. That is, although they cor-
and digesting prey has been lacking so far. In this sec-
rectly reproduce the shape of natural (type II) func-
tion, we therefore develop such a model: the steady-
tional responses, they are not able to explain the un-
state satiation (SSS) equation. It is based on the disc
derlying mechanism; or, in other words, its parameters
equation and divides the predation cycle into five stag-
cannot all be mechanistically explained. In the cases
es: search, encounter, detection, attack, and eating (Fig.
of the disc equation and the random predator equation,
2). We assume that these stages are mutually exclusive.
the parameter a (success rate) can be mechanistically
Each stage is characterized by two components: The
explained (Holling 1966, Ungar and Noy-Meir 1988,
amount of time needed for its completion and the con-
Streams 1994, Hirakawa 1997b; see also Eq. 7 below),
ditional probability that the predator reaches this stage
but not the parameter b (handling time). When fitting
given that it has reached the previous one (exception:
the disc equation or the random predator equation to
encounter; here, it is not a probability but a rate; note
an empirical curve, the resulting value for b is a mixture
that the encounter rate can have a value larger than
of different biological processes (Table 1) including
unity). Digestion is modeled as a background process
handling (attacking and eating) and digestion. As we
influencing the predator’s hunger level, which in turn
have pointed out above, handling is an active process,
determines the probability that the predator searches
whereas digestion is a background process. They can-
for prey.
not be adequately condensed into only one parameter.
In the case of the Gause-Ivlev equation, neither param-
The SSS equation components, parameters,
eter can be mechanistically explained. Its parameter a
and assumptions
(hunting success) differs in its dimension from the pa-
rameter a of the disc equation and the random predator The SSS equation components are given in Table 1
equation; a lacks a mechanistic explanation. The other and its parameters in Table 2. Like nearly every model,
parameter, ymax, is just the asymptote of the curve; there the SSS equation is a compromise between realism and
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
102
Vol. 72, No. 1
1) There is only a single predator and a single type
of prey.
2) The prey density is constant.
3) Prey are independently and randomly distributed.
4) Stages of the predation cycle exclude each other
(Fig. 2).
5) The probability that the predator searches (under
the condition that it is not handling prey), (x), is di-
rectly proportional to the predator’s hunger level h(x).
6) The hunger level h(x) depends on the fullness of
a certain part of the gut (e.g., stomach, crop).
7) The hunger level h(x) at a given prey density x is
in a steady state, which is determined by an equilibrium
of ingestion and digestion.
8) The probability of attack, , is unity, i.e., when-
ever a searching predator encounters and detects a prey,
it will attack.
9) The probability that the predator detects encoun-
tered prey, , the efficiency of attack (i.e., the pro-
portion of successful attacks), the attacking time tatt (per
prey item), the eating time teat (per prey item), and the
digestion time tdig (per prey item) are constant.
The SSS equation
To develop the SSS equation, we start with the disc
equation and modify it sequentially. In step 1, each
stage of the predation cycle is included explicitly; in
step 2, predator satiation is included by influencing the
FIG. 2. The predation cycle. We divide the predation cycle
into five stages: search, encounter, detection, attack, and eat- probability of searching.
ing. A predator enters a predation cycle under the probability
The stages of the predation cycle are (1) search, (2)
to search, (x); this is determined by the predator’s hunger
encounter, (3) detection, (4) attack, and (5) eating (Fig.
level, which in turn is influenced by digestion time. Then the
2). The probabilities that a predator reaches these stages
predator successively reaches the following stages. The prob-
ability that the predator reaches a stage under the condition are (1) the probability that a predator not occupied
that it has reached the previous stage is given in the corre- with handling searches for prey, (2) the encounter rate
sponding arrow, e.g., the probability that the predator detects
between a searching predator and an individual prey,
a prey under the condition that it has encountered that prey
(3) the probability that the predator detects an en-
is (exception: is not a probability but a rate [encounter
countered prey individual, (4) the probability that the
rate]; note that it can be larger than unity). Since is set as
unity in the SSS equation (assumption 8), it is given in pa- predator attacks a detected prey individual, and (5) the
rentheses here. Terms in circles indicate time demands of
probability that an attack is successful, i.e., the ef-
corresponding stages per prey item. We assume that the stages
ficiency of attack. We now incorporate these probabil-
are mutually exclusive (assumption 4). Terms with a super-
ities into the disc equation.
script ‘‘a’’ determine predator success rate (a).
The searching probability (x).—In the disc equa-
tion, the predator shows only two kinds of behavior:
searching for and handling prey. Therefore, the prob-
applicability. It is more realistic than the disc equation,
ability that the predator searches for prey under the
but it is reductive compared to nature. The SSS equa-
condition that it is not handling prey, (x), is unity. To
tion is a conceptual model that can, for example, be
allow values below unity, (x) has to be incorporated
used to assess how changing predator or prey charac-
explicitly into the disc equation:
teristics (e.g., defenses) qualitatively affect the func-
tional response. The point of the SSS equation is not (x)ax
y(x) . (6)
to quantitatively predict real functional response
1 (x)abx
curves. It is therefore not necessary to incorporate too
depends on prey density x because it is
many features into the model, which would render it Note that
unwieldy. However, extensions for specific predator– affected by the predator’s hunger level (see the next
prey systems are possible; these will allow us to make paragraph and assumption 5), which in turn depends
quantitative predictions with the model as well. For on prey density (see the next paragraph and assumption
this purpose, references given in Table 1 may be help- 7): (h) (h(x)) (x).
The encounter rate , the probability of detection ,
ful.
the probability of attack , and the efficiency of attack
The assumptions of the model are as follows:
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 103
c · y(x))ax
.—The product of all these terms is predator success (1
y(x) . (12)
rate a. However, for simplicity, we set 1 (as- c · y(x))abx
1 (1
sumption 8). Thus,
Solving for y(x) finally gives the following SSS equation:
a . (7)
1
The encounter rate can be calculated by various
c) 2 )
ax(b c) ax(2(b c) ax(b
1
formulae from different authors. For a three-dimen-
2abcx
sional model, e.g., in aquatic systems, one may use
the equation given by Gerritsen and Strickler (1977). a, b, c, x 0
For an analogous two-dimensional model, e.g., in ter-
ax
restrial systems, see Koopman (1956), and for a three- b c
0 0
1
y(x)
abx
dimensional model with a cylindrical instead of a
ax
spherical encounter volume, see Gigue re et al. (1982).
`
b c
0 0
For further models, see Royama (1971: Eq. 4e.6), Get- acx
1
ty and Pulliam (1991), Parsons et al. (1994), Hirakawa
ax b c 0
(1997b), and reviews from Schoener (1971) and Curio
0 a x
0 or 0
(1976). Here, for simplicity, is not calculated by
one of these formulae but is a model input; probability
(13)
of detection and efficiency of attack are also model
with success rate a , corrected handling time
inputs.
b t att / t eat , and corrected digestion time c st dig .
Explicitly incorporating efficiency of attack allows
For details on deriving Eq. 13 from Eq. 12, see Ap-
us to account for time wasted through unsuccessful
pendix A. For c 0 (i.e., no satiation), the SSS equa-
attacks. Thus, handling time b can be calculated ac-
tion simplifies to the disc equation but with the defi-
cording to Eq. 4.
nitions of Eq. 13 for a and b. For b 0 (i.e., zero
The second and final step in deriving the SSS equa-
handling time), the SSS equation simplifies to the disc
tion is to incorporate digestion. We do this by assuming
equation but with c instead of b, i.e., digestion time
that
replaces handling time in this case. Finally, without
h(x).
(x) (8)
c
any handling time or satiation (b 0), there are
no density dependent effects and so, predation rate is
This is assumption 5 and is also assumed by Rash-
directly proportional to prey density.
evsky (1959). The hunger level h(x) is the proportion
of empty volume of that part of the gut that is re-
Properties of the SSS equation
sponsible for feelings of hunger and satiation in the
The SSS equation produces type II functional re-
predator under consideration (mostly stomach or
sponses (Fig. 3). As in the disc equation, the gradient
crop); h(x) is defined for [0; 1], where h 0 means
at the origin is equal to the predator’s success rate a:
no hunger, i.e., full gut, and h 1 means 100% hunger,
i.e., empty gut. Empirical studies usually find a hy-
dy(x)
perbolic relationship between starvation time and a.
lim (14)
dx
x→0
hunger level, e.g., Holling (1966) for mantids (Hier-
odula crassa and Mantis religiosa), Antezana et al. The asymptotic maximum predation rate for prey den-
(1982) for krill (Euphausia superba), Hansen et al. sity as x approaches infinity is
(1990) for copepods (Calanus finmarchicus), and sev-
c) 2
b c (b 1
eral works on fish (reviewed by Elliott and Persson lim y(x) .
max(b; c)
2bc
1978). This hyperbolic relationship can be described x→
by the following differential equation:
where, for handling-limited predators,
dh(x) h(x)
1
1
sy(x). (9)
c ⇔ lim y(x)
b
dt tdig
b
x→
Since we assume a constant prey density (assumption
and, for digestion-limited predators,
2), the equilibrium hunger level can be obtained by
setting dh(x)/dt 0, giving 1
b ⇔ lim y(x)
c . (15)
c
(10)
h(x) s·tdig·y(x).
1 x→
Thus, the larger one of the two terms b and c determines
We define c s·tdig as ‘‘corrected digestion time’’, i.e.,
the asymptotic maximum predation rate. This is, be-
digestion time corrected for gut capacity. Therefore,
cause digestion is a ‘‘background process’’, i.e., han-
h(x) c·y(x).
1 (11) dling and digestion can be carried out simultaneously.
The slower one of these two processes is then limiting.
Inserting Eq. 11 into Eqs. 8 and 6 yields
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
104
Vol. 72, No. 1
When corrected handling time exceeds corrected di-
gestion time (b c, condition 1), the asymptotic max-
imum predation rate is 1/b. This is the same situation
as in a disc equation when attack efficiencies 100%
are considered (see Eq. 4). We call predators under this
condition ‘‘handling-limited predators.’’ Fig. 3a shows
graphs of the SSS equation for handling-limited pred-
ators. Although the asymptote is independent of c, it
is approached more slowly as digestion time becomes
more important, i.e., large digestion times result in a
slower rise of the curve. As c approaches 0, the SSS
curve approaches a disc equation curve (with a cor-
rection for attack efficiencies 100%).
When corrected digestion time exceeds corrected
handling time (c b, condition 2), the asymptotic max-
imum predation rate equals 1/c. We call predators under
this condition ‘‘digestion-limited predators.’’ Fig. 3b
shows graphs of the SSS equation for digestion-limited
predators. With larger handling times, the asymptote is
approached more slowly, yet the asymptote itself is
independent of b. As b approaches 0, the SSS curve
approaches a disc equation curve with digestion in
place of handling (c instead of b) as the limiting factor.
SSS equation curves are more flexible than disc
equation curves. Thus, it is impossible to satisfyingly
fit the disc equation to a SSS equation curve (with the
exceptions b 0 or c 0). This is, because, in the
disc equation, one parameter (b) determines the curve’s
asymptote, and two parameters (a and b) determine
how the curve reaches this asymptote, i.e., the curve’s
slope. In contrast, in the SSS equation, one parameter
(the larger one of the parameters b and c) determines
the curve’s asymptote, and three parameters (a, b, and
c) determine how the curve reaches this asymptote.
Fig. 3c illustrates how time wasted through unsuc-
cessful attacks (attack efficiency 100%) reduces
the slope of the functional response curve (and, in case
of handling-limited predators, the asymptotic maxi-
mum predation rate).
DISCUSSION
We have developed a handy mechanistic functional
response model (the SSS equation) that realistically
FIG. 3. Graphs of the SSS equation (Eq. 13). (A) Han-
dling-limited predators. Model inputs were success rate a ←
2, corrected handling time b 0.02, and corrected digestion
time c 0, 0.01, or 0.02, respectively; thus, b c. All curves (C) Effect of attack efficiency . Model inputs were a 2
0.25), b 0.01 (
are type II functional responses, and for all curves, asymptotic (corresponding 0.5) or 1 ( 0.5,
tatt 0.0025, teat 0.005; b [Eq. 13] 0.0025 / 0.5 0.005
maximum predation rate 1/b 50 (Eq. 15). However, this
0.25, tatt 0.0025, teat 0.005; b [Eq.
asymptotic maximum is approached more slowly as digestion 0.01) or 0.015 (
time becomes more important. For c 0.015), and c
0, the SSS equation 13] 0.0025 / 0.25 0.005 0.02. When
is equal to the disc equation (Eq. 2). (B) Digestion-limited attack efficiency is halved (from 0.5 to 0.25), the gradient at
predators. Model inputs were a 2, b 0, 0.01, or 0.02, the origin is halved (a 2 or 1, Eq. 14) and the predation
respectively, and c 0.02; thus, c b. All curves are type rate is decreased at almost all prey densities. However, in
II functional responses, and for all curves, asymptotic max- case of a digestion-limited predator (as in our example), as-
imum predation rate 1/c 50 (Eq. 15). However, this ymptotic maximum predation rate remains constant (1/c
asymptotic maximum is approached more slowly as handling 50, Eq. 15). In the case of a handling-limited predator (graph
time becomes more important. For b not shown), b is increased, and thus asymptotic maximum
0, the SSS equation
is equal to the disc equation, when b is replaced by c there. predation rate is decreased.
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 105
incorporates success rate, handling time, and satiation. of top predators, migration, molting, reproductive ac-
The satiation level is assumed to linearly decrease hunt- tivities, resting, sleeping, territorial behavior, thermo-
ing activities. The SSS equation thereby fills a gap in regulation, or times of slow rates of metabolism like
functional response theory, because previous models winter dormancy).
either do not treat satiation in a realistic way (since The easiest method to detect a handling-limited pred-
they do not discriminate between handling and di- ator is to directly measure corrected handling time (ac-
gesting prey or simply include satiation by a maximum cording to Eq. 4) as well as corrected digestion time
predation rate, i.e., a digestive capacity constraint) or (according to Eq. 13) and to compare them. However,
are extremely unwieldy. all predators, from whom both measurements are avail-
Like the widely used disc equation, the SSS equation able in the literature, are digestion-limited (see next
produces type II functional response curves. However, section).
there are several differences. First, because of its third Another method to detect a handling-limited pred-
parameter, the SSS equation is more flexible than the ator is:
disc equation. The differences are largest when han- 1A) Through observation, directly measure predator
handling time b according to Eq. 4.
dling time and digestion time are of the same order of
magnitude (Fig. 3). On the contrary, if one of these 1B) (Alternative to 1A) Perform short-term feeding
two factors is negligibly small, the curve becomes vir- experiments to get a short-term functional response
tually identical to that of the disc equation. Second, the without satiation effects. Fit the disc equation (if eaten
disc equation assumes an attack efficiency equal to prey was replaced) or the random predator equation (if
eaten prey was not replaced) to the data to get b (han-
100%. When this is not the case, the maximum pre-
dation rate is decreased because of time spent for un- dling time according to Eq. 4).
2A) Measure long-term maximum feeding rate ymax
successful attacks. Although mentioned by Mills
(1982), Abrams (1990a), and Streams (1994), this ef- (with satiation) at an extremely high prey density.
fect has not been incorporated into most models. It is 2B) (Alternatively to 2A) Perform long-term feeding
contrary to the basic idea of the disc equation that the experiments, ideally starting with predators in a steady
parameters a and b are independent (Holling 1965, hunger state, or do a field study. Fit the disc equation
or the random predator equation to the data to get ymax.
1966). In nature, predator attack efficiencies seldom
3) If b
reach 100% (see Curio 1976, Vermeij 1982, and Packer 1/ymax, it is likely that the predator is han-
and Ruttan 1988). Taking unsuccessful attacks into ac- dling limited.
count is especially important for predators with non- We have applied this method to available literature
negligible attacking times. Third and most important, data and have found three candidates for handling-lim-
the disc equation (with b interpreted as in Eq. 5) does ited predators. First, in the host–parasitoid system Silo
pallipes (Trichoptera: Goeridae)–Agriotypus armatus
not discriminate between handling and digesting prey.
The SSS equation, on the other hand, takes into account (Hymenoptera: Agriotypidae), Elliott (1983) directly
measured the handling time of A. armatus and found
their different nature, and as a result, the maximum
b 20.0 min. In addition, he fitted the random predator
predation rate (prey density approaches infinity) is not
19.4–20.1 min, thus b
determined by the sum of time spent for handling and equation to field data: 1/ymax
1/ymax. Second, in the predator–prey system Och-
digesting prey (as in Eq. 5), but by the maximum of
romonas sp. (a heterotrophic flagellate)–Pseudomonas
these two terms. Accordingly, we have classified pred-
ators into handling-limited and digestion-limited pred- sp. (a bacterium), Fenchel (1982a) directly measured
the handling time of Ochromonas as b
ators. Note that this classification only refers to high 20 s. In ad-
prey densities. At intermediate prey densities, our mod- dition, he performed long-term experiments (Fenchel
19 s, thus b
el shows that also handling-limited (digestion-limited) 1982b): 1/ymax 1/ymax. Third, in the
predator–prey system Polinices duplicatus (a naticid
predators experience diminished feeding rates because
of time spent for digesting (handling) prey (Figs. 3a, gastropod that drills through the shells of its prey)–
Mya arenaria (Bivalvia), the handling time of P. du-
b).
plicatus in the long-term enclosure experiments of Ed-
Handling-limited predators wards and Huebner (1977) can be estimated by data
Handling-limited predators handle (corrected for at- from Edwards and Huebner (1977) and Kitchell et al.
(1981; Appendix B): b 1.4 d; 1/ymax 1.6 d, thus b
tack efficiencies 100%) prey slower than they digest
them. For parasites and parasitoids, this means that they 1/ymax. Similarly, Boggs et al.’s (1984) results have
indicated that P. duplicatus is also handling limited
handle hosts slower than they produce eggs. In han-
when feeding on another bivalve, Mercenaria mercen-
dling-limited predators, therefore, prey uptake increas-
aria. In their study, P. duplicatus spends 75% of its
es with the amount of time spent for searching and
time in handling (i.e., drilling and eating) M. mercen-
handling prey. We consequently expect that, indepen-
aria; total foraging time (i.e., searching time plus han-
dent of prey density, handling-limited predators forage
almost all of their available time (i.e., the time not dling time) was therefore at least 75%. This exceeds
needed for nonforaging activities, such as avoidance by far corresponding values for digestion-limited pred-
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
106
Vol. 72, No. 1
ators (see Discussion: Digestion-limited predators), moschatus], Forchhammer and Boomsma [1995];
sheep [Ovis aries], Blaxter et al. [1961]; shrews [Sorex
corroborating our expectation that handling-limited
araneus, S. caecutiens, S. isodon], Saariko and Hanski
predators should spend more time in foraging than di-
gestion-limited predators. [1990]).
Further examples for handling-limited predators can For digestion-limited predators, the SSS equation,
likely be found in other parasitoids, protozoans, and contrary to Holling’s (1959b) disc equation (Eq. 2),
drilling gastropods. In general, however, handling-lim- predicts that foraging time decreases with increasing
ited predators seem to be rare. prey density. This is in accordance with empirical data,
for example from birds (Spotted Sandpipers [Actitis
Digestion-limited predators macularia], Maxson and Oring [1980]; Verdins [Au-
riparus flaviceps], Austin [1978]; Oystercatchers [Hae-
Digestion-limited predators digest prey items slower
matopus ostralegus], Drinnan [1957]; Yellow-eyed
than they handle them. For parasites and parasitoids,
Juncos [Junco phaeonotus], Caraco [1979]; humming-
this means they produce eggs slower than they handle
birds [Selasphorus rufus], Hixon et al. [1983]) and
hosts. At high prey densities, therefore, predation up-
mammals (horses [Equus caballus], Duncan [1980];
take does not further increase with the amount of time
white-tailed jackrabbits [Lepus townsendii], Rogowitz
spent for searching and handling prey. This releases
[1997]; sheep [Ovis aries], Alden and Whittaker
trade-off situations at high prey densities and closes
[1970]; mouflon [Ovis musimon], Moncorps et al.
the gap between optimal foraging and satisficing theory
(J. Jeschke, personal observation; for satisficing, see [1997]; reindeer [Rangifer tarandus tarandus], Trudell
and White [1981]; greater kudus [Tragelaphus strep-
Herbers [1981] and Ward [1992, 1993]).
siceros], Owen-Smith [1994]).
The vast majority of predators seems to be digestion
limited (see also Weiner 1992). Examples have been Finally, natural predators generally spend a major
part of their time in resting. For example, Amoeba pro-
reported from mollusks (veliger larvae: Crisp et al.
[1985]; common or blue mussel [Mytilus edulis], Bayne teus, Woodruffia metabolica, African Fish Eagles (Hal-
et al. [1989]), crustaceans (Branchipus schaefferi, iaetus vocifer), lions (Panthera leo), and wild dogs
Streptocephalus torvicornis, Dierckens et al. [1997]; (Lyaon pictus) spend only 17% of their time in hunt-
Calliopius laeviusculus, DeBlois and Leggett [1991]; ing and eating (reviewed by Curio 1976). For further
Daphnia spp., Rigler [1961], McMahon and Rigler examples, see Herbers (1981) or Bunnell and Harestad
[1963], Geller [1975]; Calanus pacificus, Frost [1972]; (1990). Since resting may be caused by satiation, this
other copepods: Paffenhofer et al. [1982], Christoffer-
¨ may suggest that such predators are digestion limited.
sen and Jespersen [1986], Head [1986], Jonsson and It is, however, more reliable, to compare predator for-
Tiselius [1990]), insects (Chaoborus spp. larvae, re- aging and nonforaging times with actual measurements
viewed by Jeschke and Tollrian [2000]; the grasshop- of handling and digestion time. This approach reveals
pers Circotettix undulatus, Dissosteira carolina, Me- that the time various herbivores spend for feeding can
lanoplus femur-rubrum, and Melanoplus sanguinipes, usually be predicted solely from their handling and
Belovsky [1986b]; dusty wing larvae [Conwentzia hag- digestion times (J. Jeschke, personal observation). In
eni], green lacewing larvae [Chrysopa californica], red other words, resting often seems to be motivated by
mite destroyer larvae [Stethorus picipes], Fleschner satiation.
[1950]), birds (Woodpigeons [Columba palumbus],
Applications of the SSS equation
Kenward and Sibly [1977]; Oystercatchers [Haema-
topus ostralegus], Kersten and Visser [1996]; hum- The SSS equation was designed as a conceptual mod-
mingbirds [Selasphorus rufus], Hixon et al. [1983], Di- el for developing general and qualitative predictions
amond et al. [1986]), and mammals (moose [Alces al- about functional responses. It can be used to predict
ces], Belovsky [1978]; pronghorn antelopes [Antilo- the effects of changing predator or prey characteristics
capra americana], bison [Bison bison], elk [Cervus by analyzing changes of the corresponding parameters.
elaphus], yellow-bellied marmots [Marmota flaviven- For example, the effects of different kinds of prey de-
tris], mule deer [Odocoileus hemionus], white-tailed fenses can be predicted. A defense that reduces the
deer [Odocoileus virginianus], bighorn sheep [Ovis predator’s success rate (e.g., camouflage) will have its
canadensis], Columbian ground squirrels [Spermophi- largest effects at low prey densities. In contrast, an
lus columbianus], Rocky Mountain cotton tails [Syl- increase in handling time due to a defense (e.g., an
vilagus nuttali], Belovsky [1986b]; cattle [Bos taurus], escape reaction [decreases success rate and increases
Campling et al. [1961]; beavers [Castor canadensis], handling time]) will lower maximum predation rate in
Belovsky [1984b], Doucet and Fryxell [1993], Fryxell handling-limited predators. In digestion-limited pred-
et al. [1994]; Thomson’s gazelles [Gazella thomsoni], ators, either predation rates will decrease or total for-
Wilmshurst et al. [1999]; human beings [Homo sapi- aging time will increase. Finally, an increase in diges-
ens], Belovsky [1987]; snowshoe hares [Lepus amer- tion time (e.g., due to barely digestible substances) will
icanus], Belovsky [1984c]; meadow voles [Microtus lower predation rates at high prey densities in diges-
pennsylvanicus], Belovsky [1984a]; muskoxen [Ovibus tion-limited predators (see also Jeschke and Tollrian
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 107
result of adaptive consumer behavior. Evolutionary Ecol-
2000). The same considerations can be used to for-
ogy 3:95–114.
mulate hypotheses about optimal investment of pred-
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Ecological Monographs
JONATHAN M. JESCHKE ET AL.
112
Vol. 72, No. 1
APPENDIX A: FROM EQ. 12 TO EQ. 13
Solving Eq. 12 for y(x) gives the two solutions y1(x) and y2(x): lim y2 (x) 0, (A1)
x→0
As the limits indicate, only the second solution, y2(x),
c) 2 ]
ax(b c) ax[2(b c) ax(b
1 1
y1 (x) , makes sense biologically. However, y2(x) is not defined for a
2abcx 0, b 0, c 0, or x 0. Eq. 12 helps to find the cor-
lim y1 (x) → , responding equations: For a 0 or x 0, Eq. 12 gives y(x)
x→0
0; for b 0, Eq. 12 gives y(x) acx); for c
(ac)/(1
c) 2 ]
ax(b c) ax[2(b c) ax(b
1 1 0, Eq. 12 gives y(x) abx); and for b c
(ab)/(1 0,
y2 (x) ,
Eq. 12 gives y(x) ax.
2abcx
APPENDIX B
An estimation of Polinices duplicatus (Gastropoda: Naticacea) handling time feeding on Mya arenaria (Bivalvia) in the
year-round experiments of Edwards and Huebner (1977).
Mya lengthi Mya shell Polinices drilling No. preyed
Mya size class i bi[d]¶ ni (
[ 5 mm]† thicknessi [mm]‡ timei [d]§ 385)†
ˆi ˆ
1 15 0.121 0.226 0.95 0.458 73
2 25 0.277 0.517 0.95 1.048 160
3 35 0.433 0.809 0.95 1.639 108
4 45 0.589 1.100 0.95 2.229 26
5 55 0.745 1.392 0.25 4.871 16
6 65 0.901 1.683 0.05 19.355 2
6
1
Resulting mean estimated handling time b bi ni 1.4 d.
ˆ ˆ
385 i 1
† Table 1 in Edwards and Huebner (1977).
‡ Table 3 in Kitchell et al. (1981): M. arenaria shell thicknessi (mm) 0.113 0.0156 lengthi (mm).
§ Kitchell et al. (1981): P. duplicatus drilling time 1.868 d/mm.
According to Kitchell et al. (1981), attack efficiency of P. duplicatus mainly depends on predator and prey size. Given
a certain predator size, is almost unity for prey below a critical size and almost zero for prey beyond that critical size.
The critical size for M. arenaria is given in Fig. 7 in Kitchell et al. (1981). The predator sizes are given in Table 3 in Edwards
and Huebner (1977). Edwards and Huebner used four individual predators with mean sizes in the relevant period (14 June–
29 August, where maximum predation rate ymax 0.63 M. arenaria/d have been reported for P. duplicatus) of 37.9 mm, 41
mm, 42.15 mm, and 50.45 mm, respectively. The corresponding critical M. arenaria lengths are roughly 53 mm for the three
small predator individuals and 60 mm for the largest one. Therefore, M. arenaria of size classes 1, 2, 3, and 4 could be
0.95), M. arenaria of size class 5 could basically only
easily attacked by all four predator individuals (ˆ 1 ˆ 2 ˆ 3 ˆ4
0.25), and M. arenaria of size class 6 could only hardly be attacked by all
be attacked by one of the four predators ( ˆ 5
four predators ( ˆ 6 0.05).
¶ Estimated P. duplicatus handling time (for M. arenaria size class i):
) 1]
bi [d] [1.5 (2 drilling timei . (B1)
ˆ i
Derivation: From Eq. 4, bi [d] (drilling time i / i) eating time i (drilling time i i) drilling time i (Kitchell et al.
ˆ
1981). However, this calculation overestimates handling time, because it is based on the assumption that unsuccessful
drills last as long as successful ones. Assuming that unsuccessful drills last, on average, half the time of successful ones,
leads to Eq. B1.
2002 by the Ecological Society of America
PREDATOR FUNCTIONAL RESPONSES: DISCRIMINATING BETWEEN
HANDLING AND DIGESTING PREY
JONATHAN M. JESCHKE,1,3 MICHAEL KOPP,2 RALPH TOLLRIAN1
AND
1Department of Ecology, Zoological Institute, Ludwig-Maximilians-Universitat Munchen, Karlstrasse 25,
¨ ¨
D-80333 Munchen, Germany
¨
¨
2Max-Planck-Institut fur Limnologie, Abteilung Okophysiologie, Postfach 165, D-24302 Plon, Germany
¨ ¨
Abstract. We present a handy mechanistic functional response model that realistically
incorporates handling (i.e., attacking and eating) and digesting prey. We briefly review
current functional response theory and thereby demonstrate that such a model has been
lacking so far. In our model, we treat digestion as a background process that does not
prevent further foraging activities (i.e., searching and handling). Instead, we let the hunger
level determine the probability that the predator searches for new prey. Additionally, our
model takes into account time wasted through unsuccessful attacks. Since a main assumption
of our model is that the predator’s hunger is in a steady state, we term it the steady-state
satiation (SSS) equation.
The SSS equation yields a new formula for the asymptotic maximum predation rate
(i.e., asymptotic maximum number of prey eaten per unit time, for prey density approaching
infinity). According to this formula, maximum predation rate is determined not by the sum
of the time spent for handling and digesting prey, but solely by the larger of these two
terms. As a consequence, predators can be categorized into two types: handling-limited
predators (where maximum predation rate is limited by handling time) and digestion-limited
predators (where maximum predation rate is limited by digestion time). We give examples
of both predator types. Based on available data, we suggest that most predators are digestion
limited.
The SSS equation is a conceptual mechanistic model. Two possible applications of this
model are that (1) it can be used to calculate the effects of changing predator or prey
characteristics (e.g., defenses) on predation rate and (2) optimal foraging models based on
the SSS equation are testable alternatives to other approaches. This may improve optimal
foraging theory, since one of its major problems has been the lack of alternative models.
Key words: consumer-resource systems; consumption rate; digestion-limited predators; digestion
time; functional response models; handling-limited predators; handling time; hunger level; predation
rate; predator–prey systems; steady-state satiation (SSS) equation.
INTRODUCTION of changing predator or prey characteristics (e.g., de-
fenses) on predation rate.
The relationship between predation rate (i.e., number
of prey eaten per predator per unit time) and prey den- PREVIOUS MODELS: A BRIEF REVIEW
sity is termed the ‘‘functional response’’ (Solomon
Scientists have been modeling functional responses
1949). It is specific for each predator–prey system. The
since the 1920s (reviewed by Holling 1966, Royama
term predator is meant in its broadest sense here, i.e.,
1971), although the term ‘‘functional response’’ was
it includes carnivores, herbivores, parasites, and par-
only introduced in 1949 by Solomon. Since, to our
asitoids. The functional response is an important char-
knowledge, the last review of functional response mod-
acteristic of predator–prey systems and an essential
els dates back to 1971 (Royama), we provide an over-
component of predator–prey models: Multiplying the
view of models published since 1959 together with the
functional response with predator population density
most important factors incorporated in each model (Ta-
and a time factor yields the total number of prey eaten
ble 1). In addition, Fig. 1 shows a ‘‘family tree’’ of
in the period of interest, e.g., one year or one prey
these functional response models. Holling (1959a) has
generation. Given further information, such as actual
categorized functional responses into three main types,
predator density and an energy conversion factor, one
which he called type I, II, and III. Our discussion will
can assess future population densities of both predator
focus on type II functional responses, since these have
and prey. With a mechanistic functional response mod-
been most frequently observed (Hassell et al. 1976,
el, as presented in this study, one can predict the effects
Begon et al. 1996). They are characterized by a hy-
perbolic curve. Starting at low prey densities on the
Manuscript received 8 May 2000; revised 14 December 2000;
abscissa, predation rate first increases almost linearly
accepted 7 February 2001; final version received 12 March 2001.
until it gradually slows down to reach an upper limit.
3 E-mail: jonathan.jeschke@gmx.net
95
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
96
Vol. 72, No. 1
TABLE 1. A selection of functional response models.
Model
A B C D E F G H I J K L
Features (C) (C) (CP) (C) (P) (C) (F) (F) (F) (P) (C) (C)
Components
Success rate1
Probability of attack
Handling time2
Searching and handling overlap-
ping3
Hunger and satiation4
Handling prey digesting prey
Adaptive behavior5
Incomplete consumption6
Nonforaging activities7
Spatial heterogeneity8
Temporal heterogeneity9
Stochasticity10
Environmental conditions11
Predator injury by prey
Inducible defenses12
Dependent on prey density
Prey density
Decreasing prey density13
Alternative prey14
Learning or switching15
Swarming effect16
Dependent on predator density
Predator density
Interference between predators17
Multiple predator effects18
Functional response types
Type I
Type II
Type III
Dome shaped
Other forms19
Notes: Small capital letters in parentheses under models indicate the kind of predator that the model was primarily designed
for: C, carnivores; F, filter feeders; H, herbivores, P, parasites or parasitoids. In the body of the table, ‘‘ ’’ means the model
includes that component, ‘‘ ’’ means the model additionally includes subcomponents, and ‘‘ ’’ means the model does not
include that component. Sources for models are as follows: (A) Gause (1934), Ivlev (1961), Eq. 1; (B) Rashevsky (1959;
no overall model but different equations); (C) Watt (1959); (D) Royama (1971: Eq. 3.12), see also Nakamura (1974: Eq.
15); (E) Royama (1971: Eq. 3.24); (F) Nakamura (1974); (G) Sjoberg (1980); (H) Lam and Frost (1976); (I) Lehman (1976);
¨
(J) Casas et al. (1993); (K) Disc equation (Holling 1959b), Eq. 2; (L) Invertebrate model (Holling 1966; see also Metz and
van Batenburg 1985a,b); (M) Vertebrate model (Holling 1965); (N) Holling and Buckingham (1976); (O) Rao and Kshiragar
(1978); (P) Metz et al. (1988; see also Metz and van Batenburg 1985a, b); (Q) Cushing (1968); (R) Tostowaryk (1972); (S)
Random predator equation (Royama 1971, Rogers 1972); (T) Random parasite equation (Royama 1971, Rogers 1972); (U)
Beddington (1975); (V) Hassell et al. (1977); (W) Longstaff (1980); (X) Mills (1982); (Y) Crowley (1973); (Z) Oaten and
Murdoch (1975); (AA) Real (1977); (BB) McNair (1980); (CC) Abrams (1982); (DD) Dunbrack and Giguere (1987); (EE)
Abrams (1990a); (FF) Descriptive equation (Fujii et al. 1986); (GG) Ungar and Noy-Meir (1988); (HH) Random patch model
˚¨
(Lundberg and Astrom 1990; see also Lundberg and Danell 1990); (II) Juliano (1989); (JJ) Fryxell (1991; see also Wilmshurst
et al. 1995, 1999, 2000); (KK) Spalinger and Hobbs (1992; see also Laca et al. 1994, Shipley et al. 1994); (LL) Farnsworth
and Illius (1996; see also Laca et al. 1994, Shipley et al. 1994); (MM) Hirakawa (1997b; see also Laca et al. 1994, Shipley
et al. 1994); (NN) Farnsworth and Illius (1998; see also Laca et al. 1994, Shipley et al. 1994); (OO) Ruxton and Gurney
(1994); (PP) Cosner et al. (1999) [This model closes a gap between density dependent and ratio dependent functional response
models. Purely ratio dependent models are not included in Table 1, but see Arditi and Ginzburg (1989). However, as Berryman
et al. (1995) have written: ‘‘Note that prey-dependent functional responses can be transformed into ratio-dependent functional
responses by substituting the prey/predator ratio for prey density in the equation.’’]; (QQ) Streams (1994); (RR) Schmitz
(1995; see also Abrams [1990c] and review by Schmitz et al. [1997]); (SS) Abrams and Schmitz (1999); (TT) Berec (2000;
see also Engen and Stenseth 1984); (UU) SSS equation (Eq. 13).
1 Success rate consists of four subcomponents: (1) encounter rate, (2) probability of detection, (3) hunger-independent
probability of attack, and (4) efficiency of attack; empirical values for the attack efficiencies of predators have been reviewed
by Curio (1976), Vermeij (1982), and Packer and Ruttan (1988).
2 Handling time (per prey item) includes attacking time (including evaluating, pursuing, and catching time) and eating
time. See also Anholt et al. (1987), Demment and Greenwood (1988), Laca et al. (1994), Parsons et al. (1994), and Shipley
et al. (1994).
3 Important for queueing predators (Juliano 1989; see also Visser and Reinders 1981, Lucas 1985, Lucas and Grafen 1985)
and vertebrate herbivores (Spalinger and Hobbs 1992, Parsons et al. 1994, Farnsworth and Illius 1996, 1998, Hirakawa
1997b; see also Laca et al. 1994, Shipley et al. 1994).
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 97
TABLE 1. Extended.
Model
M N O P Q R S T U V W X Y Z AA BB CC DD EE
(C) (C) (C) (C) (F) (C) (C) (P) (CP) (CP) (C) (C) (F) (C) (C) (C) (C) (CF) (C)
4 Some models include predator satiation via a maximum predation rate determined by the characteristics of the digestive
system (‘‘ ’’). Other models include the fact that the predator’s gut content is increased by ingestion and decreased by
digestion (‘‘ ’’). See also Campling et al. (1961), Curio (1976), Belovsky (1978, 1984a, b, c, 1986a, b, 1987), Mayzaud and
Poulet (1978), Bernays and Simpson (1982), Murtaugh (1984), Crisp et al. (1985), Demment and Greenwood (1988), Verlinden
and Wiley (1989), Illius and Gordon (1991), Doucet and Fryxell (1993), Forchhammer and Boomsma (1995), Henson and
Hallam (1995), Hirakawa (1997a), and Wilmshurst et al. (2000).
5 See also Belovsky (1978, 1984a, b, c, 1986a, b, 1987), Cook and Cockrell (1978), Sih (1980, 1984), Owen-Smith and
Novellie (1982), McNair (1983), Abrams (1984, 1987, 1989, 1990b, c, 1991, 1992, 1993), Engen and Stenseth (1984),
Formanowicz (1984), Lucas (1985), Wanink and Zwarts (1985), Stephens and Krebs (1986), Anholt et al. (1987), Demment
˚¨
and Greenwood (1988), Belovsky et al. (1989), Verlinden and Wiley (1989), Astrom et al. (1990), Lundberg and Danell
(1990), Mitchell and Brown (1990), Abrams and Matsuda (1993), Doucet and Fryxell (1993), Werner and Anholt (1993),
McNamara and Houston (1994), Forchhammer and Boomsma (1995), Hirakawa (1995, 1997a), Fryxell and Lundberg (1997),
Leonardsson and Johansson (1997), Rothley et al. (1997), Schmitz et al. (1997), and Wilmshurst et al. (2000).
6 See also Buckner (1964), Johnson et al. (1975), Curio (1976), Cook and Cockrell (1978), Sih (1980), Owen-Smith and
Novellie (1982), McNair (1983), Formanowicz (1984), Lucas (1985), Lucas and Grafen (1985), Metz and van Batenburg
˚¨
(1985a, b), Astrom et al. (1990), Lundberg and Danell (1990), and Fryxell and Lundberg (1997).
7 For example, avoidance of top predators, migration, molting, reproductive activities, resting, sleeping, territorial behavior,
thermoregulation, and times of slow rates of metabolism like winter dormancy; see also Belovsky (1978, 1984a, b, c, 1986a,
b, 1987), Caraco (1979), Herbers (1981), Bernays and Simpson (1982), Owen-Smith and Novellie (1982), Abrams (1984,
1991, 1993), Stephens and Krebs (1986), Belovsky et al. (1989), Verlinden and Wiley (1989), Bunnell and Harestad (1990),
Mitchell and Brown (1990), McNamara and Houston (1994), Forchhammer and Boomsma (1995), Hirakawa (1997a), Leon-
ardsson and Johansson (1997), and Rothley et al. (1997).
8 See also Griffiths and Holling (1969), Paloheimo (1971a, b), Oaten (1977), May (1978), Real (1979), McNair (1983),
Belovsky et al. (1989), Blaine and DeAngelis (1997), Fryxell and Lundberg (1997), and Wilmshurst et al. (2000), among others.
9 For example, diel or annual periodicity (Curio 1976, Bernays and Simpson 1982, Belovsky et al. 1989, Forchhammer
and Boomsma 1995).
10 See also Paloheimo (1971a, b), Curry and DeMichele (1977), Curry and Feldman (1979), McNair (1983), Lucas (1985),
and Metz and van Batenburg (1985a, b).
11 For example, precipitation, temperature (Fedorenko 1975, Thompson 1978, Bernays and Simpson 1982), and wind.
12 Behavioral and morphological defenses, that are not permanently present but are induced by the predator (e.g., Fryxell
and Lundberg 1997, Karban and Baldwin 1997, Tollrian and Harvell 1999, Jeschke and Tollrian (2000).
13 See also Curry and DeMichele (1977).
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
98
Vol. 72, No. 1
TABLE 1. Extended.
Model
FF GG HH II JJ KK LL MM NN OO PP QQ RR SS TT UU
Features (CP) (H) (H) (C) (H) (H) (H) (H) (H) (C) (C) (C) (CH) (H) (C) (C)
Components
Success rate1
Probability of attack
Handling time2
Searching and handling over-
lapping3
Hunger and satiation4
Handling prey digesting prey
Adaptive behavior5
Incomplete consumption6
Nonforaging activities7
Spatial heterogeneity8
Temporal heterogeneity9
Stochasticity10
Environmental conditions11
Predator injury by prey
Inducible defenses12
Dependent on prey density
Prey density
Decreasing prey density13
Alternative prey14
Learning or switching15
Swarming effect16
Dependent on predator density
Predator density
Interference between predators17
Multiple predator effects18
Functional response types
Type I
Type II
Type III
Dome shaped
Other forms19
14 See also Belovsky (1978, 1984a, b, c, 1986a, b, 1987), Owen-Smith and Novellie (1982), Engen and Stenseth (1984),
Metz and van Batenburg (1985a), Wanink and Zwarts (1985), Abrams (1987, 1989, 1990b, c), Belovsky et al. (1989), Abrams
and Matsuda (1993), Doucet and Fryxell (1993), Parsons et al. (1994), Forchhammer and Boomsma (1995), Fryxell and
Lundberg (1997), Rothley et al. (1997), Schmitz et al. (1997), and Wilmshurst et al. (2000).
15 Learning includes training effects; switching means either switching between prey types (in this case, there is a ‘‘ ’’
at ‘‘alternative prey’’) or behavioral switching, e.g., from sitting and waiting to cruising. Only those models that explicitly
consider learning or switching have ‘‘ ’’ here. Optimal foraging models where switching is a simulation result have ‘‘ ’’
here. For an experimental example of the interaction between learning and spatial distribution see Real (1979); see also
Fryxell and Lundberg (1997) and Kaiser (1998).
16 A swarming effect decreases predation rate with increasing prey density. It can be the result of (1) a better or earlier
detection of the predator by prey, (2) a worse detection of prey by the predator, (3) a better active prey defense, (4) predator
confusion which usually decreases probability or efficiency of attack, (5) clogging of filters (in case of filter feeders), or (6)
accumulation of toxic prey substances. The form of the functional response can be dome shaped in this case. See Miller
(1922), Brock and Riffenburgh (1960), Mori and Chant (1966), Tostowaryk (1972), Halbach and Halbach-Keup (1974), Neill
and Cullen (1974), Nelmes (1974), Milinski and Curio (1975), Bertram (1978), Lazarus (1979), Williamson (1984), Morgan
and Godin (1985), Landeau and Terborgh (1986), and Inman and Krebs (1987).
17 Interference also includes prey exploitation by other predators. Only those models that consider interference inclusively
and prey exploitation explicitly have ‘‘ ’’ here. Models that account for a decreasing prey density through predation and the
number of predators present and include prey exploitation in an implicit way have ‘‘ ’’ here. See also models by Griffiths and
Holling (1969), Hassell and Varley (1969), Royama (1971, model in §4i), DeAngelis et al. (1975), Curry and DeMichele (1977),
Parker and Sutherland (1986), Korona (1989), Ruxton et al. (1992), Holmgren (1995), Fryxell and Lundberg (1997), and
Doncaster (1999); for empirical studies, see Norris and Johnstone (1998), Triplet et al. (1999), or references in Holmgren (1995).
18 Soluk (1993), Sih et al. (1998).
19 Crowley (1973) and Farnsworth and Illius (1996), intermediate type I/II; Nakamura (1974), type II similar; Lam and
Frost (1976), Fujii et al. (1986), type I similar; Lehman (1976), partly type I similar; Metz et al. (1988) and random patch
˚¨
model (Lundberg and Astrom 1990), hyperbolic (type II similar) functional response without an asymptote; Abrams (1982),
Juliano (1989), Fryxell (1991), Schmitz (1995), Hirakawa (1997b), Farnsworth and Illius (1998), Abrams and Schmitz (1999),
and Berec (2000), various forms; see also Parsons et al. (1967, type II with a threshold prey density, corresponding empirical
curves in the same study and in Parsons et al. (1969), Cook and Cockrell (1978; double plateau functional responses), Abrams
(1987, 1989; decreasing functional responses), and Fryxell and Lundberg (1997; various forms).
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 99
FIG. 1. A ‘‘family tree’’ of functional response models.
All functional response models include a factor that Although purely phenomenological, the Gause-Ivlev
determines the curve’s gradient at the origin (‘‘success equation (Gause 1934, Ivlev 1961) has usually been
rate’’ which is a measure of the predator’s hunting ef- viewed as the classical satiation model
ficiency; it has been termed the ‘‘rate of successful
y(x) ymax (1 exp [ a x]) (1)
search’’ by Holling [1959a, b, 1965, 1966]).
where a is hunting success (dimension in SI units: m2
Handling and digestion for a two-dimensional system, e.g., a terrestrial system,
and m3 for a three-dimensional system, e.g., an aquatic
Limitation of predation rate at high prey densities
system), x is prey density (individuals/m2 or individ-
has usually been attributed to either handling time or
uals/m3, respectively), y is predation rate (s 1), and ymax
satiation. However, the exact nature of these two factors
is asymptotic maximum predation rate as x approaches
and their relationship has been modeled in a variety of
infinity (s 1). In the common interpretation, the diges-
different ways and this has led to considerable con-
tive system determines ymax, and the functional response
fusion. The point is that handling prey is an active
curve gradually rises to this value. Rashevsky (1959)
process whereas digestion is a background process. As
has extended the Gause-Ivlev equation by modeling
a consequence, in contrast to handling prey, digestion
satiation more mechanistically: the predator’s gut con-
does not directly prevent the predator from further
tent is increased by ingestion and decreased by diges-
searching or handling. Rather, digestion influences the
tion. Other models including satiation but not handling
predator’s hunger level, which in turn influences the
time have been developed by Watt (1959), Parsons et
probability that the predator searches for new prey. It
al. (1967), Royama (1971), Nakamura (1974), Lam and
is thus necessary to discriminate digestion from han-
Frost (1976), Lehman (1976), Sjoberg (1980), Crisp et
¨
dling in a functional response model. In the following,
al. (1985), Metz and van Batenburg (1985b), Metz et
we briefly review existing models with respect to their
al. (1988), Abrams (1990c), Casas et al. (1993; para-
treatment of these two factors. In our opinion, no com-
pletely satisfying solution to the problem exists to date. sitoid egg load as analogous to hunger level), Henson
Models including satiation but not handling time.— and Hallam (1995), and Abrams and Schmitz (1999).
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
100
Vol. 72, No. 1
Models including handling time but not satiation.— b tatt / teat
In contrast, there are a number of models that include
⇔ asymptotic maximum predation rate
handling time but no predator satiation effects (Holling
1959b, Cushing 1968, Royama 1971, Rogers 1972, teat ) 1.
(tatt / (4)
Tostowaryk 1972, Beddington 1975, Hassell et al.
In using this definition, handling time includes time
1977, Real 1977, Cook and Cockrell 1978, Curry and
wasted through unsuccessful attacks (see also Mills
Feldman 1979, Longstaff 1980, McNair 1980, Visser
1982, Abrams 1990a, Streams 1994).
and Reinders 1981, Abrams 1982, 1987, 1990a, Fujii
Models including both handling time and satia-
et al. 1986, Dunbrack and Giguere 1987, Ungar and
`
tion.—One approach to include both handling time and
˚¨
Noy-Meir 1988, Juliano 1989, Lundberg and A strom
digestion time is to sum them up or to increase handling
1990, Spalinger and Hobbs 1992, Parsons et al. 1994,
time by a ‘‘digestive pause’’ (Crowley 1973, Rao and
Ruxton and Gurney 1994, Streams 1994, Farnsworth
Kshirsagar 1978, Mills 1982, Henson and Hallam
and Illius 1996, Fryxell and Lundberg 1997, Cosner et
1995), i.e., an inactive time period related to digestion
al. 1999, Berec 2000). The most popular functional
(Holling 1965, 1966). When modeled this way, diges-
response model today, Holling’s (1959b) disc equation,
tion is not distinguished from handling. Mills (1982)
belongs to this class:
used this concept to extend the disc equation by inter-
ax preting its parameter b as
y(x) (2)
abx
1
b tatt teat stdig
where a is success rate (dimension in SI units: m2/s or
⇔ asymptotic maximum predation rate
m3/s, respectively; note that the dimensions of a and
a [Gause-Ivlev equation] differ), b is predator han- 1
teat stdig )
(tatt (5)
dling time per prey item (s), x is prey density (indi-
where s is satiation per prey item (dimensionless) and
viduals/m2 or individuals/m3, respectively), and y is
tdig is digestion time per prey item (s; see Table 2).
predation rate (s 1). The curve’s gradient at the origin
A second way to consider both handling and diges-
is equal to a, and the asymptotic maximum for x as x
tion time is to combine the disc equation (which already
approaches infinity is 1/b. The disc equation is math-
includes handling time) with a digestive capacity con-
ematically equivalent to the Michaelis-Menten model
straint (Fryxell 1991, Schmitz 1995, Hirakawa 1997b,
of enzyme kinetics and the Monod formula for bacterial
Farnsworth and Illius 1998). This constraint limits
growth. The Royama-Rogers random predator equation
maximum predation rate but does not otherwise affect
(Royama 1971, Rogers 1972) is a modification of the
the functional response. These models therefore dis-
disc equation that accounts for a decreasing prey den-
criminate between handling and digesting prey. How-
sity in the course of an experiment or between discrete
ever, neither the process of digestion, nor the predator
prey generations. In the original paper (Holling 1959b),
satiation level are considered. The approach to combine
the parameter b of the disc equation denoted the general
handling time with a digestive capacity constraint has
meaning of ‘‘handling time’’ at that time, i.e., the sum
its origins in linear programming models (e.g., Belov-
of attacking time tatt (per prey item; including evalu-
sky 1978, 1984a, b, c, 1986a, b, 1987, Doucet and
ating, pursuing, and catching time) and eating time teat
Fryxell 1993, Forchhammer and Boomsma 1995).
(per prey item):
The only family of models that treats digestion as a
b tatt teat background process, which influences foraging activ-
⇔ asymptotic maximum predation rate ities but does not prevent them, is Holling’s (1966)
invertebrate model and its extensions (Holling 1965,
1
teat )
(tatt (3) Holling and Buckingham 1976, Curry and DeMichele
1977, Metz and van Batenburg 1985a). In the inver-
with handling time b as it was originally defined by
tebrate model, the predation cycle is subdivided into
Holling (1959b).
several stages, and each stage depends on predator hun-
Holling originally developed the disc equation as a
ger level. After a meal, the predator is assumed to
mechanistic model for an artificial predator–prey sys-
undergo a digestive pause and then continues searching
tem: humans ‘‘preying’’ on paper discs (Holling
when it is hungry again. While searching, the predator
1959b). Compared to natural predator–prey systems,
simultaneously continues digestion of its last meal. The
however, it is now clear that the underlying assump-
invertebrate model therefore discriminates between
tions are unrealistic (Hassell et al. 1976). Two points
handling and digesting prey. Here, the length of the
have met the most severe criticism: First, the predator
digestive pause depends on hunger level. Since hunger
does not become satiated, and second, the disc equation
level in turn depends on prey density, the length of the
assumes that every attack by the predator is successful,
digestive pause depends on prey density. This is in
i.e., attack efficiency 100%. Attack efficiencies
100% can be incorporated into the disc equation by contrast to the models mentioned above (Eq. 5), where
defining b as the length of the digestive pause is unrealistically as-
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 101
TABLE 2. The SSS equation parameters.
Parameter Description Dimension† Defined for
m3/s
Encounter rate number of encounters between a searching predator and [0; ]
a single prey item; an encounter is defined as an arrival of a prey item
in the predator’s encounter volume
Probability that the predator detects encountered prey [0; 1]
Efficiency of attack proportion of successful attacks [0; 1]
s Satiation per prey item reciprocal capacity of the hunger-determining [0; ]
part of the gut (mostly stomach or crop); example: if the stomach ca-
pacity of a human is equal to 10 potatoes, then s 0.1
tatt Attacking time per prey item time between prey detection and end of s [0; ]
attack
tdig Digestion time per prey item food transit time ( 50% emptying time) s [0; ]
for the hunger-determining part of the gut, e.g., stomach transit time
for humans
teat Eating time per prey item time between capture and finished ingestion s [0; ]
Note: The parameters can be summarized by a (success rate [m3/s]), b (corrected handling time [s]), and c (corrected
digestion time [s]), see Eq. 13.
† In SI units and given for a three-dimensional system, e.g., an aquatic system; in the case of a two-dimensional system,
e.g., a terrestrial system, m3 must be replaced by m2.
sumed to be constant. The term ‘‘digestive pause’’ re- is no mechanistic linkage to the processes of ingestion
lates to foraging activities only: predators may well use and digestion.
the digestive pause for nonforaging activities, for ex- Their mathematical simplicity renders the disc, the
ample, for looking out for top predators or for sleeping. random predator, and the Gause-Ivlev equation as func-
However, because of its 22 parameters, the invertebrate tional response submodels in predator–prey population
model is extremely unwieldy, and its extensions are models. However, for a deeper understanding of the
even more elaborate. functional response, mechanistic models are necessary.
The parameters of mechanistic models can all be mech-
Phenomenological vs. mechanistic models anistically explained. These models can thus, for ex-
ample, be used to calculate the effects of changing
Probably because of their mathematical simplicity,
predator or prey characteristics (e.g., defenses) on pre-
the Holling (1959b) disc equation (Eq. 2), the Royama-
dation rate.
Rogers random predator equation (Royama 1971, Rog-
ers 1972), and the Gause-Ivlev equation (Gause 1934,
THE STEADY-STATE SATIATION (SSS) EQUATION
Ivlev 1961; Eq. 1) have been the most popular func-
We have shown that a handy mechanistic functional
tional response models. However, they must be con-
response model that realistically incorporates handling
sidered phenomenological. That is, although they cor-
and digesting prey has been lacking so far. In this sec-
rectly reproduce the shape of natural (type II) func-
tion, we therefore develop such a model: the steady-
tional responses, they are not able to explain the un-
state satiation (SSS) equation. It is based on the disc
derlying mechanism; or, in other words, its parameters
equation and divides the predation cycle into five stag-
cannot all be mechanistically explained. In the cases
es: search, encounter, detection, attack, and eating (Fig.
of the disc equation and the random predator equation,
2). We assume that these stages are mutually exclusive.
the parameter a (success rate) can be mechanistically
Each stage is characterized by two components: The
explained (Holling 1966, Ungar and Noy-Meir 1988,
amount of time needed for its completion and the con-
Streams 1994, Hirakawa 1997b; see also Eq. 7 below),
ditional probability that the predator reaches this stage
but not the parameter b (handling time). When fitting
given that it has reached the previous one (exception:
the disc equation or the random predator equation to
encounter; here, it is not a probability but a rate; note
an empirical curve, the resulting value for b is a mixture
that the encounter rate can have a value larger than
of different biological processes (Table 1) including
unity). Digestion is modeled as a background process
handling (attacking and eating) and digestion. As we
influencing the predator’s hunger level, which in turn
have pointed out above, handling is an active process,
determines the probability that the predator searches
whereas digestion is a background process. They can-
for prey.
not be adequately condensed into only one parameter.
In the case of the Gause-Ivlev equation, neither param-
The SSS equation components, parameters,
eter can be mechanistically explained. Its parameter a
and assumptions
(hunting success) differs in its dimension from the pa-
rameter a of the disc equation and the random predator The SSS equation components are given in Table 1
equation; a lacks a mechanistic explanation. The other and its parameters in Table 2. Like nearly every model,
parameter, ymax, is just the asymptote of the curve; there the SSS equation is a compromise between realism and
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
102
Vol. 72, No. 1
1) There is only a single predator and a single type
of prey.
2) The prey density is constant.
3) Prey are independently and randomly distributed.
4) Stages of the predation cycle exclude each other
(Fig. 2).
5) The probability that the predator searches (under
the condition that it is not handling prey), (x), is di-
rectly proportional to the predator’s hunger level h(x).
6) The hunger level h(x) depends on the fullness of
a certain part of the gut (e.g., stomach, crop).
7) The hunger level h(x) at a given prey density x is
in a steady state, which is determined by an equilibrium
of ingestion and digestion.
8) The probability of attack, , is unity, i.e., when-
ever a searching predator encounters and detects a prey,
it will attack.
9) The probability that the predator detects encoun-
tered prey, , the efficiency of attack (i.e., the pro-
portion of successful attacks), the attacking time tatt (per
prey item), the eating time teat (per prey item), and the
digestion time tdig (per prey item) are constant.
The SSS equation
To develop the SSS equation, we start with the disc
equation and modify it sequentially. In step 1, each
stage of the predation cycle is included explicitly; in
step 2, predator satiation is included by influencing the
FIG. 2. The predation cycle. We divide the predation cycle
into five stages: search, encounter, detection, attack, and eat- probability of searching.
ing. A predator enters a predation cycle under the probability
The stages of the predation cycle are (1) search, (2)
to search, (x); this is determined by the predator’s hunger
encounter, (3) detection, (4) attack, and (5) eating (Fig.
level, which in turn is influenced by digestion time. Then the
2). The probabilities that a predator reaches these stages
predator successively reaches the following stages. The prob-
ability that the predator reaches a stage under the condition are (1) the probability that a predator not occupied
that it has reached the previous stage is given in the corre- with handling searches for prey, (2) the encounter rate
sponding arrow, e.g., the probability that the predator detects
between a searching predator and an individual prey,
a prey under the condition that it has encountered that prey
(3) the probability that the predator detects an en-
is (exception: is not a probability but a rate [encounter
countered prey individual, (4) the probability that the
rate]; note that it can be larger than unity). Since is set as
unity in the SSS equation (assumption 8), it is given in pa- predator attacks a detected prey individual, and (5) the
rentheses here. Terms in circles indicate time demands of
probability that an attack is successful, i.e., the ef-
corresponding stages per prey item. We assume that the stages
ficiency of attack. We now incorporate these probabil-
are mutually exclusive (assumption 4). Terms with a super-
ities into the disc equation.
script ‘‘a’’ determine predator success rate (a).
The searching probability (x).—In the disc equa-
tion, the predator shows only two kinds of behavior:
searching for and handling prey. Therefore, the prob-
applicability. It is more realistic than the disc equation,
ability that the predator searches for prey under the
but it is reductive compared to nature. The SSS equa-
condition that it is not handling prey, (x), is unity. To
tion is a conceptual model that can, for example, be
allow values below unity, (x) has to be incorporated
used to assess how changing predator or prey charac-
explicitly into the disc equation:
teristics (e.g., defenses) qualitatively affect the func-
tional response. The point of the SSS equation is not (x)ax
y(x) . (6)
to quantitatively predict real functional response
1 (x)abx
curves. It is therefore not necessary to incorporate too
depends on prey density x because it is
many features into the model, which would render it Note that
unwieldy. However, extensions for specific predator– affected by the predator’s hunger level (see the next
prey systems are possible; these will allow us to make paragraph and assumption 5), which in turn depends
quantitative predictions with the model as well. For on prey density (see the next paragraph and assumption
this purpose, references given in Table 1 may be help- 7): (h) (h(x)) (x).
The encounter rate , the probability of detection ,
ful.
the probability of attack , and the efficiency of attack
The assumptions of the model are as follows:
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 103
c · y(x))ax
.—The product of all these terms is predator success (1
y(x) . (12)
rate a. However, for simplicity, we set 1 (as- c · y(x))abx
1 (1
sumption 8). Thus,
Solving for y(x) finally gives the following SSS equation:
a . (7)
1
The encounter rate can be calculated by various
c) 2 )
ax(b c) ax(2(b c) ax(b
1
formulae from different authors. For a three-dimen-
2abcx
sional model, e.g., in aquatic systems, one may use
the equation given by Gerritsen and Strickler (1977). a, b, c, x 0
For an analogous two-dimensional model, e.g., in ter-
ax
restrial systems, see Koopman (1956), and for a three- b c
0 0
1
y(x)
abx
dimensional model with a cylindrical instead of a
ax
spherical encounter volume, see Gigue re et al. (1982).
`
b c
0 0
For further models, see Royama (1971: Eq. 4e.6), Get- acx
1
ty and Pulliam (1991), Parsons et al. (1994), Hirakawa
ax b c 0
(1997b), and reviews from Schoener (1971) and Curio
0 a x
0 or 0
(1976). Here, for simplicity, is not calculated by
one of these formulae but is a model input; probability
(13)
of detection and efficiency of attack are also model
with success rate a , corrected handling time
inputs.
b t att / t eat , and corrected digestion time c st dig .
Explicitly incorporating efficiency of attack allows
For details on deriving Eq. 13 from Eq. 12, see Ap-
us to account for time wasted through unsuccessful
pendix A. For c 0 (i.e., no satiation), the SSS equa-
attacks. Thus, handling time b can be calculated ac-
tion simplifies to the disc equation but with the defi-
cording to Eq. 4.
nitions of Eq. 13 for a and b. For b 0 (i.e., zero
The second and final step in deriving the SSS equa-
handling time), the SSS equation simplifies to the disc
tion is to incorporate digestion. We do this by assuming
equation but with c instead of b, i.e., digestion time
that
replaces handling time in this case. Finally, without
h(x).
(x) (8)
c
any handling time or satiation (b 0), there are
no density dependent effects and so, predation rate is
This is assumption 5 and is also assumed by Rash-
directly proportional to prey density.
evsky (1959). The hunger level h(x) is the proportion
of empty volume of that part of the gut that is re-
Properties of the SSS equation
sponsible for feelings of hunger and satiation in the
The SSS equation produces type II functional re-
predator under consideration (mostly stomach or
sponses (Fig. 3). As in the disc equation, the gradient
crop); h(x) is defined for [0; 1], where h 0 means
at the origin is equal to the predator’s success rate a:
no hunger, i.e., full gut, and h 1 means 100% hunger,
i.e., empty gut. Empirical studies usually find a hy-
dy(x)
perbolic relationship between starvation time and a.
lim (14)
dx
x→0
hunger level, e.g., Holling (1966) for mantids (Hier-
odula crassa and Mantis religiosa), Antezana et al. The asymptotic maximum predation rate for prey den-
(1982) for krill (Euphausia superba), Hansen et al. sity as x approaches infinity is
(1990) for copepods (Calanus finmarchicus), and sev-
c) 2
b c (b 1
eral works on fish (reviewed by Elliott and Persson lim y(x) .
max(b; c)
2bc
1978). This hyperbolic relationship can be described x→
by the following differential equation:
where, for handling-limited predators,
dh(x) h(x)
1
1
sy(x). (9)
c ⇔ lim y(x)
b
dt tdig
b
x→
Since we assume a constant prey density (assumption
and, for digestion-limited predators,
2), the equilibrium hunger level can be obtained by
setting dh(x)/dt 0, giving 1
b ⇔ lim y(x)
c . (15)
c
(10)
h(x) s·tdig·y(x).
1 x→
Thus, the larger one of the two terms b and c determines
We define c s·tdig as ‘‘corrected digestion time’’, i.e.,
the asymptotic maximum predation rate. This is, be-
digestion time corrected for gut capacity. Therefore,
cause digestion is a ‘‘background process’’, i.e., han-
h(x) c·y(x).
1 (11) dling and digestion can be carried out simultaneously.
The slower one of these two processes is then limiting.
Inserting Eq. 11 into Eqs. 8 and 6 yields
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
104
Vol. 72, No. 1
When corrected handling time exceeds corrected di-
gestion time (b c, condition 1), the asymptotic max-
imum predation rate is 1/b. This is the same situation
as in a disc equation when attack efficiencies 100%
are considered (see Eq. 4). We call predators under this
condition ‘‘handling-limited predators.’’ Fig. 3a shows
graphs of the SSS equation for handling-limited pred-
ators. Although the asymptote is independent of c, it
is approached more slowly as digestion time becomes
more important, i.e., large digestion times result in a
slower rise of the curve. As c approaches 0, the SSS
curve approaches a disc equation curve (with a cor-
rection for attack efficiencies 100%).
When corrected digestion time exceeds corrected
handling time (c b, condition 2), the asymptotic max-
imum predation rate equals 1/c. We call predators under
this condition ‘‘digestion-limited predators.’’ Fig. 3b
shows graphs of the SSS equation for digestion-limited
predators. With larger handling times, the asymptote is
approached more slowly, yet the asymptote itself is
independent of b. As b approaches 0, the SSS curve
approaches a disc equation curve with digestion in
place of handling (c instead of b) as the limiting factor.
SSS equation curves are more flexible than disc
equation curves. Thus, it is impossible to satisfyingly
fit the disc equation to a SSS equation curve (with the
exceptions b 0 or c 0). This is, because, in the
disc equation, one parameter (b) determines the curve’s
asymptote, and two parameters (a and b) determine
how the curve reaches this asymptote, i.e., the curve’s
slope. In contrast, in the SSS equation, one parameter
(the larger one of the parameters b and c) determines
the curve’s asymptote, and three parameters (a, b, and
c) determine how the curve reaches this asymptote.
Fig. 3c illustrates how time wasted through unsuc-
cessful attacks (attack efficiency 100%) reduces
the slope of the functional response curve (and, in case
of handling-limited predators, the asymptotic maxi-
mum predation rate).
DISCUSSION
We have developed a handy mechanistic functional
response model (the SSS equation) that realistically
FIG. 3. Graphs of the SSS equation (Eq. 13). (A) Han-
dling-limited predators. Model inputs were success rate a ←
2, corrected handling time b 0.02, and corrected digestion
time c 0, 0.01, or 0.02, respectively; thus, b c. All curves (C) Effect of attack efficiency . Model inputs were a 2
0.25), b 0.01 (
are type II functional responses, and for all curves, asymptotic (corresponding 0.5) or 1 ( 0.5,
tatt 0.0025, teat 0.005; b [Eq. 13] 0.0025 / 0.5 0.005
maximum predation rate 1/b 50 (Eq. 15). However, this
0.25, tatt 0.0025, teat 0.005; b [Eq.
asymptotic maximum is approached more slowly as digestion 0.01) or 0.015 (
time becomes more important. For c 0.015), and c
0, the SSS equation 13] 0.0025 / 0.25 0.005 0.02. When
is equal to the disc equation (Eq. 2). (B) Digestion-limited attack efficiency is halved (from 0.5 to 0.25), the gradient at
predators. Model inputs were a 2, b 0, 0.01, or 0.02, the origin is halved (a 2 or 1, Eq. 14) and the predation
respectively, and c 0.02; thus, c b. All curves are type rate is decreased at almost all prey densities. However, in
II functional responses, and for all curves, asymptotic max- case of a digestion-limited predator (as in our example), as-
imum predation rate 1/c 50 (Eq. 15). However, this ymptotic maximum predation rate remains constant (1/c
asymptotic maximum is approached more slowly as handling 50, Eq. 15). In the case of a handling-limited predator (graph
time becomes more important. For b not shown), b is increased, and thus asymptotic maximum
0, the SSS equation
is equal to the disc equation, when b is replaced by c there. predation rate is decreased.
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 105
incorporates success rate, handling time, and satiation. of top predators, migration, molting, reproductive ac-
The satiation level is assumed to linearly decrease hunt- tivities, resting, sleeping, territorial behavior, thermo-
ing activities. The SSS equation thereby fills a gap in regulation, or times of slow rates of metabolism like
functional response theory, because previous models winter dormancy).
either do not treat satiation in a realistic way (since The easiest method to detect a handling-limited pred-
they do not discriminate between handling and di- ator is to directly measure corrected handling time (ac-
gesting prey or simply include satiation by a maximum cording to Eq. 4) as well as corrected digestion time
predation rate, i.e., a digestive capacity constraint) or (according to Eq. 13) and to compare them. However,
are extremely unwieldy. all predators, from whom both measurements are avail-
Like the widely used disc equation, the SSS equation able in the literature, are digestion-limited (see next
produces type II functional response curves. However, section).
there are several differences. First, because of its third Another method to detect a handling-limited pred-
parameter, the SSS equation is more flexible than the ator is:
disc equation. The differences are largest when han- 1A) Through observation, directly measure predator
handling time b according to Eq. 4.
dling time and digestion time are of the same order of
magnitude (Fig. 3). On the contrary, if one of these 1B) (Alternative to 1A) Perform short-term feeding
two factors is negligibly small, the curve becomes vir- experiments to get a short-term functional response
tually identical to that of the disc equation. Second, the without satiation effects. Fit the disc equation (if eaten
disc equation assumes an attack efficiency equal to prey was replaced) or the random predator equation (if
eaten prey was not replaced) to the data to get b (han-
100%. When this is not the case, the maximum pre-
dation rate is decreased because of time spent for un- dling time according to Eq. 4).
2A) Measure long-term maximum feeding rate ymax
successful attacks. Although mentioned by Mills
(1982), Abrams (1990a), and Streams (1994), this ef- (with satiation) at an extremely high prey density.
fect has not been incorporated into most models. It is 2B) (Alternatively to 2A) Perform long-term feeding
contrary to the basic idea of the disc equation that the experiments, ideally starting with predators in a steady
parameters a and b are independent (Holling 1965, hunger state, or do a field study. Fit the disc equation
or the random predator equation to the data to get ymax.
1966). In nature, predator attack efficiencies seldom
3) If b
reach 100% (see Curio 1976, Vermeij 1982, and Packer 1/ymax, it is likely that the predator is han-
and Ruttan 1988). Taking unsuccessful attacks into ac- dling limited.
count is especially important for predators with non- We have applied this method to available literature
negligible attacking times. Third and most important, data and have found three candidates for handling-lim-
the disc equation (with b interpreted as in Eq. 5) does ited predators. First, in the host–parasitoid system Silo
pallipes (Trichoptera: Goeridae)–Agriotypus armatus
not discriminate between handling and digesting prey.
The SSS equation, on the other hand, takes into account (Hymenoptera: Agriotypidae), Elliott (1983) directly
measured the handling time of A. armatus and found
their different nature, and as a result, the maximum
b 20.0 min. In addition, he fitted the random predator
predation rate (prey density approaches infinity) is not
19.4–20.1 min, thus b
determined by the sum of time spent for handling and equation to field data: 1/ymax
1/ymax. Second, in the predator–prey system Och-
digesting prey (as in Eq. 5), but by the maximum of
romonas sp. (a heterotrophic flagellate)–Pseudomonas
these two terms. Accordingly, we have classified pred-
ators into handling-limited and digestion-limited pred- sp. (a bacterium), Fenchel (1982a) directly measured
the handling time of Ochromonas as b
ators. Note that this classification only refers to high 20 s. In ad-
prey densities. At intermediate prey densities, our mod- dition, he performed long-term experiments (Fenchel
19 s, thus b
el shows that also handling-limited (digestion-limited) 1982b): 1/ymax 1/ymax. Third, in the
predator–prey system Polinices duplicatus (a naticid
predators experience diminished feeding rates because
of time spent for digesting (handling) prey (Figs. 3a, gastropod that drills through the shells of its prey)–
Mya arenaria (Bivalvia), the handling time of P. du-
b).
plicatus in the long-term enclosure experiments of Ed-
Handling-limited predators wards and Huebner (1977) can be estimated by data
Handling-limited predators handle (corrected for at- from Edwards and Huebner (1977) and Kitchell et al.
(1981; Appendix B): b 1.4 d; 1/ymax 1.6 d, thus b
tack efficiencies 100%) prey slower than they digest
them. For parasites and parasitoids, this means that they 1/ymax. Similarly, Boggs et al.’s (1984) results have
indicated that P. duplicatus is also handling limited
handle hosts slower than they produce eggs. In han-
when feeding on another bivalve, Mercenaria mercen-
dling-limited predators, therefore, prey uptake increas-
aria. In their study, P. duplicatus spends 75% of its
es with the amount of time spent for searching and
time in handling (i.e., drilling and eating) M. mercen-
handling prey. We consequently expect that, indepen-
aria; total foraging time (i.e., searching time plus han-
dent of prey density, handling-limited predators forage
almost all of their available time (i.e., the time not dling time) was therefore at least 75%. This exceeds
needed for nonforaging activities, such as avoidance by far corresponding values for digestion-limited pred-
Ecological Monographs
JONATHAN M. JESCHKE ET AL.
106
Vol. 72, No. 1
ators (see Discussion: Digestion-limited predators), moschatus], Forchhammer and Boomsma [1995];
sheep [Ovis aries], Blaxter et al. [1961]; shrews [Sorex
corroborating our expectation that handling-limited
araneus, S. caecutiens, S. isodon], Saariko and Hanski
predators should spend more time in foraging than di-
gestion-limited predators. [1990]).
Further examples for handling-limited predators can For digestion-limited predators, the SSS equation,
likely be found in other parasitoids, protozoans, and contrary to Holling’s (1959b) disc equation (Eq. 2),
drilling gastropods. In general, however, handling-lim- predicts that foraging time decreases with increasing
ited predators seem to be rare. prey density. This is in accordance with empirical data,
for example from birds (Spotted Sandpipers [Actitis
Digestion-limited predators macularia], Maxson and Oring [1980]; Verdins [Au-
riparus flaviceps], Austin [1978]; Oystercatchers [Hae-
Digestion-limited predators digest prey items slower
matopus ostralegus], Drinnan [1957]; Yellow-eyed
than they handle them. For parasites and parasitoids,
Juncos [Junco phaeonotus], Caraco [1979]; humming-
this means they produce eggs slower than they handle
birds [Selasphorus rufus], Hixon et al. [1983]) and
hosts. At high prey densities, therefore, predation up-
mammals (horses [Equus caballus], Duncan [1980];
take does not further increase with the amount of time
white-tailed jackrabbits [Lepus townsendii], Rogowitz
spent for searching and handling prey. This releases
[1997]; sheep [Ovis aries], Alden and Whittaker
trade-off situations at high prey densities and closes
[1970]; mouflon [Ovis musimon], Moncorps et al.
the gap between optimal foraging and satisficing theory
(J. Jeschke, personal observation; for satisficing, see [1997]; reindeer [Rangifer tarandus tarandus], Trudell
and White [1981]; greater kudus [Tragelaphus strep-
Herbers [1981] and Ward [1992, 1993]).
siceros], Owen-Smith [1994]).
The vast majority of predators seems to be digestion
limited (see also Weiner 1992). Examples have been Finally, natural predators generally spend a major
part of their time in resting. For example, Amoeba pro-
reported from mollusks (veliger larvae: Crisp et al.
[1985]; common or blue mussel [Mytilus edulis], Bayne teus, Woodruffia metabolica, African Fish Eagles (Hal-
et al. [1989]), crustaceans (Branchipus schaefferi, iaetus vocifer), lions (Panthera leo), and wild dogs
Streptocephalus torvicornis, Dierckens et al. [1997]; (Lyaon pictus) spend only 17% of their time in hunt-
Calliopius laeviusculus, DeBlois and Leggett [1991]; ing and eating (reviewed by Curio 1976). For further
Daphnia spp., Rigler [1961], McMahon and Rigler examples, see Herbers (1981) or Bunnell and Harestad
[1963], Geller [1975]; Calanus pacificus, Frost [1972]; (1990). Since resting may be caused by satiation, this
other copepods: Paffenhofer et al. [1982], Christoffer-
¨ may suggest that such predators are digestion limited.
sen and Jespersen [1986], Head [1986], Jonsson and It is, however, more reliable, to compare predator for-
Tiselius [1990]), insects (Chaoborus spp. larvae, re- aging and nonforaging times with actual measurements
viewed by Jeschke and Tollrian [2000]; the grasshop- of handling and digestion time. This approach reveals
pers Circotettix undulatus, Dissosteira carolina, Me- that the time various herbivores spend for feeding can
lanoplus femur-rubrum, and Melanoplus sanguinipes, usually be predicted solely from their handling and
Belovsky [1986b]; dusty wing larvae [Conwentzia hag- digestion times (J. Jeschke, personal observation). In
eni], green lacewing larvae [Chrysopa californica], red other words, resting often seems to be motivated by
mite destroyer larvae [Stethorus picipes], Fleschner satiation.
[1950]), birds (Woodpigeons [Columba palumbus],
Applications of the SSS equation
Kenward and Sibly [1977]; Oystercatchers [Haema-
topus ostralegus], Kersten and Visser [1996]; hum- The SSS equation was designed as a conceptual mod-
mingbirds [Selasphorus rufus], Hixon et al. [1983], Di- el for developing general and qualitative predictions
amond et al. [1986]), and mammals (moose [Alces al- about functional responses. It can be used to predict
ces], Belovsky [1978]; pronghorn antelopes [Antilo- the effects of changing predator or prey characteristics
capra americana], bison [Bison bison], elk [Cervus by analyzing changes of the corresponding parameters.
elaphus], yellow-bellied marmots [Marmota flaviven- For example, the effects of different kinds of prey de-
tris], mule deer [Odocoileus hemionus], white-tailed fenses can be predicted. A defense that reduces the
deer [Odocoileus virginianus], bighorn sheep [Ovis predator’s success rate (e.g., camouflage) will have its
canadensis], Columbian ground squirrels [Spermophi- largest effects at low prey densities. In contrast, an
lus columbianus], Rocky Mountain cotton tails [Syl- increase in handling time due to a defense (e.g., an
vilagus nuttali], Belovsky [1986b]; cattle [Bos taurus], escape reaction [decreases success rate and increases
Campling et al. [1961]; beavers [Castor canadensis], handling time]) will lower maximum predation rate in
Belovsky [1984b], Doucet and Fryxell [1993], Fryxell handling-limited predators. In digestion-limited pred-
et al. [1994]; Thomson’s gazelles [Gazella thomsoni], ators, either predation rates will decrease or total for-
Wilmshurst et al. [1999]; human beings [Homo sapi- aging time will increase. Finally, an increase in diges-
ens], Belovsky [1987]; snowshoe hares [Lepus amer- tion time (e.g., due to barely digestible substances) will
icanus], Belovsky [1984c]; meadow voles [Microtus lower predation rates at high prey densities in diges-
pennsylvanicus], Belovsky [1984a]; muskoxen [Ovibus tion-limited predators (see also Jeschke and Tollrian
PREDATOR FUNCTIONAL RESPONSES
Febraury 2002 107
result of adaptive consumer behavior. Evolutionary Ecol-
2000). The same considerations can be used to for-
ogy 3:95–114.
mulate hypotheses about optimal investment of pred-
Abrams, P. A. 1990a. The effects of adaptive behavior on
ators in raising success rate, handling efficiency, or the type-2 functional response. Ecology 71:877–885.
digestive capacity. More generally, the SSS equation Abrams, P. A. 1990b. Mixed responses to resource densities
can be linked with cost–benefit models to investigate and their implications for character displacement. Evolu-
tionary Ecology 4:93–102.
predator and prey evolution using predation rate as an
Abrams, P. A. 1990c. Adaptive responses of generalist her-
indirect measure of fitness.
bivores to competition: Convergence or divergence. Evo-
Since the basic SSS equation contains many sim- lutionary Ecology 4:103–114.
plifying assumptions, it should not primarily be viewed Abrams, P. A. 1991. Life history and the relationship between
food availability and foraging effort. Ecology 72:1242–
as a model for quantitatively predicting functional re-
1252.
sponses. However, the model is open to modifications
Abrams, P. A. 1992. Predators that benefit prey and prey that
to better match the properties of specific predator–prey harm predators: unusual effects of interacting foraging ad-
systems (using numerical analyses when necessary). aptations. American Naturalist 140:573–600.
For example, making attack efficiency a decreasing Abrams, P. A. 1993. Why predation rates should not be pro-
portional to predator density. Ecology 74:726–733.
function of prey density allows the modeling of a
Abrams, P. A., and H. Matsuda. 1993. Effects of adaptive
swarming effect due to predator confusion. After in-
predatory and anti-predator behavior in a two prey–one
corporating this confusion effect and accounting for a predator system. Evolutionary Ecology 7:312–326.
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may be used to improve predator–prey theory in gen- of factors influencing herbage intake and availability. Aus-
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APPENDIX A: FROM EQ. 12 TO EQ. 13
Solving Eq. 12 for y(x) gives the two solutions y1(x) and y2(x): lim y2 (x) 0, (A1)
x→0
As the limits indicate, only the second solution, y2(x),
c) 2 ]
ax(b c) ax[2(b c) ax(b
1 1
y1 (x) , makes sense biologically. However, y2(x) is not defined for a
2abcx 0, b 0, c 0, or x 0. Eq. 12 helps to find the cor-
lim y1 (x) → , responding equations: For a 0 or x 0, Eq. 12 gives y(x)
x→0
0; for b 0, Eq. 12 gives y(x) acx); for c
(ac)/(1
c) 2 ]
ax(b c) ax[2(b c) ax(b
1 1 0, Eq. 12 gives y(x) abx); and for b c
(ab)/(1 0,
y2 (x) ,
Eq. 12 gives y(x) ax.
2abcx
APPENDIX B
An estimation of Polinices duplicatus (Gastropoda: Naticacea) handling time feeding on Mya arenaria (Bivalvia) in the
year-round experiments of Edwards and Huebner (1977).
Mya lengthi Mya shell Polinices drilling No. preyed
Mya size class i bi[d]¶ ni (
[ 5 mm]† thicknessi [mm]‡ timei [d]§ 385)†
ˆi ˆ
1 15 0.121 0.226 0.95 0.458 73
2 25 0.277 0.517 0.95 1.048 160
3 35 0.433 0.809 0.95 1.639 108
4 45 0.589 1.100 0.95 2.229 26
5 55 0.745 1.392 0.25 4.871 16
6 65 0.901 1.683 0.05 19.355 2
6
1
Resulting mean estimated handling time b bi ni 1.4 d.
ˆ ˆ
385 i 1
† Table 1 in Edwards and Huebner (1977).
‡ Table 3 in Kitchell et al. (1981): M. arenaria shell thicknessi (mm) 0.113 0.0156 lengthi (mm).
§ Kitchell et al. (1981): P. duplicatus drilling time 1.868 d/mm.
According to Kitchell et al. (1981), attack efficiency of P. duplicatus mainly depends on predator and prey size. Given
a certain predator size, is almost unity for prey below a critical size and almost zero for prey beyond that critical size.
The critical size for M. arenaria is given in Fig. 7 in Kitchell et al. (1981). The predator sizes are given in Table 3 in Edwards
and Huebner (1977). Edwards and Huebner used four individual predators with mean sizes in the relevant period (14 June–
29 August, where maximum predation rate ymax 0.63 M. arenaria/d have been reported for P. duplicatus) of 37.9 mm, 41
mm, 42.15 mm, and 50.45 mm, respectively. The corresponding critical M. arenaria lengths are roughly 53 mm for the three
small predator individuals and 60 mm for the largest one. Therefore, M. arenaria of size classes 1, 2, 3, and 4 could be
0.95), M. arenaria of size class 5 could basically only
easily attacked by all four predator individuals (ˆ 1 ˆ 2 ˆ 3 ˆ4
0.25), and M. arenaria of size class 6 could only hardly be attacked by all
be attacked by one of the four predators ( ˆ 5
four predators ( ˆ 6 0.05).
¶ Estimated P. duplicatus handling time (for M. arenaria size class i):
) 1]
bi [d] [1.5 (2 drilling timei . (B1)
ˆ i
Derivation: From Eq. 4, bi [d] (drilling time i / i) eating time i (drilling time i i) drilling time i (Kitchell et al.
ˆ
1981). However, this calculation overestimates handling time, because it is based on the assumption that unsuccessful
drills last as long as successful ones. Assuming that unsuccessful drills last, on average, half the time of successful ones,
leads to Eq. B1.