Journal of Animal Ecology 2012, 81, 185–189
doi: 10.1111/j.1365-2656.2011.01876.x
Flexible components of functional responses
Toshinori Okuyama*
Department of Entomology, National Taiwan University, Taipei 106, Taiwan
Summary
1. The functional response of predators describes the rate at which a predator consumes prey and
is an important determinant of community dynamics. Despite the importance, most empirical
studies have considered a limited number of models of functional response. In addition, the models
often make strong assumptions about the pattern of predation processes, even though functional
responses can potentially exhibit a wide variety of patterns.
2. In addition to the limited model consideration, model selections of functional response models
cannot tease apart the components of predation (i.e. capture rate and handling time) when flexible
traits are considered because it is always possible that many different combinations of the capture
rate and handling time can lead to the same predation rate.
3. This study directly examined the capture rate and handling time of functional response in a mite
community. To avoid the model selection problem, the searching and handling behaviour data
were collected. The model selection was applied directly to these two components of predation
data. Commonly used functional response models and models that allow for more flexible patterns
were compared.
4. The results indicated that assumptions of the commonly used models were not supported by the
data, and a flexible model was selected as the best model. These results suggest the need to consider
a wider variety of predation patterns when characterizing a functional response. Without making a
strong assumption (e.g. static handling time), model selections on functional response models cannot be used to make reliable inferences on the predation mechanisms.
Key-words: adaptive behaviour, capture rate, density dependence, flexible traits, functional
response, handling time, model selection
Introduction
The functional response describes the rate at which a predator consumes prey. In predator–prey studies, the functional
response plays an important role in that it connects behavioural-level processes (e.g. foraging behaviour) and community-level processes. For example, in theoretical studies,
community dynamics (e.g. persistence and stability) are sensitive to the choice of the functional response model (Murdoch,
Briggs & Nisbet 2003; Turchin 2003). It is important that
functional response models adequately capture real predation patterns to make reliable community-level predictions.
Despite this importance, we may not have a clear idea about
the relationship between the components of predation (i.e.
capture rates and handling time) and predation rates (i.e.
functional response).
Ecologists are increasingly aware of the importance of flexible traits in functional responses (see Abrams 2010, for
review). One example of functional response models with a
flexible trait, based on Holling type II functional response, is
*Correspondence author. E-mail: okuyama@ntu.edu.tw
aCN
1 þ aChN
eqn 1
where a, h, and N are the attack rate, handling time and prey
density, respectively. C is a function that represents the flexible
trait (e.g. foraging effort) of the predator. For example, the
predator-dependent model (Arditi & Akaçakaya 1990) is an
example of a flexible trait model in which it is assumed that
C(P) ¼ P)m, where P is the predator density and m is the
interference parameter. When the trait expression C is constant, eqn 1 is equivalent to the original Holling type II functional response. Theoretical studies have used a variety of
ways to model the trait expression for C. For example, some
studies have assumed a specific functional form (Křivan &
Schmitz 2004), while others used optimal foraging approaches
to derive the optimal expression (Křivan & Sirot 2004). However, although ecologists recognize that the trait expression C
may be flexible, empirical studies have not explored much of
the possibilities (Kratina et al. 2009; Hauzy et al. 2010).
Many functional response models are associated with their
mechanistic interpretations. For example, Holling type II
model assumes that the prey capture rate increases linearly
with the prey density and the handling time is constant.
Ó 2011 The Author. Journal of Animal Ecology Ó 2011 British Ecological Society
186 T. Okuyama
However, explicit examinations of these mechanistic assumptions are rare. Instead, empirical studies focus on describing
the pattern of the predation rate. The common experimental
design records the number of prey consumed in a given time
interval (Juliano & Williams 1987; Juliano 1993) and does
not directly examine the components of the predation such as
the searching and handling behaviour (but see Tully, Cassey
& Ferriére 2005; Kratina, Vos & Anholt 2007). This causes a
serious problem especially when we start considering flexible
traits. For example, although most of the functional response
models assume that the handling time is constant, if it were
flexible (e.g. Okuyama 2010), we can no longer make mechanistic interpretations of models based on predation rates
because different mechanisms (i.e. capture rate and handling
time) can lead to an identical predation rate.
The main purposes of this study were to explicitly characterize a functional response with a flexible formulation at the
levels of capture rate and handling time and then to compare
this model against common predator-dependent and predator-independent models. In particular, this study investigates
the functional response of the predatory mite Phytoseiulus
persimilis on the prey mite Tetranychus urticae and shows
that a model that allows for flexible patterns can best describe
the data. An important consideration for the common functional response studies (model selections in particular) is also
discussed.
Materials and methods
MODELS AND ANALYSIS
Common functional response models have the form
k
;
1 þ kh
eqn 2
where k is the prey capture rate of a predator given that the predator
is searching for prey (search process). Thus, k is potentially a function
of environmental variables such as the densities of interacting species.
The expected time to capture a prey is k)1. Once a prey is captured,
the predator handles the prey (handling process). The average duration of handling a single prey is h. Once the predator finishes handling
the prey, it starts searching for a new prey (search process). Based on
these assumptions, eqn 2 is derived (Stephens & Krebs 1986). For
example, Holling type II model assumes k ¼ aN and Holling type III
model assumes k ¼ aN2, while both models assume that the handling
time does not change on average.
In this study, I considered six models: Holling’s type II model
(k ¼ aN), Holling’s type III model (k ¼ aN2), h-model (k ¼ aNh),
Arditi–Akaçakaya model (k ¼ aNP)m), a phenomenological model
(logit(k/a) ¼ b0 + b1N + b2P) and a flexible trait model. For the
flexible trait model, I considered k ¼ aCN, where C(N,P) ¼
exp{aN + bP}. In this model, for example, negative a and b indicate
that the expression of the flexible trait will decrease with the density
of the prey and predator, respectively. The phenomenological model
uses the logit function that is commonly used in the logistic regres0 þ b1 N þ b2 Pg
sion. In particular, k ¼ 1 aexpfb
þ expfb0 þ b1 N þ b2 Pg. All of these models make
the same assumption about the handling process (i.e. constant handling time), while they make the variable assumptions about the
search process (i.e. capture rate).
Most functional response studies indirectly estimate the parameters of the models using the number of prey captured in a given time.
However, when both the capture rate and handling time are flexible,
we cannot make mechanistic interpretations of the models based on
predation rates. For any functional response with values kA and hA,
we can always find other values kB and hB that satisfy
kA
kB
¼
:
1 þ kA hA 1 þ kB hB
eqn 3
For example, suppose kA ¼ aR and hA is constant, and kB ¼
AR2. Then, we can find a density-dependent handling time
1
hB ¼ ð1 þ kA hA Þk1
A kB that satisfies eqn 3. In other words, for
any functional response model (kA and hA) that may be chosen based
on an experiment and its model selection, there are infinitely many
combinations of kB and hB that give the identical predation rate.
Thus, it is not possible to make an inference about the mechanisms
(i.e. capture rate and handling time) of the predation unless we make
a strong assumption such as a static handling time (but see Okuyama
2010). For example, even when a model selection selects Holling type
II model as the best model (and even if the predation rate is accurate),
it may not mean that the capture rate increases linearly nor that the
handling time is constant despite the model’s mechanistic assumptions. The same argument applies to other models such as predatordependent models. The selection of a predator-independent model
does not imply that both capture rate and handling time are predator-independent. Ecologists are familiar that it is not possible to
distinguish between Holling type II model [aR/(1 + ahR)] and the
corresponding Michaelis–Menten form [cR/(d + R)]. The reason
discussed above is basically the same here (i.e. because two functions
are equivalent).
The objective of this study was to understand the capture rate and
handling time of predation, not to characterize a predation rate phenomenologically. Thus, to solve the model selection problem, this
study directly recorded data from the search process and the handling
behaviour. Because for all models considered here, the expected time
required to capture the first prey is (kP))1, the actual time to capture
the first prey Ts can be used to test the capture rate assumption. In
reality, functional response is influenced by many other factors such
as digestion and satiation (Jeschke, Kopp & Tollrian 2002; van Rijn
et al. 2005) and previous experience (McCoy & Bolker 2008). While
the consideration of those factors are important, the data were
collected from the first prey capture to minimize the effect of
confounded factors.
STUDY ANIMALS
The predatory mite, Phytoseiulus persimilis, and the prey mite,
Tetranychus urticae, were used as the study subjects. The mites were
reared in a temperature-controlled room (26 °C) on soybean plants.
The subject predatory mites were removed from a prey-abundant
culture 24 h before they were used in the experiment for the starvation control of the predators.
In this experiment, a foraging bout consists of a search process
and the subsequent handling process. To examine how the prey
density and the predator density affect the foraging bouts, three
levels (1, 2 and 3 individuals) of the prey and predator densities
were created, and all possible density combinations were examined. The predator(s) was/were first placed on a similar-sized fresh
soybean leaf (mean ± SD; 2Æ89 ± 0Æ158 cm in the length) for
30 min for acclimation. Subsequently, the prey mite(s) was/were
released on the leaves, and their interactions were recorded using
Ó 2011 The Author. Journal of Animal Ecology Ó 2011 British Ecological Society, Journal of Animal Ecology, 81, 185–189
Flexible functional responses 187
dence intervals of the parameters of the best model are,
a (0Æ037,0Æ06), a ()0Æ407,)0Æ209) and b ()0Æ545,)0Æ361).
Likelihood ratio test also shows that both a (P < 0Æ001) and
b (P < 0Æ001) are different from zero. The coefficient of
determination may be interpreted as the fit of models, sometimes referred as pseudo R2 (Anderson 2008). Based on
Nagelkerke’s (1991) measure, the coefficient of determination of the best model is 0Æ16.
The handling time data were fitted to a generalized linear
model with gamma-distributed errors and the log link function. In other words, the handling time were assumed to
follow a gamma distribution whose mean l is described as
log(l) ¼ b0 + b1N + b2P + b3NP where b0, b1, b2 and b3
are the parameters of the model. For example, b1 < 0
indicates that average handling time decreases with the prey
density N. The estimated parameters are b0 ¼ 8Æ3
(<0Æ001),b1 ¼ )0Æ13 (0Æ077),b2 ¼ )0Æ11 (0Æ15) and b3 ¼ 0Æ04
(0Æ24) (the values in the parentheses are P-values). Thus, the
data marginally support that the average handling time
decreases with the prey density (P ¼ 0Æ077). Although the
specific interaction effect b3 was not detected (P ¼ 0Æ24), the
visual inspection of the data (Fig. 2) suggests that the relationship between the handling time and prey density vary
among the predator density. If the effect of the prey density
were independently analysed for each predator density, when
the predator density is one (P ¼ 1), the average handling
time decreased with the prey density (b1 ¼ )0Æ13,P ¼ 0Æ015)
a digital video camera. Each leaf was placed on a water-saturated
woven fabric to prevent the mites moving off of the leaf. All trials
were carried out in a temperature-controlled room (26 °C). Each
trial lasted until a predator captured and consumed (i.e. handled)
a prey. The time to capture a prey, Ts, and the subsequent handling time, h, were recorded. When there were multiple predators,
the data for the first predator that captured a prey were recorded.
All model organisms were used only once in the experiment. Each
treatment combination was replicated 56 times. The models were
compared using Akaike Information Criterion (AIC) (Burnham &
Anderson 2002).
Results
The raw search time data and handling time data are shown
in Figs 1 and 2, respectively. The mean l and variance r2
relationship of the search time roughly followed that of the
exponential distribution (i.e. l2 ¼ r2), and thus, the exponential distribution was used to compute the likelihoods. The
maximum-likelihood estimates and the AIC of the capture
rate models are shown in Table 1. Although AIC is known to
lenient toward parameter-rich models (Ripplinger & Sullivan
2007), the AIC of the two most parameter-rich models is by
far the best compared with the other models. The AIC of the
flexible trait model is the smallest among the models, indicating the model best describes the data based on this criterion.
However, the logit model is basically equivalent to the flexible trait model (i.e. small d AIC). The 95% bootstrap confi-
P=1
P=2
P=3
Search time (min)
102
101·5
101
100·5
100
10–0·5
10–1
Fig. 1. Box plots of the search time. The panels describe the different predator densities,
P. The square points indicate the means.
1
2
3
1
2
3
1
2
3
Number of prey (N)
P=1
P=2
P=3
Handling time (min)
102·2
Fig. 2. Box plots of the handling time. The
panels describe the different predator
densities, P. The square points indicate the
means.
102
101·8
101·6
101·4
101·2
101
100·8
1
2
3
1
2
3
Number of prey (N)
Ó 2011 The Author. Journal of Animal Ecology Ó 2011 British Ecological Society, Journal of Animal Ecology, 81, 185–189
1
2
3
188 T. Okuyama
Table 1. Maximum-likelihood estimates and Akaike Information Criterion (AIC) of the models. k shows the number of parameters. The flexible
trait model consists of C ¼ exp{aN + bP}. The logit model is logit(k/a) ¼ b0 + b1N + b2P
Model
k
AIC
d AIC
Parameters
aCN
logit
aNP)m
aNh
aN
aN2
3
4
2
2
1
1
4413Æ287
4414Æ192
4444Æ457
4473Æ187
4496Æ945
4677Æ413
0
0Æ905
31Æ17
59Æ9
83Æ66
264Æ13
a
a
a
a
a
a
but the same pattern is absent for the higher predator
densities. However, when there were multiple predators,
more than one predator sometimes ate the same prey
simultaneously. Thus, the handling time data for the multiple-predator treatment are confounded with this detail (i.e.
prey sharing behaviour).
Discussion
Models come with a variety of implicit and explicit assumptions that are rarely empirically examined. This study examined two components of the functional response models, and
a parameter-rich flexible trait model was selected as the best
model for describing the capture rate. Handling time also did
not follow the common assumption and was density-dependent. Although these results may not appear surprising, the
majority of functional response studies assume otherwise
(without testing) and make mechanistic inferences based on
the indirect measurements (i.e. number of prey eaten in a
given time). If the characterization of predation rates is the
ultimate goal, this may not be a problem, but as the importance of flexible traits is recognized (Bolker et al. 2003;
Abrams 2010), detailed examinations about how individual behaviour relates to predation rates become more
important.
Many functional response models assume that the capture
rate increases linearly with the prey density (i.e. k increases
linearly with N). However, the capture rate increased slower
than expected by the linear model in the mite community (e.g.
a < 0 and h < 1). The same qualitative result was reported
in other studies (e.g. Mols et al. 2004). Some mechanisms
have been postulated to explain this potentially common pattern (Ruxton 2005; Okuyama 2009). Predators usually use
some cues associated with prey when searching. For example,
predatory mites are chemosensory foragers (Dicke & Sabelis
1987). The value of prey detection would be the highest for the
first prey, but it may decrease for the successive prey
(Okuyama 2009). This provides one possible explanation of
the decelerating capture rate. Furthermore, it is also known
that prey mites exhibit antipredator behaviour Škaloudová,
Zemek & Křivan 2007). In the model, the variable trait C was
discussed as the foraging effort of the predator, but it may also
be influenced by potential adaptive behaviour of the prey.
When there was only one predator per leaf, the average
handling time decreased with the prey density. When there
were multiple predators, there was no association between
¼
¼
¼
¼
¼
¼
0Æ0477,a ¼ )0Æ3117,b ¼ )0Æ4504
0Æ0732,b0 ¼ )0Æ4887,b1 ¼ 0Æ3429,b2 ¼ )0Æ6049
0Æ0035, m ¼ 0Æ1889
0Æ01326, h ¼ 0Æ4894
0Æ0095
0Æ0044
the handling time and the prey density. This may be because
predators sometimes simultaneously fed on the same prey,
which would have affected the handling time. Although there
have not been many studies that directly examined the relationship between the handling time and the prey density, studies that examined this relationship also found the negative
correlation (e.g. Cook & Cockrell 1978; Giller 1980; Collins,
Ward & Dixon 1981; Cooper & Anderson 2006; Okuyama
2010). Because whether handling time is constant or flexible is
an important factor in understanding the predation mechanism, directly recording handling time is advisable.
Functional responses (predation rates) are typically
described based on the combinations of capture rate (k) and
handling time (h). Previously, I discussed that two different
models of functional response components (e.g. k1 and k2)
can result in the identical functional response depending on
the corresponding handling times (h1 and h2). Even if two
models are not identical, these confounded factors will make
it difficult to tease apart two models statistically (Trexler,
McCulloch & Travis 1998). For example, given k1,k2 and h1,
a specific h2 would make the two functional responses identical, but even if h2 is not exactly the same as the specific model
but at least qualitatively similar, the same problem would
remain. It is important for us to be able to estimate nonlinear
parameters associated with mechanistic models (Trexler,
McCulloch & Travis 1998).
A weakness of the direct method used in the study is that
even if we understand the capture rate k and the handling
time h, the resulting predation rate may not be what we
expect (eqn 2). To examine the validity of the mechanistic
assumption (eqn 2), we need the combination of the direct
examination of the components (e.g. capture rate and handling time) and the predation rate. When the extrapolation of
the components does not match the predicted predation rate,
it may reveal some hidden important factors in the predation
process. For example, when each predator behaves according
to specific k and h, the resulting predation rate may be different from the expectation when a spatial structure is considered (Okuyama 2009). Understanding how the behavioural
components such as the searching and handling behaviour
relate to the predation rate is an essential information to scale
individual behaviour up to community dynamics. The best
model selected in the study is associated with a relatively low
coefficient of determination [but also see Hoetker (2007) for
the interpretation], which suggests the possibility for other
models. The attempts to better understand the relationship
Ó 2011 The Author. Journal of Animal Ecology Ó 2011 British Ecological Society, Journal of Animal Ecology, 81, 185–189
Flexible functional responses 189
would reveal the model that may better describe the predation rates.
It is important to note that although a flexible trait model
was selected, this does not necessarily mean that the predators adjusted their trait (e.g. foraging effort) in a densitydependent manner. For example, suppose k ¼ aN2 is
selected. This can be reparameterized k ¼ aCN, where C ¼
N to imply a different mechanism (although this may be a
pathological example). In other words, any model may be
considered a flexible trait model based on a reparameterization. This is the same model selection problem discussed earlier. This study was able to characterize the patterns of
capture rate and handling time, which was not possible based
on the conventional design (i.e. number of prey consumed in
a given interval). Nevertheless, in this study, the lower mechanisms cannot be understood because any capture rate k can
be decomposed into a variety of static and flexible components. How much detail is needed in a study would depend
on its objective. But a more elaborate experimental design is
needed when one wants to understand the mechanism of predation rather than to phenomenologically characterize the
pattern. This is an important consideration when we start
characterizing flexible trait models, and discussions for developing the standard empirical protocol is needed.
While many empirical studies have characterized functional responses, most of them considered only few models
that do not encompass potential predation patterns.
Furthermore, even simplest assumptions (e.g. constant handling time) are usually not empirically verified, which causes
a problem in the model selection if we want to interpret them
mechanistically. This study showed that the common
assumptions regarding the handling time and capture rate
are not supported by the data, and a model that allows for
flexible patterns was selected. A functional response model
allows us to connect behavioural processes and communitylevel processes. To extrapolate lower-level dynamics to community dynamics, those details must be explicitly considered
when quantifying functional responses.
Acknowledgements
I thank Bob Ruyle and two anonymous reviewers for their insightful comments
on the manuscript. Dr Chi-Yang Lee provided the mite populations and helped
me to set up the culture. Fu-Chyun Chu collected the majority of the data. This
study was supported by the National Science Council of Taiwan (97-2321-B002-036-MY2,99-2628-B-002-051-MY3).
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Received 5 September 2010; accepted 9 May 2011
Handling Editor: Frank van Veen
Ó 2011 The Author. Journal of Animal Ecology Ó 2011 British Ecological Society, Journal of Animal Ecology, 81, 185–189