Flexible components of functional responses


Correspondence author. E-mail: okuyama@ntu.edu.tw


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.


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

image(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) = Pm, 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. 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

image(eqn 2)

where λ is the prey capture rate of a predator given that the predator is searching for prey (search process). Thus, λ is potentially a function of environmental variables such as the densities of interacting species. The expected time to capture a prey is λ−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 λ = aN and Holling type III model assumes λ = 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 (λ = aN), Holling's type III model (λ = aN2), θ-model (λ = aNθ), Arditi–Akaçakaya model (λ = aNPm), a phenomenological model (logit(λ/a) = b0 + b1N + b2P) and a flexible trait model. For the flexible trait model, I considered λ = aCN, where C(N,P) =exp{αN + βP}. In this model, for example, negative α and β 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 regression. In particular, inline image. 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 λA and hA, we can always find other values λB and hB that satisfy

image(eqn 3)

For example, suppose λA = aR and hA is constant, and λB = AR2. Then, we can find a density-dependent handling time inline image that satisfies eqn 3. In other words, for any functional response model (λA and hA) that may be chosen based on an experiment and its model selection, there are infinitely many combinations of λB 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 predator-dependent 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 (λP)−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; 2·89 ± 0·158 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 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).


The raw search time data and handling time data are shown in Figs 1 and 2, respectively. The mean μ and variance σ2 relationship of the search time roughly followed that of the exponential distribution (i.e. μ2 = σ2), 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 δ AIC). The 95% bootstrap confidence intervals of the parameters of the best model are, a (0·037,0·06), α (−0·407,−0·209) and β (−0·545,−0·361). Likelihood ratio test also shows that both α (P < 0·001) and β (P < 0·001) 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 0·16.

Figure 1.

 Box plots of the search time. The panels describe the different predator densities, P. The square points indicate the means.

Figure 2.

 Box plots of the handling time. The panels describe the different predator densities, P. The square points indicate the means.

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 = exp{αN + βP}. The logit model is logit(λ/a) = b0 + b1N + b2P
ModelkAICδ AICParameters
aCN34413·2870a = 0·0477,α = −0·3117,β = −0·4504
logit44414·1920·905a = 0·0732,b0 = −0·4887,b1 = 0·3429,b2 = −0·6049
aNPm24444·45731·17a = 0·0035, m = 0·1889
aNθ24473·18759·9a = 0·01326, θ = 0·4894
aN14496·94583·66a = 0·0095
aN214677·413264·13a = 0·0044

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 μ is described as log(μ) = 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 = 8·3 (<0·001),b1 = −0·13 (0·077),b2 = −0·11 (0·15) and b3 = 0·04 (0·24) (the values in the parentheses are P-values). Thus, the data marginally support that the average handling time decreases with the prey density (P = 0·077). Although the specific interaction effect b3 was not detected (P = 0·24), 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 = 1), the average handling time decreased with the prey density (b1 = −0·13,P = 0·015) 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).


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. λ increases linearly with N). However, the capture rate increased slower than expected by the linear model in the mite community (e.g. α < 0 and θ < 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 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 (λ) and handling time (h). Previously, I discussed that two different models of functional response components (e.g. λ1 and λ2) 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 λ1,λ2 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 λ 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 λ 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 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 density-dependent manner. For example, suppose λ = aN2 is selected. This can be reparameterized λ = aCN, where C = 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 λ 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 community-level processes. To extrapolate lower-level dynamics to community dynamics, those details must be explicitly considered when quantifying functional responses.


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-B-002-036-MY2,99-2628-B-002-051-MY3).