1The predator-dependent Beddington–DeAngelis functional response model can be considered as an extension of the prey-dependent Holling's type II functional response model, since it includes, apart from the states ‘searching for prey’ and ‘handling prey’, a third behavioural state, namely ‘mutual interference with competitors’. The model is further based upon the underlying idea of mass action, which means that it is assumed that predator and prey numbers are infinitely large.
2This latter assumption casts doubt on the applicability of the model to experimental situations, which have been used to estimate the underlying behavioural parameters, because such experiments are usually performed with very few competitors.
3Therefore, a stochastic version of the Beddington–DeAngelis model is presented which overcomes these problems. A maximum-likelihood procedure for parameter estimation is presented and applied to shore crabs foraging on blue mussels.
4In passing, a mistake in the derivation of the deterministic Beddington–DeAngelis model is corrected, resulting in a slightly different solution.
The prey-dependent Holling's type II functional response, also known as the disc equation (Holling 1959), is still the most widely used functional response model (Skalski & Gilliam 2001; Jeschke, Kopp & Tollrian 2002). The model is based upon the idea that the predator can be in either one of two mutually exclusive states: it is either searching for prey or handling prey. The model furthermore assumes that the capture rate of the predator (and hence the transition from the searching to the handling state) is proportional to the number of prey in the system. Beddington (1975) extended Holling's approach by incorporating the possibility of multiple predators that can mutually interfere. He assumed that predators when meeting a competitor start to interfere, that is they then enter an interference state and he arrived at a functional response in which the predator's per capita intake rate is predator dependent, that is it is a function of both prey density and predator density. More or less at the same time, DeAngelis et al. (1975) independently proposed the same function, but considered it as an empirical relationship and did not attempt to derive the term. The model is often referred to as the Beddington–DeAngelis model (Skalski & Gilliam 2001). Later Ruxton, Guerney & De Roos (1992) embroidered upon Beddington's approach and corrected some major inconsistencies in the derivation of the model. Yet, they arrived at the same equation.
Skalski & Gilliam (2001) took a practical approach to the question of how relevant predator-dependent functional responses actually are, and collected 19 data sets from the literature where predator intake rates were measured for at least two prey densities and two predator densities. These data sets were used to examine which model, Holling or Beddington–DeAngelis, gives a better description of predator intake rates. In their comparison, they also included few other functional response models, for example, the Hassell–Varley model, that were not derived from mechanistic principles. They remarked that mechanistic models do, however, lead to clearer science, reiterating a plea we earlier made (Van der Meer & Ens 1997). From their analysis, they concluded that in most cases the Beddington–DeAngelis model provided a better description of predator feeding than Holling's type II functional response.
Skalski & Gilliam (2001) used only experimental data where both predator and prey numbers were under experimental control. Experiments are indeed to be preferred over observational field data where many factors may be confounded with prey and predator densities. However, such experimental data are usually (and, certainly for larger animals, almost necessarily) from experiments with very few predators, quite often as few as two (Swennen, Leopold & De Bruijn 1989; Mansour & Lipcius 1991; Van Gils & Piersma 2004; Vahl et al. 2005; Smallegange, Van der Meer & Kurvers 2006; Smallegange & Van der Meer 2007). This practice contradicts the assumption of infinitely large prey and predator populations, made in the derivation by Ruxton et al. (1992) and, as we show below, also made by Beddington (1975). So here, we do have a problem: the theory is based on infinitely large populations, but the experimental data required to link the theory to reality are usually from experiments with very few predators. Proper analysis of such experiments requires a functional response model valid for a finite number of predators. Here we present such a model using the theory of stochastic processes. The underlying behavioural rules within the model are entirely analogous to those used by Ruxton et al. (1992). In passing, we correct a mistake in Ruxton et al.'s derivation for infinitely large populations, with the consequence that the resulting model is not entirely equivalent (though rather similar) to the Beddington–DeAngelis model. Finally, using experimental data on shore crabs feeding on blue mussels, we show how the stochastic model can be fitted to data.
The infinitely large population approach
Holling's type II functional response can be based on a simple behavioural model in which each individual predator can be in one of two mutually exclusive states: the predator is either searching or handling prey. The transition rate from the searching state to the handling state follows, under the assumption of infinitely large populations, the rules of mass action, as in chemical reaction kinetics. This means that the rate at which collisions between predators and prey occur is proportional to the product of the number of searching predators and the number of prey in the system. At each collision, the predator enters the handling state. The (per capita) transition rate from handling back to searching is assumed constant. The model can be written as a set of two differential equations, where the equilibrium can be derived by setting the differential equations equal to zero. It can be shown that the equilibrium is stable. The equilibrium density of searching predators determines the overall intake rate as a function of prey density, that is, the functional response. The number of handling (H) and searching (S) predators is given by the following set of differential equations
where D is the prey abundance (dimensionless, or when expressed per surface area or volume with dimension L−2 or L−3), ν is the searching rate, also called the rate of successful search or the success rate (T−1, L2T−1 or L3T−1), and λ is the transition rate from handling to searching (T−1). At equilibrium (sometimes called pseudo-equilibrium, because D, which is the number of prey, is assumed constant), the equations simplify to vS*D = λH*, where S* and H* refer to the equilibrium numbers. The per capita intake rate W (dimension T−1), which equals vD times the fraction of searchers at equilibrium, is then
which is equivalent to Holling's type II functional response, with an expected handling time equal to 1/λ.
Holling's model can be extended by introducing a second type of ‘prey’, with which the predators can ‘collide’. Assume that this type of prey is not edible. It only costs time to ‘handle’. One might think of a filter feeder whose feeding apparatus is temporarily clogged with inedible silt particles that have to be removed (Kooijman 2006). The relevant differential equations are now
where F is the number of predators wasting time with the inedible prey type, E is the abundance of the inedible prey which is assumed constant, µ is the searching rate for this inedible prey, and ϕ is the transition rate from the wasting-time state back to the searching state. Again, the intake rate in equilibrium is obtained by setting all differential equations equal to zero:
and a functional response equation appears that is similar to Beddington's. Indeed, Beddington (1975) followed exactly this model, with competing predators in the role of inedible prey consuming time. As was shown by Ruxton et al. (1992), this approach is not entirely consistent, because it fails to consider that these competing predators themselves can also be in different states. If it is assumed that only searching predators can enter the wasting-time state, one should take into account that the competing predators are not always searching. They can be handling prey as well, or they can already be involved in an interference event.
Ruxton et al. (1992) improved upon Beddington's approach and modelled explicitly that searching predators will start to interfere both when they meet up with another searching and another handling predator. In order to keep track of whether or not an interfering predator possesses a prey item, the model distinguishes two different interference states. One state is reached from the searching state, the other from the handling state (Fig. 1). The equilibrium fraction of searching predators now determines the overall intake rate as a function of prey density and predator density, which we earlier referred to as the generalized functional response (Van der Meer & Ens 1997). The number of handling (H), searching (S), interfering after searching (F) and interfering after handling (G) predators is now given by the following set of differential equations
The equations given here differ from the ones given by Ruxton et al. (1992) in that they used the term 2µS2 instead of µS2 under the reasoning that at each ‘collision’ between searching predators, two (and not one) searching predators leave the searching state S and enter the interference state F (G.D. Ruxton, personal communication). Although it is correct that at each collision, two predators move to another state, the fact was overlooked that the rate at which collisions between two searching predators occur equals which for infinitely large populations is equal to µS2/2. The multiplication by two and the division by two cancel out.
Setting all four differential equations equal to zero gives the equilibrium densities and subsequently the functional response equation (see Appendix S1).
This is a rather awkward expression, but it can be considerably simplified if interference is a rare event (see Appendix S1). In that case, the functional response is (approximately) given by
The equation differs from the model at which Ruxton et al. (1992) arrived, which was equivalent to the Beddington–DeAngelis model. Apparently, this similarity was due to the erroneous use of the term 2µS2 instead of µS2. The mechanistic underpinning that Ruxton et al. (1992) provided for the widely used Beddington–DeAngelis model (Skalski & Gilliam 2001) appears to be not entirely correct. Note further that if ϕ→∞, in which case the interference state is immediately left, the predicted intake rate is exactly the same as in Holling's type II functional response.
The finite population approach
The foraging process within a small predator population can be considered as a continuous time Markov chain, which is a particular type of stochastic process (Ross 1989). Markov chains, which have often been used for modelling animal behaviour (Metz & Van Batenburg 1985; Haccou & Meelis 1994), possess the property that the probability of being in some specific state in the future only depends on the present state and is independent of past states. In a continuous-time Markov chain, transitions from one state to another can occur at any moment in time. The duration between transitions is therefore exponentially distributed and the expected time until the next transition depends upon the so-called transition rate parameters. The probability that a continuous-time Markov chain will be in state j at time t (generally) converges (when t gets infinitely large) to a limiting value, or limiting probability, independent of the initial state. When the Markov chain is a foraging process, the limiting probability of each state can be considered as the fraction of time that the foraging process is in that state.
If we consider a system with only two predators in which the foraging process is analogous to that in the Ruxton model, then seven states can be distinguished: SS (both predators are searching), SH (the first animal is searching, the second handling), HS (vice versa), HH (both are handling), FF (both are interfering after searching), FG (the first is interfering after searching, the second is interfering after handling), and finally GF (the first is interfering after handling, the second is interfering after searching). Since it is assumed that two handling predators will not interfere, the state GG is non-existent. The transition rates follow from the rate at which a searching predator starts handling (vD), a handling predator finishes its meal (λ), a pair of predators starts interfering (µ), and an interference interaction ends (ϕ). Knowing the transition rates (Fig. 2a), the limiting probabilities Pj can be derived for each state j from the so-called balance equations (Table 1). The balance equations state that in the long run, the rate at which the process enters state j equals the rate at which the process leaves state j. For example, the rate at which the process can enter the first state SS equals the sum of the transition rates from the second state SH to the first (λP2), from the third HS to the first (λP3), and from the fifth FF to the first (ϕP5). For simple processes, the balance equations might directly reveal the limiting probabilities, but for more complex processes numerical approximation methods have to be used. See, for example, Ross (1989, pp. 284–287) for details.
Table 1. Balance equations and limiting probabilities, relative to the limiting probability of the ‘all predators are searching’ state. (a) Markov chain with two predators, in which the states indicate the behaviour of each specific individual; the first letter refers to the behaviour of the first individual, the second letter to that of the second individual; (b) Markov chain with two predators, but without information on specific individuals; a letter indicates the behaviour and is followed by a figure which shows how many individuals are involved in that type of behaviour. See text for explanations of the behaviours
Rate at which leave
Rate at which enter
Relative limiting probability
(2vD + µ)P1
λ (P2 + P3) + ϕP5
(vD + λ + µ)P2
λP4 + ϕP6
(vD + λ + µ)P3
λP4 + ϕP7
vD (P3 + P4)
Rate at which leave
Rate at which enter
Relative limiting probability
SS, both predators are searching; SH, the first animal is searching, the second handling; HS, vice versa; HH, both are handling; FF, both are interfering after searching; FG, the first is interfering after searching, the second is interfering after handling; and finally GF, the first is interfering after handling, the second is interfering after searching.
S2, both predators are searching; S1H1, one animal is searching, the other handling; H2, both are handling; F2, both are interfering after searching; G2, one is interfering after handling, the other after searching.
(2vD + µ)P1
λP2 + ϕP4
(vD + λ + µ)P2
2vDP1 + 2λP3 + ϕP5
The Markov chain presented in Fig. 2a can be simplified by pooling specific states: SH and HS can be pooled in a single state and the same is true for FG and GF (Fig. 2b). Information on which specific predator is doing what is lost and in defining the transition rates one should now take explicitly into account the number of possible transitions. For example, the transition from the first state (now called S2, which means two searching predators) to the second state (now called S1H1, which means that one predator is searching and the other is handling, without taking into account which of the two is searching or handling) can occur in two different ways, because there are two different predators that can encounter a prey. The transition rate thus equals 2vD. Similarly, when both predators are handling, the transition rate to the state S1H1 equals 2λ.
The set of balance equations for this Markov chain (Table 1b) can be considerably simplified by adding the balance equation for the fourth state F2 to the balance equation for the first state S2. This reveals the equation (2vD + µ)P1 + ϕP4 = λP2 + ϕP4 + µP1, which can be simplified to 2vDP1 = λP2. This new equation implies that the transition rate from state S2 to state S1H1 equals the transition rate from state S1H1 to S2. Adding the balance equations for the third state H2 and the fifth state G2 to the balance equation for the second state S1H1 reveals the same result. What basically happens is that the balance equations simplify to a set of equations which say that for all connected states i and j the transition rate from state i to state j should be equal to the transition rate from state j to state i.
This result can be used to obtain the limiting probabilities for all states relative to the limiting probability for one specific state, for example the first state which is the all-predators-are-searching state S2 (Table 1). Since 2vDP1 = λP2, it follows that the limiting probability for the second state S1H1 is given by Similarly, µP1 = ϕP4, so the limiting probability for the fourth state F2 equals The limiting probabilities for the third state H2 and the fifth state G2 can first be expressed relative to the second state S1H1 and next to the first state S2. For example, and next The limiting probabilities themselves can now be obtained by dividing each relative limiting probability by the sum of all relative limiting probabilities.
The overall limiting probability that a focal predator is searching follows from first multiplying for each state the limiting probability by the fraction of searchers and subsequently summing over all states. For two predators, the overall relative limiting probability that a focal predator is searching equals 1 (for the S2 state) plus 1/2 times 2vD/λ (for the S1H1 state), which gives 1 + vD/λ. The sum of all relative limiting probabilities equals 1 + 2vD/λ + (vD/λ)2 + µ/ϕ + 2(µ/ϕ)(vD/λ) or (1 + vD/λ)2 + µ/ϕ(1 + 2vD/λ). The ratio of these two relative probabilities gives the limiting probability that a focal predator is searching, or in other words the fraction of searchers, and directly yields the functional response. The per capita intake rate W, which equals vD times the fraction of searchers, is thus
which is exactly the same result as the approximate result obtained by the deterministic model using the number of predators P = 1.
In Appendix S2, it is shown that the property that for all connected states i and j, the transition rate from state i to state j is equal to the transition rate from state j to state i, also holds when processes with more than two predators are considered. Hence, for all connected states i and j, it holds that qijPi = qjiPj, where qij is the transition rate when in state i to state j. As before, this implies that for all states the limiting probabilities, relative to the all-predators-are-searching state, follow from the property Pk = (qik/qki) ... (q1j/qj1)P1. Knowledge of all these relative limiting probabilities implies that the fraction of searchers can be derived and subsequently the functional response equation. We worked out this whole procedure for systems up to seven predators. It appeared that the results can be generalized to any number of predators (see Appendix S3), and the average intake W(k) of a predator in a k-predator system equals
where Q(k) is a complicated but explicit expression, which can easily be computerized.
where α ≡ vD/λ and β ≡ µ/ϕ.
Comparing the two approaches
Assuming that searching predators interfere with both searching as well as handling predators, the infinitely large population approach revealed a functional response model (equation 3) and an approximation (equation 4) that both predict that intake rate decreases homogeneously with increasing number of predators. The finite population approach, introduced here, resulted in a model that makes a different prediction. Particularly, if interference is strong, a zigzag type of functional response is predicted, with relatively low intake rates when the number of predators is even and high intake rates with an uneven number of predators (Fig. 3): two dogs fight for a bone, and a third runs away with it. The finite population functional response (equation 6) fluctuates around equation (3), but the zig-zags decrease in amplitude as the number of predators increases. Hence, for a larger number of predators than say five or six, equation (3) seems to suffice. The approximation given by equation (4) is yet rather poor in case of strong interference.
Practicalities: fitting the model to real data
A maximum likelihood parameter estimation procedure can be applied to the specific Markov chains discussed in this paper (see Appendix S4). For example, for the two-predator case (with states 1–5: S2, S1H1, H2, F2, and G2, see Fig. 2b) the maximum likelihood estimators (plus or minus standard errors) are
where nij is the number of transitions from i to j and yi is the total time spent in state i.
We performed experiments in which we observed the foraging and interference behaviour of two male adult shorecrabs Carcinus maenas with a major chela length between 29 mm and 35 mm, feeding on mussels Mytilus edulis with a length of 12–14 mm in an experimental tank of 0·25 m2 surface area. Consumed prey were immediately replaced to keep prey density constant. Here we use, as an example, the data from seven trials with a total duration of approximately one and a half hour and in which prey density was equal to 32 mussels per tank. Four trials are from the first experiment of Smallegange et al. (2006). Three trials are from a similar unpublished experiment. Details of the experimental procedure can be found in Smallegange et al. (2006). In all trials we tracked the sequence of behaviours (searching, handling or interfering) of both predators, resulting in five system states (as given above), and recorded all transition times. Combining the results for all seven trials (Table 2) and using the maximum-likelihood procedure described above, we arrived at estimates (including SE) of 0·0110 ± 0·00145 (s−1) for D, 0·0192 ± 0·00237 (s−1) for , 0·0185 ± 0·00245 (s−1) for , and 0·0788 ± 0·00969 (s−1) for . These results imply a searching rate for prey of 0·86 cm2 s−1, a searching rate for predators of 48 cm2 s−1, an expected handling time of 54 s, and an expected fighting time of 12·7 s. Probability plots show that the expectation that the state durations follow exponential distributions is indeed reasonable. Two examples are shown in Fig. 4. One should, however, realize that the notion of exponentially distributed searching, interference and handling times is not very stringent. A normally distributed handling time, for example, would have yielded a similar estimator, that is mean handling time would have been estimated by (y2 + 2y3)/(n21 + n32).
Table 2. Observed number of transitions in all seven ‘two-crabs-feeding-on-mussels’ trials from each of the five possible states (rows) to another one (columns). The states are S2, S1H1, H2, F2 and G2, respectively. Last column gives the total state durations in seconds
The ongoing debate on the usefulness of predator-dependent functional responses in ecological theory (Abrams & Ginzburg 2000) might become less of an arm-chair ecology exercise if more empirical studies on the importance of mutual interference in true predator-prey systems are performed, similar to those studies reviewed by Skalski & Gilliam (2001). In order to avoid the confounding effects of all sorts of external factors that are unavoidably associated with observational field studies, an experimental approach seems to be a prerequisite for such studies. For example, field studies on interference in wading birds foraging on intertidal mudflats are difficult to interpret as high competitor densities usually only occur during short periods when the tide goes out or comes in. Such high-density periods thus occur either right at the start of each six-and-a-quarter-hour feeding period (at least in Western Europe the period lasts that long), following a six-and-a-quarter-hour resting period, or at the end of it, which may seriously affect the generality of the observations taken during these short periods. The obvious alternative is the experimental approach where small numbers of predators are often a necessity. Using wading birds again as an example, the only experiments that we are aware of were performed with numbers varying between 2 and 14 (Swennen et al. 1989; Van Gils & Piersma 2004; Vahl et al. 2005). The stochastic version (i.e. framed in a continuous time Markov chain) of the Beddington–DeAngelis mutual interference model, which we developed here, is therefore a more useful and appropriate tool to analyse the results of such experiments than the deterministic version. Apart from the limited use of the deterministic model to analyse experimental results, one may even wonder to what extent the idea of mass action applies to true field situations. For systems where predators forage in small isolated patches with at most only very few competitors within the same patch, the stochastic version might also be the more appropriate tool. Simulation studies of such spatial systems might also gain credibility when using the stochastic version (or at least its equilibrium state). Since we derived an explicit expression for the functional response, the use of the stochastic version instead of the deterministic one comes at almost no (computational) costs. A final reservation about the use of the deterministic Beddington–DeAngelis function as it has been hitherto applied (equation 2), and basically the same is true for its improved version equation (4), is that these functions are not very good approximations of the proper solution presented in equation (3), when interference is not a rare event, as has been illustrated in Fig. 3.
The continuous time Markov chain model can be used in two different ways to analyse the data and estimate the four parameters of the functional response. Above, we advocated the use of a maximum likelihood procedure, which requires information on the total time spent in each state and the number of transitions between states. This implies that the behaviour of all animals in the system must be continuously recorded, a practice quite common in ethological research, but not always in ecological foraging studies (Mansour & Lipcius 1991). If information is only available on the intake rate of a focal individual, then the functional response equation, given by equation (6) W(k) = vD · Q(k − 1)/Q(k), has to be used in a nonlinear regression of intake rate versus prey and predator density. However, such an approach assumes that the Markov chain has reached its equilibrium state and that the system behaviour is no longer dependent upon the initial condition. It furthermore requires that observations are performed at multiple prey densities, a requirement that is not necessary in the first option (but nevertheless always a sensible thing to do).
Earlier we made a plea for the use of mechanistic models of mutual interference over phenomenological models, such as, for example, the Hassell–Varley model (Van der Meer & Ens 1997), and a similar point of view was recently expressed by others (Skalski & Gilliam 2001; Turchin 2003). In our view, such a plea implies that the estimation of the parameters of the mechanistic model should also be as closely linked to the underlying mechanisms as possible. For this reason, we advocate the use of behavioural observations in which all state durations and transitions are recorded instead of just measuring intake rates at various predator densities. The plea also implies that a hierarchy exists in terms of ‘closeness’ to the assumed mechanisms among the various versions of the mutual interference functional response model: eqn 6 > eqn 3 > eqn 4 > eqn 2. Of course, depending upon the application, this hierarchy should be weighed against, for example, mathematical tractability, but uncritical use of lowest ranked version, that is, the ‘traditional’ version of the Beddington–DeAngelis model can no longer be defended solely on the basis of being a mechanistically based approach.
Finally, it should be stressed that the use of mechanistically based behavioural models enables one to go beyond the very simple interference rules applied here. One possible extension is a system with different types of predators, for example, dominants and subordinates, where the first may kleptoparasitize the seconds. Work on this topic is in progress.
We thank Göran Englund and Graeme Ruxton for constructive comments. I.M. Smallegange was financially supported by a research grant from the Netherlands Organization for Scientific Research (NWO).