Stochastic modelling of prey depletion processes
Introduction
The number of prey a predator kills per unit of time, termed the functional response (Solomon, 1949), is a central parameter for population dynamics, as its form governs the global stability of systems. In the biocontrol context, its shape can determine whether a predator is able to regulate its prey (Murdoch and Oaten, 1975).
A classical terminology for functional response types was proposed by Holling (1959). Apart from a few ecologists (see, e.g. Turchin, 2003, p. 82), most authors use Holling's classification of functional responses into three types, which can in principle be distinguished by plotting the functional response against the number of available prey. A type I functional response increases linearly up to a maximum and then remains constant. This form of predation seems to occur very rarely in nature and is thus almost always ignored for modelling. A type II functional response has a concave shape: it increases smoothly and tends asymptotically to a constant with a negative second derivative. In type III functional responses, the second derivative is first positive and then negative, but the curve is also asymptotically constant. Such a functional response is often called sigmoid or S-shaped (Fig. 1).
Generally, a type III functional response is said to reflect the ability of a predator to “learn” or to “switch” prey. Such predators focus on alternative prey species when the density of the focal species is low, but switch to hunting it when this density becomes high (Murdoch, 1969). These generalist predators, unlike specialist ones, are desirable for biocontrol, so it is of high interest to determine whether a given predator has a type II or type III functional response.
Not only did Holling (1959) classify functional response types, but he also proposed models for the functional response, under which the predator is either searching for or is “handling” a prey. The two parameters involved in Holling's model are the searching rate, sometimes also called the attack rate, and interpreted as the area covered by a predator per unit of time, and the handling time, understood as the time needed for chasing, killing and eating a single prey. The general model iswhere N and P represent the prey and the predator density, respectively, and and denote the searching rate and the handling time.
When is constant, the Holling type II functional response arises, whereas for type III the situation is less clear. Although Hassell (1978, p. 40) and more recently Juliano (2001) claim that type III functional responses can arise whenever increases with the prey density N, this assertion is misleading. Even for the very general searching function of Juliano (2001),the monotone increase condition is not sufficient. As explained in Appendix A, the supplementary condition assures type III functional response in the general model (Eq. (1)).
Many standard functional response models can be derived from the general model (Eq. (1)) with a Juliano searching rate function (Eq. (2)). The model proposed by Hassell (1978, p. 45) arises when . When and , the functional response becomes a standard form of Holling type II functional response, whereas when and , it reduces to a simple form of Holling type III functional response (Gotelli, 2001, p. 137). In the extreme case where every parameter is zero except d, the functional response is said to be linear.
Functional response modelling is itself exploited in the broader field of prey–predator system modelling. The classical models are represented using systems of ordinary differential equations (ODEs) such aswhere the functions and stand for the intrinsic dynamics of each respective population, and the functions and represent the functional response and the numerical response (Gotelli, 2001, pp. 127–128). As the numerical response is usually considered to be simply a fraction of the functional response and the intrinsic functions are easy to model, research has mainly focused on the functional response.
Almost all the (bitrophic) models developed so far in population dynamics share this common structure.
Although ODE systems have given valuable insight into population dynamics, they have many drawbacks. The variables and are taken to be continuous but represent discrete numbers of predators and prey. When the abundances are low, however, the distinction may be important. Moreover, the trajectories of populations under ODE models are deterministic: if the initial conditions are given, the dynamics are entirely known. Nature is not like this, so stochastic models may better account for the random nature of secretive processes like predation. In their book, Curry and Feldman (1987, p. 5) wrote: “In general, a stochastic model is more realistic and facilitates a proper validation analysis, but [] is at present a greatly needed but much underdeveloped research area”. More than 20 years later, the situation has changed overall, but although many biologists use stochastic models, most areas of ecological modelling remain deterministic. In the modelling of functional response, Fenlon and Faddy (2006) reported only two early contributions resting on stochastic theory (Curry and DeMichele, 1977, Curry and Feldman, 1979).
The present paper integrates the classical Holling models for functional responses into a stochastic prey depletion model, thus enabling a stochastic reappraisal of data first analysed through an ODE model. We will show it provides other interesting and useful results, all likelihood-based and thus inaccessible through ODE models.
Section snippets
Data
Many classical analyses of the functional response have used the data produced by Hassell et al. (1977) in laboratory conditions. The data from the experiment conducted by Schenk and Bacher (2002) that we shall use are of another kind. They studied the depletion of the shield beetle Cassida rubiginosa due to its exclusive predator, the paper wasp Polistes dominulus, in natural conditions. For the field biologist, this prey–predator system has two advantages: nearly 100% of the predation events
Classical approach based on ODE
Schenk and Bacher (2002) used the functional response model proposed by Juliano (2001), which improves Holling's (1959) original model by taking into account the continuous depletion of prey throughout the experiment (suggested by Rogers, 1972) and by using a very general function for modelling the searching rate (see Eq. (2)). The number of prey eaten per predator iswhere stands for the initial number of prey and T stands for the duration of the experiment.
Model comparison and parameter estimates
The full, 4-parameter, Holling–Juliano model results in maximum log-likelihoods of and of when fitted to the data for the years 2000 and 2001, respectively. The traditional Holling type II model, where the parameters b and c are zero, gives a far poorer fit, with maximum log-likelihoods equal to and (likelihood ratio test, and ). Removing the parameters c and d does not change the fit significantly, since the maximum log-likelihoods are and
Discussion
The stochastic approach is an appealing way to model natural processes, and can be regarded as a good alternative to classical modelling based on ODEs. From a conceptual point of view, it provides a systematic approach to model biological phenomena such as the dynamics of small populations, which are naturally represented as random. From a practical point of view, the possibility of building a likelihood function enables statistical analyses that would be impossible with an ODE model.
The
Acknowledgement
This work was funded by the Swiss National Science Foundation (FNS) through the National Centre of Competence in Research “Plant Survival” (http://www2.unine.ch/nccr).
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2023, Ecological ModellingCitation Excerpt :Thus according to Rogers, parasitoids such as female wasps do not face host depletion, which occurs only during predation. Other later authors such as Morales-Ramos, et al. (1996), Kratina et al. (2009), Clerc et al. (2009), Okuyama & Ruyle (2011), and Okuyama (2013), always approached the subject from the side of the predator. According to the Random Parasite Equation, any host can accommodate an infinite number of parasitoid eggs without the parasitoid noticing it.