Elsevier

Ecological Modelling

Volume 166, Issues 1–2, 1 August 2003, Pages 135-146
Ecological Modelling

Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability

https://doi.org/10.1016/S0304-3800(03)00131-5Get rights and content

Abstract

In this work, we use an analytical approach to study the dynamic consequences of the simplest forms of refuge use by the prey. Although this problem is not new, there are surprisingly few intents to clarify the role of prey refuges in simple predator–prey models other than the original Lotka–Volterra equations. Here we incorporate prey refuges in a widely known continuous model that satisfies the principle of biomass conversion. We will evaluate the effects with regard to the local stability of equilibrium points in the first quadrant, equilibrium density values, and the long-term persistence of the populations.

We show that there is a trend from limit cycles through non-zero stable points up to predator extinction and prey stabilizing at high densities. This transitions occur as hidden prey increase in number or proportion, and/or increases the ratio of mortality to conversion efficiency of predators. The domains of stability in terms of the parameter space differ between the two modes of refuge use analyzed.

Introduction

The study of the consequences of individual behavior on the population phenomena can be recognized as a major issue in contemporary theoretical ecology. Nevertheless, linking behavior to population dynamics has received comparatively little attention (Real and Levin, 1991).

In the frame of predator–prey systems, most of the empirical and theoretical work has considered the behavior of predators and its implications on the population dynamics (Harrison, 1979, Holling, 1959, Murdoch and Oaten, 1975, Ruxton, 1995, Sih et al., 1988). At a comparatively smaller extent, the hiding behavior of preys has been incorporated as a new ingredient of simple predator–prey models and its major consequences on the system stability have been studied. This was initially done by modifying the original Lotka–Volterra predator–prey equations and the most widely reported conclusion was the community equilibrium being stabilized by the addition of refuges for preys, and prey extinction being prevented (Harrison, 1979, Holling, 1959, Maynard Smith, 1974, Murdoch and Oaten, 1975, Sih, 1987).

The traditional ways in which the effect of refuge use by the preys has been incorporated in predator–prey models is to consider either a constant number or a constant proportion of the prey population being protected from predation (Taylor, 1984). The effect itself of refuge use (and probably of any antipredator strategy) on the population growth is complex in nature, but for modeling purposes it can be understood as constituted by two components: the first one is a primary effect, comprising the reduction of prey mortality due to reduction in predation success. Therefore, primary effects affect positively the population growth of preys and negatively that of predators. Secondary effects could include trade-offs and by-products of the prey behavior that could be either advantageous or detrimental for the involved populations. A classic secondary effect is the reduction in prey’s birth rate due to the sub-optimal states of resources and/or conditions in the refuge. Furthermore, a different kind of effects—indirect effects—could arise in more complex food-web models as a result of the feedback structure of the system.

More complex models incorporating prey refuges have been introduced and analyzed by McNair (1986), Sih (1987), Ruxton (1995), and Scheffer and de Boer (1995), among others, which implicitly assume the existence of cost on the prey growth rate, and other secondary effects such as predator dependence of refuge use. The conclusions commonly are referred to as changes in the stability properties of the system explained by the addition of refuge. Nevertheless, the results are often ambiguous and difficult to interpret in biological terms.

An accepted strategy is to study the most simple but plausible models before to move toward more complex ones if empirical or theoretical evidence justifies this. On the basis of the Lotka–Volterra formulation, more realism has been added to obtain simple predator–prey models, through including self-limitation in the lowest-level population, and by making saturating the functional and numerical responses (May, 1974). Nevertheless, to our knowledge there are surprisingly few studies in which the primary effect of refuge is incorporated into simple continuous predator–prey models other than the original Lotka–Volterra.

An exception is the work of Collings (1995), which introduces and analyzes a model, attributed to May (1973) in its original form, where the population growth of both preys and predators is logistic in absence of predation, and the functional response is hyperbolic. They incorporate the refuge as a constant fraction of the prey, by which the predator carrying capacity and the prey mortality due to predation are affected. The logistic model used by Collings belongs to a family that do not conform to the principle of biomass conversion. That is, the functional and numerical responses are not explicitly related. The model we will use here to incorporate the primary refuge effect, on the other hand, is a natural extension of the Lotka–Volterra model with the inclusion of prey self-limitation and a Holling II functional response. Here the predator reproductive rate responds only to the rate of prey killed per predator, thus obeying the principle of biomass conversion (Ginzburg, 1988). This model has a long tradition in theoretical ecology (Ginzburg, 1988, May, 1974, Maynard Smith, 1974, Murray, 1989, Yodzis, 1989) and a systematic study of the effects of prey behavior on the system dynamics should not disregard its use as a starting point.

Recognizing that there is a huge variety of predator–prey models in the ecological literature, those best known and understood in mathematical and biological terms are likely to be the Lotka–Volterra model, the May model, and the Rosenzweig–MacArthur model (Rosenzweig, 1971) which we use here. As mentioned above, the theoretical study of the population consequences of prey’s refuge in its simplest form has been done on the first two only and this work intends to fill this gap.

For consistency with the previous works on this field, we will consider refuge as an environmental place where predation rate is lower. Likewise, we will consider, as usual, the effect of having a constant number of prey and a constant proportion of preys using refuges. The mathematical analysis will be done separately in each case.

We will evaluate the effects of refuge use by the prey, through the analysis of the following model responses: (a) local stability of equilibrium points in the first quadrant, (b) equilibrium density values, and (c) long-term persistence of each population.

Section snippets

The basic model

We assume the populations sizes changing continuously with time, uniform distribution over space and neither age or sex structure. The model, we will analyze here belong to the general Gause model (Freedman, 1980), which is represented by the second-order differential equations system:X:dxdt=xg(x)−yφ(x)dydt=(pφ(x)−c)y

We denote by x=x(t) the population size of prey (measured in biomass or number density), and by y=y(t) the population size of predators. The following assumptions are implicit in

A constant proportion of prey using refuges

When considering xr=βx, the model that represent this situation is given by the Kolmogorov type system:Xμβ:dxdt=r1−xKq(1−β)y(1−β)x+axdydt=bp(1−β)x(1−β)x+a−cywhich can be written asXμβ:dxdt=r1−xKqyx+(a/(1−β))xdydt=bpxx+(a/(1−β))−cythat is, the only change relative to system (2) is the new value of the half saturation constant a′=(a/(1−β)). It is easy to see that the dynamics of the system (3) is topologically equivalent to the original system (Andronov et al., 1973, Sotomayor, 1979). When β→0,

A constant number of prey using refuges

A different dynamics is expected when a fixed quantity of prey γ>0 uses refuges, since the y-axis is not an invariant set. Under this assumption, we get the systemXμγ:dxdt=r1−xKx−q(x−γ)yx−γ+adydt=bp(x−γ)x−γ+a−cywhere the equilibrium points are O=(0,0), PK=(K,0) and Pe=(xe,ye) with xe=(ac+(pc)γ/(pc)) and ye=(pr(ac+(pc)γ)((Kγ)(pc)−ac)/cqK(pc)2).

Different cases appears, for γ>a, γ=a and γ<a. If (Kγ)(pc)−ac=0, the point Pe collapses with PK. We distinguish two cases in this situation: γa

Discussion

Earlier theoretical work suggest that the use of refuges by prey has a stabilizing effect on the predator–prey dynamics, particularly when a fixed number of hidden prey is considered (Maynard Smith, 1974, Murdoch and Oaten, 1975, Harrison, 1979, Sih, 1987, Ives and Dobson, 1987, Ruxton, 1995). Nevertheless, other results in the context of different models show no such simple pattern (McNair, 1986, Collings, 1995). On the other hand, it has been shown that refuges could also increase the

Acknowledgements

This work was supported by grants FONDECYT 1010399 and DI UCV No. 124 778/2001 to E.G.O. and FONDECYT 3000051 to R.R.J. The authors wish to thank Dr. Jaime Mena-Lorca, Dr. Jorge González-Guzmán and other members of the Mathematical Ecology Group of the Catholic University of Valparaı́so, for their valuable comments and suggestions.

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