A predator–prey refuge system: Evolutionary stability in ecological systems

Abstract

A refuge model is developed for a single predator species and either one or two prey species where no predators are present in the prey refuge. An individual’s fitness depends on its strategy choice or ecotype (predators decide which prey species to pursue and prey decide what proportion of their time to spend in the refuge) as well as on the population sizes of all three species. It is shown that, when there is a single prey species with a refuge or two prey species with no refuge compete only indirectly (i.e. there is only apparent competition between prey species), that stable resident systems where all individuals in each species have the same ecotype cannot be destabilized by the introduction of mutant ecotypes that are initially selectively neutral. In game-theoretic terms, this means that stable monomorphic resident systems, with ecotypes given by a Nash equilibrium, are both ecologically and evolutionarily stable. However, we show that this is no longer the case when the two indirectly-competing prey species have a refuge. This illustrates theoretically that two ecological factors, that are separately stabilizing (apparent competition and refuge use), may have a combined destabilizing effect from the evolutionary perspective. These results generalize the concept of an evolutionarily stable strategy (ESS) to models in evolutionary ecology. Several biological examples of predator–prey systems are discussed from this perspective.

Introduction

It is well recognized that there is a fundamental connection between ecology and evolution, expressed succinctly through the metaphor “The ecological theater and the evolutionary play” by Hutchinson (1976). For instance, ecological traits that describe animal behavior such as habitat usage and foraging strategy are the object and result of natural selection. Thus, to study the stability of an ecological system, one must not only consider ecological perturbations that disturb the densities of species in the ecosystem but also evolutionary perturbations when new mutant ecotypes arise and potentially invade the ecosystem. When these systems contain several interacting species, the fitness of any individual will depend on its own ecotype as well as on those of all other interacting individuals which coexist in this ecosystem. That is, the evolutionary play contains an underlying game-theoretical model (c.f. Brown et al., 1999).

Evolutionary game theory is one of the most successful tools for understanding the evolution of animal behavior when population densities are fixed. Its classical stability notion of an evolutionarily stable strategy (ESS) demands that there is no mutant behavior which can invade a resident population where individuals behave according to the ESS (Maynard Smith, 1982). In population ecology, it is also natural to ask whether new “ecotypes” can successfully invade an existing stable resident ecosystem. In previous work (Cressman and Garay, 2003a, Cressman and Garay, 2003b), we introduced a multi-species density-dependent evolutionary stability notion in a general theoretical setting which requires that all possible mutant ecotypes will die out under the ecological dynamics. This notion of evolutionary stability is based on two main assumptions; namely, that the behavior of each individual is passed on to its offspring and that mutations are rare.¹ Both of these assumptions are generally accepted in models of evolutionary ecology (e.g. Marrow et al., 1996, Vincent and Brown, 2005, Cressman and Garay, 2006, Kun and Scheuring, 2006). Moreover, there is empirical evidence that genetic inheritance plays an important role in prey foraging and predator search behaviors (e.g. Hughes and Taylor, 1997, Lister and Neff, 2006) and that prey response to predator behavior is an inherited trait (Abjörnsson et al., 2004).

In order to have an evolutionary play in an ecological theater, at least one of the species must have two (or more) behavioral choices (called two pure strategies in game-theoretic terminology). In our predator–prey models, either prey have a refuge (and so a behavioral choice of whether or not to use their refuge) or there are two prey species (in which case the predator has a choice of which prey to search for). In fact, our final model (Section 4.1) combines both aspects since it consists of one predator species and two prey species, each with a refuge. From an ecological point of view, this is a minimal food web with three species where each species acts in an evolutionary play.

Our approach to study stability in these multi-species predator–prey ecosystems is an evolutionary ecological one that combines the game-theoretic ESS approach of Maynard Smith (1982) with ecological dynamics. We assume that the ecotype of each individual in the system that belongs to a species engaged in an evolutionary play is characterized by the amount of time spent in each habitat (for a prey) and the dietary intake (for a predator). By introducing rare mutant ecotypes for each of these species into a monomorphic resident system, we examine whether ecological selection will eventually fix the best ecotypes in each species. After a brief general discussion in Section 2 of models (based on multi-habitat multi-species predator–prey systems) that combine evolution and ecology as well as of literature related to refuge systems, Sections 3 The refuge model with one prey species, 4 The model with one predator and two prey populations deal with specific models that incorporate prey refuges.

In Section 3, where there is a single prey species, we find that the refuge can stabilize a predator–prey system that would be unstable without it but can never destabilize an otherwise stable system. This phenomenon is well-known for ecological models (McNair, 1987). What is new here is that the stabilizing effect of a refuge continues to operate when behavioral evolution is combined with ecology. In Section 4, where we assume that there is no direct competition between the two prey species, we show that evolutionary effects cannot destabilize a stable ecosystem when there is no refuge. However, we go on to show that combining a refuge with apparent competition (Section 4.1) may produce instability, a surprising result given the stabilizing effect of either factor on its own. The results and methods are summarized and discussed further in the final section.