Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis

Abstract

On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.

Introduction

Understanding the mechanism of evolutionary diversification of consumers remains an important problem in evolutionary ecology [1]. Although it is well known that two predator species with linear functional responses can stably coexist on two competing prey species on an ecological timescale [2], [3]. It is, however, far less understood whether such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. The ecological models that demonstrate equilibrium coexistence do not address evolutionary stability, whereas the evolutionary models that show the evolutionarily stable coexistence usually fix the environmental conditions [4]. When an eco-evolutionary feedback loop is taken into account, the environmental conditions necessarily co-evolve and accordingly the spectrum of possible dynamical behavior becomes a lot richer [5]. This viewpoint affects not only the intuition of evolutionary biologists, but also their theoretical tools.

When ecological and evolutionary dynamics occur on a similar timescale, we should couple the evolutionary and population dynamics, this may lead to very complex dynamics, including periodic switching and chaos [2], [6], [7], [8], [9], [10]. However, if mutations occur infrequently such that a mutant strategy either has spread or has been excluded, and the populations have reached their ecological equilibriums by the time the next mutants come along, then the two dynamics can be decoupled. This separation of ecological and evolutionary timescales is widely used in theoretical study [11], [12], [13], [14]. In fact, in this case, the evolutionary processes can be described by an adaptive dynamical system [1], [11], [15]. Adaptive dynamics is a mathematical theory that explicitly links population dynamics to a long-term evolution driven by mutation and natural selection. A particularly intriguing phenomenon revealed by the theory of adaptive dynamics is evolutionary branching, that is, a change from a monomorphic to a dimorphic population. Evolutionary branching occurs at particular phenotypes (called branching points) that are attractors of monomorphic evolution but are not evolutionarily stable in the sense that they are not immune to invasion by mutants. Near such a phenotype, the resident and mutant populations can coexist and selection becomes disruptive, the population will become dimorphic and diverge in trait values [16]. Additional quantitative information about the speed of adaptive movement is embodied in the ‘canonical equation’ which describes how the average value of trait will change on a very long timescale when mutations are very small and rare [5], [11]. When the ecological and evolutionary timescales are separated, complicated evolutionary dynamics can also occur if the population dynamical attractor is not a fixed point [17], [18].

By using the evolutionary game theory, Brown and Vincent [19] found that, in the case of one-predator-two-prey coevolution, the evolutionary outcome depends on the magnitude of niche breadth. When predators have a broad niche breadth, the evolutionarily stable strategy contains two prey and a single predator species. If the consumer niche breadth is narrow, then the evolutionarily stable strategy may consist of two specialist predators. With the method of adaptive dynamics, Ma and Levin [1] studied the evolution of resource adaptation and found that generalist strategies evolve if there is a switching benefit; specialists evolve if there is a switching cost. White and Bowers [20] investigated the adaptive dynamics of Lotka–Volterra systems with trade-offs and found that for evolutionary branching to occur we require that one (or both) of the traded-off parameters includes an interspecific parameter dependency and that the trade-off function has weakly accelerating costs. Rueffler et al. [21] analyzed a model of one evolving consumer feeding on two resources and found that if selection is frequency dependent, then the population can become dimorphic through evolutionary branching at the trait value of the generalist. Such analysis highlights a rich lode of theoretical issues, but they need for a framework in which to address them [1]. In addition, after branching in predator phenotype, they did not address whether the coexistence of the two predator species can be maintained over evolutionary time and how this evolution depends on the trade-off function.

The present paper has two aims. One is to investigate under what conditions a predator population will change from monomorphism to dimorphism. The other is to address whether the two predator species evolving from the single ancestor can continue to coexist on the long-term evolutionary timescale. Focusing at first on a monomorphic predator population, we use the new method of critical function analysis [22] to identify the general properties of trade-off functions that can induce evolutionary branching in the predator strategy and show how this evolution depends on the strength of trade-off. Compared with previous methods by assuming a particular trade-off function [1], [21], [23], this new method can identify all possible evolutionary outcomes under various trade-offs and it is also helpful in estimating how likely these evolutionary outcomes may be. After branching has occurred in the predator phenotype, we proceed to investigate the final evolutionary outcomes of such a dimorphic predator population and identify the trade-off properties that support an evolutionarily stable dimorphism. We extend the existing theory and models studied by Ma and Levin [1] and Rueffler et al. [21]. Our main approach is based on the theory of adaptive dynamics [11], [15], [24]. In this approach, evolutionary dynamics is studied by using the concept of invasion fitness [24]. We also present a geometrical method, due to Rueffler et al. [25], for analyzing the adaptive dynamics of dimorphisms. But we limit ourselves to phenotypic evolution under clonal reproduction, that is, we simply ignore the importance of genes and sex. This paper may serve as a reader-friendly tutorial on adaptive dynamics (with special attention for the relatively new critical function analysis).

The organization of this paper is as follows. In the next section, we present the formulations of mathematical models and derive the invasion fitness for mutant predators. In Section 3, by using the method of critical function analysis, we investigate the monomorphic evolutionary dynamics. In Section 4, we study the dimorphic evolution and give two examples with evolutionarily stable coexistence. Numerical simulations are also presented respectively in Sections 3 Critical function analysis, 4 Dimorphic evolutionary dynamics to illustrate the feasibility of our main results. Brief conclusion remarks are given in Section 5.