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

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.     

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

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.     

Stability in N-species coevolutionary systems

Stability criteria have recently been developed for coevolutionary Lotka-Volterra systems where individual fitness functions are assumed to be linear in the population state. We extend these criteria as part of a general theory of coevolution (that combines effects of ecology and evolution) based on arbitrary (i.e. nonlinear) fitness functions and a finite number of individual phenotypes. The central role of the stationary density surface where species' densities are at equilibrium is emphasized. In particular, for monomorphic resident systems, it is shown coevolutionary stability is equivalent to ecological stability combined with evolutionary stability on the stationary density surface. Also discussed is how our theory relates to recent treatments of phenotypic coevolution via adaptive dynamics when there is a continuum of individual phenotypes.     

The ideal free distribution as an evolutionarily stable state in density‐dependent population games

In classical games that have been applied to ecology, individual fitness is either density independent or population density is fixed. This article focuses on the habitat selection game where fitness depends on the population density that evolves over time. This model assumes that changes in animal distribution operate on a fast time scale when compared to demographic processes. Of particular interest is whether it is true, as one might expect, that resident phenotypes who use density‐dependent optimal foraging strategies are evolutionarily stable with respect to invasions by mutant strategies. In fact, we show that evolutionary stability does not require that residents use the evolutionarily stable strategy (ESS) at every population density; rather it is the combined resident–mutant system that must be at an evolutionary stable state. That is, the separation of time scales assumption between behavioral and ecological processes does not imply that these processes are independent. When only consumer population dynamics in several habitats are considered (i. e. when resources do not undergo population dynamics), we show that the existence of optimal foragers forces the resident‐mutant system to approach carrying capacity in each habitat even though the mutants do not die out. Thus, the ideal free distribution (IFD) for the single‐species habitat selection game becomes an evolutionarily stable state that describes a mixture of resident and mutant phenotypes rather than a strategy adopted by all individuals in the system. Also discussed is how these results are affected when animal distribution and demographic processes act on the same time scale.     

Evolutionary stability concepts for N-species frequency-dependent interactions.

The classical static concept of an evolutionarily stable strategy (ESS) for a single species gives rise to two new notions when there are more than two species (called an N-species ESS and RL-stability). The paper relates these to the dynamic stability of monomorphic and polymorphic evolutionary systems. It is shown that RL-stability implies the global asymptotic stability of either system with or without mutations. However, the N-species ESS only implies stability of the monomorphic system.     

Stability in -species coevolutionary systems

Stability criteria have recently been developed for coevolutionary Lotka–Volterra systems where individual fitness functions are assumed to be linear in the population state. We extend these criteria as part of a general theory of coevolution (that combines effects of ecology and evolution) based on arbitrary (i.e. nonlinear) fitness functions and a finite number of individual phenotypes. The central role of the stationary density surface where species’ densities are at equilibrium is emphasized. In particular, for monomorphic resident systems, it is shown coevolutionary stability is equivalent to ecological stability combined with evolutionary stability on the stationary density surface. Also discussed is how our theory relates to recent treatments of phenotypic coevolution via adaptive dynamics when there is a continuum of individual phenotypes.     

Ecological and evolutionary responses in complex communities: implications for invasions and eco‐evolutionary feedbacks

It is easier to predict the ecological and evolutionary outcomes of interactions in less diverse communities. As species are added to communities, their direct and indirect interactions multiply, their niches may shift, and there may be increased ecological redundancy. Accompanying this complexity in ecological interactions, is also complexity in selection and subsequent evolution, which may feed back to affect the ecology of the system, as species with different traits may play different ecological roles. Drawing from my own work and that of many others, I first discuss what we currently understand about ecology and evolution in light of simple and diverse communities, and suggest the importance of escape from community complexity per se in the success of invaders. Then, I examine how community complexity may influence the nature and magnitude of eco‐evolutionary feedbacks, classifying eco‐evolutionary dynamics into three general types: those generating alternative stable states, cyclic dynamics, and those maintaining ecological stasis and stability. The latter may be important and yet very hard to detect. I suggest future directions, as well as discuss methodological approaches and their potential pitfalls, in assessing the importance and longevity of eco‐evolutionary feedbacks in complex communities. Synthesis The ecology, evolution and eco‐evolutionary dynamics of simple and diverse communities are reviewed. In more diverse communities, direct and indirect interactions multiply, species’ niches often shift, ecological redundancy can increase, and selection may be less directional. Community complexity may influence the magnitude and nature of eco‐evolutionary dynamics, which are classified into three types: those generating alternative stable states, cyclic dynamics, and those maintaining ecological stasis and stability. Strengths and pitfalls of approaches to investigating eco‐evolutionary feedbacks in complex field communities are discussed.     

Species packing in eco‐evolutionary models of seasonally fluctuating environments

As ecology and evolution become ever more entwined, many areas of ecological theory are being re‐examined. Eco‐evolutionary analyses of classic coexistence mechanisms are yielding new insights into the structure and stability of communities. We examine fluctuation‐dependent coexistence models, identifying communities that are both ecologically and evolutionarily stable. Members of these communities possess distinct environmental preferences, revealing widespread patterns of limiting similarity. This regularity leads to consistent changes in the structure of communities across fluctuation regimes. However, at high amplitudes, subtle differences in the form of fluctuations dramatically affect the collapse of communities. We also show that identical fluctuations can support multiple evolutionarily stable communities – a novel example of alternative stable states within eco‐evolutionary systems. Consequently, the configuration of communities will depend on historical contingencies, including details of the adaptive process. Integrating evolution into the study of coexistence offers new insights, while enriching our understanding of ecology.     

Uninvadability in N-species frequency models for resident-mutant systems with discrete or continuous time

The uninvadability concept, that was originally introduced through static comparisons of individual fitness in resident-mutant systems for a single species, is developed for multi-species models with frequency-dependent fitness by extending its equivalent single-species dynamic characterization. This multi-species definition is then reinterpreted in terms of individual fitness functions based on intra and interspecific interactions. The resultant concept is discussed in relation to that of an N-species ESS (evolutionarily stable strategy) and to dynamic stability of monomorphic and polymorphic evolutionary systems.     

Evolutionary Dynamics of Seed Size and Seedling Competitive Ability

We present a model for the evolutionary dynamics of seed size when there is a trade-off between seed size and seed number, and seedlings from large seeds are better competitors and have a higher precompetitive survival than seedlings from small seeds. We find that strong competitive asymmetry, high resource levels, and intermediate harshness of the precompetitive environment favor coexistence of plants with different seed sizes. If the evolution of seed size is mutation-limited and single mutations have only a small phenotypic effect, then an initially monomorphic population reaches the final evolutionarily stable polymorphic state through one or more discrete evolutionary branching events. At each such branching event, a given lineage already present in the population divides into two phenotypically diverging daughter lines, each with its own seed size. If the precompetitive survival of seeds and seedlings is high for small and large seeds alike, however, evolutionary branching may be followed by the extinction of one or more lineages. Various results presented here are model-independent and point the way to a more general evolutionary bifurcation theory describing how the number and stability properties of evolutionary equilibria may change as a consequence of changes in model parameters.     

Evolution restricts the coexistence of specialists and generalists: the role of trade-off structure

Environmental variability and adaptive foraging behavior have been shown to favor coexistence of specialists and generalists on an ecological timescale. This leaves unaddressed the question of whether such coexistence can also be expected on an evolutionary timescale. In this article, we study the attainability, through gradual evolution, of specialist-generalist coexistence, as well as the evolutionary stability of such communities when allowing for immigration. Our analysis shows that the potential for specialist-generalist coexistence is much more restricted than originally thought and strongly depends on the trade-off structure assumed. We establish that ecological coexistence is less likely for species facing a trade-off between per capita reproduction in different habitats than when the trade-off acts on carrying capacities alone. We also demonstrate that coexistence is evolutionarily stable whenever it is ecologically stable but that in most cases, such coexistence cannot be reached through gradual evolution. We conclude that an evolutionarily stable community of specialists and generalists may be created only through immigration from elsewhere or through mutations of large effect. Our results highlight that trade-offs in fitness-determining traits can have counterintuitive effects on the evolution of specialization.     

Evolutionary branching of virulence in a single-infection model

This study explores the evolutionary dynamics of pathogen virulence in a single-infection model with density-dependent mortality. Although virulence is not an adaptation of the pathogen per se, it is generally believed to be an inevitable by-product of a pathogen's need to propagate and transmit to new hosts: an increase in virulence will parallel an increase in transmission efficacy. The exact characteristics of the trade-off curve defined by this relationship are important with respect to possible evolutionary scenarios. We conduct a critical function analysis, a method that exposes the evolutionary outcome resulting from trade-offs of arbitrary shape, and find that this simple model can display a wide variety of evolutionary dynamics; comprising multiple stable attractors, evolutionary repellors, and most notably, evolutionary branching points. We identify the conditions under which the different evolutionary outcomes are realised. Our analysis furthermore considers the evolution of coexisting strains, and identifies the trade-off characteristics that will support an evolutionarily stable dimorphic state. We find that an evolutionarily stable dimorphism may exist also in the absence of a branching point in the monomorphic state. The analysis reveals that an evolutionarily stable dimorphism will always be attracting and that no further branching is possible under this model. We discuss our results in relation to the dimension of the environmental feedback inherent in the model, and to results from previous studies and models of evolution of virulence.     

Evolutionary game theory as a framework for studying biological invasions

Although biological invasions pose serious threats to biodiversity, they also provide the opportunity to better understand interactions between the ecological and evolutionary processes structuring populations and communities. However, ecoevolutionary frameworks for studying species invasions are lacking. We propose using game theory and the concept of an evolutionarily stable strategy (ESS) as a conceptual framework for integrating the ecological and evolutionary dynamics of invasions. We suggest that the pathways by which a recipient community may have no ESS provide mechanistic hypotheses for how such communities may be vulnerable to invasion and how invaders can exploit these vulnerabilities. We distinguish among these pathways by formalizing the evolutionary contexts of the invader relative to the recipient community. We model both the ecological and the adaptive dynamics of the interacting species. We show how the ESS concept provides new mechanistic hypotheses for when invasions result in long- or short-term increases in biodiversity, species replacement, and subsequent evolutionary changes.     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

Stable persistence of relict populations involved evolutionary shifts of reproductive characters in the genus Tanakaea (Saxifragaceae)

Tertiary relicts often show evolutionary stasis in morphology and ecology and have been hypothesized to retain stable population sizes in refugia. However, recent studies have reported that some relicts evolutionarily shifted their physiology, ecology, and morphology and experienced various patterns of demography. To understand the historical survival of relict plants, a multidimensional study investigating the evolution of ecological and morphological traits as well as population demographic history is needed. The genus Tanakaea (Saxifragaceae) comprises two species in China and Japan. These species share most vegetative characteristics and are sometimes treated as a single species. The distribution pattern is relictual, as the populations are confined to small areas in mesic warm temperate forests less influenced by Quaternary glacial climates. Focusing on the relictual plant group, this study tested the hypotheses of evolutionary stasis and population stability in long-term refugia. Genetic analyses using plastome sequences and genome-wide single-nucleotide polymorphisms revealed divergence of the two species approximately 6.8 million years ago and strong genetic differentiation of the regional populations. Demographic analysis revealed that almost all populations retained stable population sizes during glacial–interglacial climate changes, supporting the traditional view. However, morphological assessments revealed a simultaneous shift in breeding systems (from hermaphrodite to dioecy/non-clonal to clonal reproduction) in Japanese species and intraspecific differentiation of leaf traits. Therefore, the relict species do not show evolutionary stasis in every aspect. Changes in reproductive characteristics may have contributed to their long-term in situ survival.     

Adaptation and habitat selection in the eco-evolutionary process

The struggle for existence occurs through the vital rates of population growth. This basic fact demonstrates the tight connection between ecology and evolution that defines the emerging field of eco-evolutionary dynamics. An effective synthesis of the interdependencies between ecology and evolution is grounded in six principles. The mechanics of evolution specifies the origin and rules governing traits and evolutionary strategies. Traits and evolutionary strategies achieve their selective value through their functional relationships with fitness. Function depends on the underlying structure of variation and the temporal, spatial and organizational scales of evolution. An understanding of how changes in traits and strategies occur requires conjoining ecological and evolutionary dynamics. Adaptation merges these five pillars to achieve a comprehensive understanding of ecological and evolutionary change. I demonstrate the value of this world-view with reference to the theory and practice of habitat selection. The theory allows us to assess evolutionarily stable strategies and states of habitat selection, and to draw the adaptive landscapes for habitat-selecting species. The landscapes can then be used to forecast future evolution under a variety of climate change and other scenarios.     

Evolutionarily stable strategies in food selection models with fitness sets

Most current models for optimal food selection apply to ecological and behavioural optimization. In this paper optimal food selection theory is extended to apply to evolutionary optimization. A general evolutionary model for optimal food selection must incorporate the concept of fitness sets--or that variables, changing as a result of natural selection in evolutionary time, cannot, in general, vary independently of each other. A "Charnov type" optimal food selection model with a fitness set is investigated, and evolutionarily stable strategy (ESS) solutions of the evolutionary variables (i.e., the efficiencies of using available food types) are found. From this analysis it follows that the relative frequency of various food types in the environment may, under specified conditions, influence the evolutionarily optimal diet. Secondly, the analysis demonstrates that a food type not in the optimal diet may, in evolutionary time, be added to this by becoming more abundant. Thirdly, it follows from the analysis that the ecological result of MacArthur and Pianka, that food types are worth eating even if there is competition for them, is not generally applicable when referring to an evolutionary time scale. Finally, it is pointed out that for the diet to be an ESS, it is necessary that the consumer's density is stable and that the consumer's population dynamics are subjected to some density-dependent factor.     

The adaptive dynamics of Lotka-Volterra systems with trade-offs

We analyse the adaptive dynamics of a generalised type of Lotka-Volterra model subject to an explicit trade-off between two parameters. A simple expression for the fitness of a mutant strategy in an environment determined by the established, resident strategy is obtained leading to general results for the position of the evolutionary singular strategy and the associated second-order partial derivatives of the mutant fitness with respect to the mutant and resident strategies. Combinations of these results can be used to determine the evolutionary behaviour of the system. The theory is motivated by an example of prey evolution in a predator-prey system in which results show that only (non-EUS) evolutionary repellor dynamics, where evolution is directed away from a singular strategy, or dynamics where the singular strategy is an evolutionary attractor, are possible. Moreover, the general theory can be used to show that these results are the only possibility for all Lotka-Volterra systems in which aside from the trade-offs all parameters are independent and in which the interaction terms are of quadratic order or less. The applicability of the theory is highlighted by examining the evolution of an intermediate predator in a tri-trophic model.     

On the Sympatric Evolution and Evolutionary Stability of Coexistence by Relative Nonlinearity of Competition

If two species exhibit different nonlinear responses to a single shared resource, and if each species modifies the resource dynamics such that this favors its competitor, they may stably coexist. This coexistence mechanism, known as relative nonlinearity of competition, is well understood theoretically, but less is known about its evolutionary properties and its prevalence in real communities. We address this challenge by using adaptive dynamics theory and individual-based simulations to compare community stabilization and evolutionary stability of species that coexist by relative nonlinearity. In our analysis, evolution operates on the species'' density-compensation strategies, and we consider a trade-off between population growth rates at high and low resource availability. We confirm previous findings that, irrespective of the particular model of density dependence, there are many combinations of overcompensating and undercompensating density-compensation strategies that allow stable coexistence by relative nonlinearity. However, our analysis also shows that most of these strategy combinations are not evolutionarily stable and will be outcompeted by an intermediate density-compensation strategy. Only very specific trade-offs lead to evolutionarily stable coexistence by relative nonlinearity. As we find no reason why these particular trade-offs should be common in nature, we conclude that the sympatric evolution and evolutionary stability of relative nonlinearity, while possible in principle, seems rather unlikely. We speculate that this may, at least in part, explain why empirical demonstrations of this coexistence mechanism are rare, noting, however, that the difficulty to detect relative nonlinearity in the field is an equally likely explanation for the current lack of empirical observations, and that our results are limited to communities with non-overlapping generations and constant resource supply. Our study highlights the need for combining ecological and evolutionary perspectives for gaining a better understanding of community assembly and biogeographic patterns.     

Fluctuation Induces Evolutionary Branching in a Mathematical Model of Ecosystems

Ecological systems are always subjected to various environmental fluctuations. They evolve under these fluctuations and the resulting systems are robust against them. The diversity in ecological systems is also acquired through the evolution. How do the fluctuations affect the evolutionary processes? Do the fluctuations have direct impact on the species diversity in ecological systems? In the present paper, we investigate the relation between the environmental fluctuation and the evolution of species diversity with a mathematical model of evolutionary ecology. In the model, individual organisms compete for a single restricted resource and the temporal fluctuation in the resource supply is introduced as the environmental fluctuation. The evolutionary process is represented by the mutational change of genotypes which determines their resource utilization strategies. We found that when the environmental state is switched form static to fluctuating conditions, the initial closely related population distributed around the genotype adapted for the static environment is destabilized and divided into two groups in the genotype space; i.e., the evolutionary branching is induced by the environmental fluctuation. The consequent multiple species structures is evolutionary stable at the presence of the fluctuation. We perform the evolutionary invasion analysis for the phenomena and illustrate the mechanisms of the branchings. The results indicate a novel process of increasing the species diversity via evolutionary branching, and the analysis reveals the mechanisims of the branching preocess as the response to the environmental fluctuation. The robustness of the evolutionary process is also discussed.