Adaptive evolution of foraging-related traits in a predator-prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.     

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

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.     

Adaptive Evolution of Defense Ability Leads to Diversification of Prey Species

In this paper, by using the adaptive dynamics approach, we investigate how the adaptive evolution of defense ability promotes the diversity of prey species in an initial one-prey–two-predator community. We assume that the prey species can evolve to a safer strategy such that it can reduce the predation risk, but a prey with a high defense ability for one predator may have a low defense ability for the other and vice versa. First, by using the method of critical function analysis, we find that if the trade-off is convex in the vicinity of the evolutionarily singular strategy, then this singular strategy is a continuously stable strategy. However, if the trade-off is weakly concave near the singular strategy and the competition between the two predators is relatively weak, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the prey strategy, if the trade-off curve is globally concave, then the prey species might eventually evolve into two specialists, each caught by only one predator species. However, if the trade-off curve is convex–concave–convex, the prey species might eventually branch into two partial specialists, each being caught by both of the two predators and they can stably coexist on the much longer evolutionary timescale.     

Evolutionary branching and long-term coexistence of cycling predators: critical function analysis

It is well known that two predators with different functional responses can coexist on one prey when the system exhibits nonequilibrium dynamics. In this paper, we investigate under which conditions such coexistence is evolutionarily stable, and whether the two predators may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in handling time, and hence in the shape of their Holling type II functional response. Longer handling times are costly in terms of lost foraging time, but allow the predator to extract more nutrients from the prey and therefore to produce more offspring per consumed prey. In the analysis, we apply a new method to accommodate arbitrary trade-off functions between handling time and offspring production. Contrary to previous results obtained assuming a particular trade-off [Kisdi, E. and Liu, S., 2006. J. Evol. Biol. 19, 49-58], we find that evolutionary branching of handling time is possible, although it does not appear to be very likely and can be excluded for a class of trade-offs. Evolutionarily stable coexistence of two predators occurs under less restrictive conditions, which are always satisfied when the trade-off function has two strongly concave parts connected by a convex piece.     

The evolution of phenotypic traits in a predator-prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.     

Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction

It has recently been demonstrated that ecological feedback mechanisms can facilitate the emergence and maintenance of cooperation in public goods interactions: the replicator dynamics of defectors and cooperators can result, for example, in the ecological coexistence of cooperators and defectors. Here we show that these results change dramatically if cooperation strategy is not fixed but instead is a continuously varying trait under natural selection. For low values of the factor with which the value of resources is multiplied before they are shared among all participants, evolution will always favour lower cooperation strategies until the population falls below an Allee threshold and goes extinct, thus evolutionary suicide occurs. For higher values of the factor, there exists a unique evolutionarily singular strategy, which is convergence stable. Because the fitness function is linear with respect to the strategy of the mutant, this singular strategy is neutral against mutant invasions. This neutrality disappears if a nonlinear functional response in receiving benefits is assumed. For strictly concave functional responses, singular strategies become uninvadable. Evolutionary branching, which could result in the evolutionary emergence of cooperators and defectors, can occur only with locally convex functional responses, but we illustrate that it can also result in coevolutionary extinction.     

Continuously stable strategies as evolutionary branching points

Evolutionary branching points are a paradigmatic feature of adaptive dynamics, because they are potential starting points for adaptive diversification. The antithesis to evolutionary branching points are continuously stable strategies (CSS's), which are convergent stable and evolutionarily stable equilibrium points of the adaptive dynamics and hence are thought to represent endpoints of adaptive processes. However, this assessment is based on situations in which the invasion fitness function determining the adaptive dynamics have non-zero second derivatives at CSS. Here we show that the scope of evolutionary branching can increase if the invasion fitness function vanishes to higher than first order at CSS. Using classical models for frequency-dependent competition, we show that if the invasion fitness vanishes to higher orders, a CSS may be the starting point for evolutionary branching. Thus, when invasion fitness functions vanish to higher than first order at equilibrium points of the adaptive dynamics, evolutionary diversification can occur even after convergence to an evolutionarily stable strategy.     

The geometric theory of adaptive evolution: trade-off and invasion plots

The purpose of this paper is to take an entirely geometrical path to determine the evolutionary properties of ecological systems subject to trade-offs. In particular we classify evolutionary singularities in a geometrical fashion. To achieve this, we study trade-off and invasion plots (TIPs) which show graphically the outcome of evolution from the relationship between three curves. The first invasion boundary (curve) has one strain as resident and the other strain as putative invader and the second has the roles of the strains reversed. The parameter values for one strain are used as the origin with those of the second strain varying. The third curve represents the trade-off. All three curves pass through the origin or tip of the TIP. We show that at this point the invasion boundaries are tangential. At a singular TIP, in which the origin is an evolutionary singularity, the invasion boundaries and trade-off curve are all tangential. The curvature of the trade-off curve determines the region in which it enters the singular TIP. Each of these regions has particular evolutionary properties (EUS, CS, SPR and MI). Thus we determine by direct geometric argument conditions for each of these properties in terms of the relative curvatures of the trade-off curve and invasion boundaries. We show that these conditions are equivalent to the standard partial derivative conditions of adaptive dynamics. The significance of our results is that we can determine whether the singular strategy is an attractor, branching point, repellor, etc. simply by observing in which region the trade-off curve enters the singular TIP. In particular we find that, if and only if the TIP has a region of mutual invadability, is it possible for the singular strategy to be a branching point. We illustrate the theory with an example and point the way forward.     

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.     

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.     

The evolution of resource use

 The evolution of a consumer exploiting two resources is investigated. The strategy x under selection represents the fraction of time or energy an individual invests into extracting the first resource. In the model, a dimensionless parameter α quantifies how simultaneous consumption of both resources influences consumer growth; α<0 corresponds to hemi-essential resources, 0<α<1 corresponds to complementary resources, α=1 corresponds to perfectly substitutable resources, and α>1 corresponds to antagonistic resources. An analysis of the ecological and evolutionary dynamics leads to five conclusions. First, when α≤1, there is a unique singular strategy x ^* for the adaptive dynamics and it is evolutionarily stable and globally convergent stable. Second, when α=1, the singular strategy x ^* corresponds to the populations exhibiting an ideal free distribution and a population playing this strategy can invade and displace populations playing any other strategy. Third, when α>1, the strategies x=0 and x=1 are evolutionarily stable and convergent stable. Hence, if the populations initially specialize on one resource, evolution amplifies this specialization. Fourth, when α is slightly larger than one (i.e. the resources are slightly antagonistic), there is a convergent stable singular strategy whose basin of attraction is almost the entire strategy space (0,1). This singular strategy is evolutionarily unstable and serves as an evolutionary branching point. Following evolutionary branching, our analysis and numerical simulations suggest that evolutionary dynamics are driven toward an end state consisting of two populations specializing on different resources. Fifth, when α>>1, there is only one singular strategy and it is convergent unstable and evolutionarily unstable. Hence, if resources are overly antagonistic, evolutionary branching does not occur and ultimately only one resource is exploited. Received: 8 June 2002 / Revised version: 28 November 2002 / Published online: 23 April 2003 This work was supported by NSF Grant DMS-0077986 Key words or phrases: Consumer-resource interactions – Adaptive dynamics – Evolutionary branching     

Evolution of conditional dispersal: evolutionarily stable strategies in spatial models

We consider a two-species competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be three or more evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.     

Evolution of handling time can destroy the coexistence of cycling predators

Several consumers (predators) with Holling type II functional response may robustly coexist even if they utilize the same resource (prey), provided that the population exhibits nonequilibrium dynamics and the handling time of predators is sufficiently different. We investigate the evolution of handling time and, in particular, its effect on coexistence. Longer handling time is costly in terms of lost foraging time, but allows more nutrients to be extracted from a captured prey individual. Assuming a hyperbolically saturating relationship between handling time and the number of new predators produced per prey consumed, we obtain three results: (i) There is a globally evolutionarily stable handling time; (ii) At most two predator strategies can coexist in this model; (iii) When two predators coexist, a mutant with intermediate handling time can always invade. This implies that there is no evolutionarily stable coexistence, and the evolution of handling time eventually leads to a single evolutionarily stable predator. These results are proven analytically and are valid for arbitrary (not only small) mutations; they however depend on the relationship between handling time and offspring production and on the assumption that predators differ only in their prey handling strategy.     

Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree

We present a general framework for modelling adaptive trait dynamics in which we integrate various concepts and techniques from modern ESS-theory. The concept of evolutionarily singular strategies is introduced as a generalization of the ESS-concept. We give a full classification of the singular strategies in terms of ESS-stability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategy's neighbourhood. Of particular interest is a type of singular strategy that is an evolutionary attractor from a great distance, but once in its neighbourhood a population becomes dimorphic and undergoes disruptive selection leading to evolutionary branching. Modelling the adaptive growth and branching of the evolutionary tree can thus be considered as a major application of the framework. A haploid version of Levene's soft selection model is developed as a specific example to demonstrate evolutionary dynamics and branching in monomorphic and polymorphic populations.     

The Balance of Terror: An Alternative Mechanism for Competitive Trade-Offs and Its Implications for Invading Species

This article uses models to propose an explanation for three observations in community ecology: the apparent overreaction of prey to attack by specialist predators, the existence of a common trade-off among components of competitive ability in communities of unrelated competitors, and the ability of invading species to break the native trade-off. Strategies that increase resource collection ability are assumed to increase vulnerability to attack by specialist consumers according to a vulnerability function. If competitors compete for a common resource and share the same form of the vulnerability function, then they are favored to converge on the same evolutionarily stable level of competitiveness or trade-off curve even if the parameters describing their specialized consumers differ. The position of the common strategy or trade-off curve depends on the whole guild, with more speciose guilds tending to favor higher levels of competitiveness. Invaders can break the native trade-off if they come from a guild with a higher trade-off curve, an effect possibly enhanced evolutionarily by escape from specialist consumers.     

The evolutionary stability of automimicry

Internal defences such as toxins cannot be detected from a distance by a predator, and are likely to be costly to produce and maintain. Populations of well-defended prey may therefore be vulnerable to invasion from rare 'cheater' mutants that do not produce the toxin themselves but obtain some protection from their resemblance to their better defended conspecifics (automimicry). Although it is well established that well-defended and weakly defended morphs may coexist stably in protected dimorphisms, recent theoretical work suggests that such dimorphisms would not be resistant to invasion by novel mutants with defence levels intermediate to those present. Given that most defences (including toxins) are likely to be continuous traits, this implies that automimicry may tend to be a transitory phenomenon, and thus less likely to explain variation in defence levels in nature. In contrast to this, we show that automimicry can also be evolutionarily stable for continuous traits, and that it may evolve under a wide range of conditions. A recently developed geometric method allows us to determine directly from a trade-off curve whether an evolutionarily stable defence dimorphism is at all possible, and to make some qualitative inferences about the ecological conditions that may favour it.     

A comparison of foraging strategies in a patchy environment.

In this paper we compare foraging strategies that might be used by predators seeking prey in a patchy environment. The strategies differ in the extent to which predators aggregate in response to prey density. The approach to the comparison is suggested by the idea of evolutionarily stable strategies. A strategy is said to be evolutionarily stable if it cannot be invaded by another strategy. Thus we examine scenarios where a small number of individuals using one strategy are introduced into a situation where a large number of individuals using the other strategy are already present. However, our foraging models do not explicitly incorporate predator population dynamics, so we use net energy uptake as a surrogate for reproductive fitness. In cases where all of the patches visited by predators sustain prey populations, we find that for any pair of strategies one of them will have a higher net energy uptake than the other whether it is the resident or the introduced strain. However, which one is higher will typically depend on the total predator population, which is determined by the resident strain. If the predators leave prey densities high, the more aggregative strain will have the advantage. If the predators reduce prey densities to low levels the less aggregative strain will have the advantage. In cases where one strain of predators aggregates in response to prey density and the other does not, then there might be patches which do not contain prey but do contain (non-aggregating) predators. In those cases, there is the possibility that whichever strategy is used by the introduced strain will yield a higher energy uptake than that used by the resident strain. This suggests that if some patches are empty of prey then aggregative and non-aggregative strategies may be able to coexist.     

Dimorphism by Singularity Theory in a Model for River Ecology

Geritz, Gyllenberg, Jacobs, and Parvinen show that two similar species can coexist only if their strategies are in a sector of parameter space near a nondegenerate evolutionarily singular strategy. We show that the dimorphism region can be more general by using the unfolding theory of Wang and Golubitsky near a degenerate evolutionarily singular strategy. Specifically, we use a PDE model of river species as an example of this approach. Our finding shows that the dimorphism region can exhibit various different forms that are strikingly different from previously known results in adaptive dynamics.     

The Evolution of Resource Adaptation: How Generalist and Specialist Consumers Evolve

Why and how specialist and generalist strategies evolve are important questions in evolutionary ecology. In this paper, with the method of adaptive dynamics and evolutionary branching, we identify conditions that select for specialist and generalist strategies. Generally, generalist strategies evolve if there is a switching benefit; specialists evolve if there is a switching cost. If the switching cost is large, specialists always evolve. If the switching cost is small, even though the consumer will first evolve toward a generalist strategy, it will eventually branch into two specialists.     

Evolutionary Behaviour,Trade-Offs and Cyclic and Chaotic Population Dynamics

Many studies of the evolution of life-history traits assume that the underlying population dynamical attractor is stable point equilibrium. However, evolutionary outcomes can change significantly in different circumstances. We present an analysis based on adaptive dynamics of a discrete-time demographic model involving a trade-off whose shape is also an important determinant of evolutionary behaviour. We derive an explicit expression for the fitness in the cyclic region and consequently present an adaptive dynamic analysis which is algebraic. We do this fully in the region of 2-cycles and (using a symbolic package) almost fully for 4-cycles. Simulations illustrate and verify our results. With equilibrium population dynamics, trade-offs with accelerating costs produce a continuously stable strategy (CSS) whereas trade-offs with decelerating costs produce a non-ES repellor. The transition to 2-cycles produces a discontinuous change: the appearance of an intermediate region in which branching points occur. The size of this region decreases as we move through the region of 2-cycles. There is a further discontinuous fall in the size of the branching region during the transition to 4-cycles. We extend our results numerically and with simulations to higher-period cycles and chaos. Simulations show that chaotic population dynamics can evolve from equilibrium and vice-versa.