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.     

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.     

Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games

We study the evolutionary effect of rare mutations causing global changes in traits. We consider asymmetric binary games between two players. The first player takes two alternative options with probability x and 1−x; and the second player takes options with probability y and 1−y. Due to natural selection and recurrent mutation, the population evolves to have broad distributions of x and y. We analyze three cases showing qualitatively different dynamics, exemplified by (1) vigilance-intrusion game, (2) asymmetric hawk-dove game and (3) cleaner-client game. We found that the evolutionary outcome is strongly dependent upon the distribution of mutants’ traits, more than the mutation rates. For example in the vigilance-intrusion game, the evolutionary dynamics show a perpetual stable oscillation if mutants are always close to the parent (local-mutation mode), whilst the population converges to a stable equilibrium distribution if mutants can be quite different from the parent (global-mutation mode), even for extremely low mutation rate. When common local mutations and rare global mutations occur simultaneously, the evolutionary outcome is controlled by the latter.     

Mutation in evolutionary games can increase average fitness at equilibrium

We study game dynamical interactions between two strategies, A and B, and analyse whether the average fitness of the population at equilibrium can be increased by adding mutation from A to B. Classifying all two by two games with payoff matrix [(a,b),(c,d)], we show that mutation from A to B enhances the average fitness of the whole population (i) if both a and d are less than (b + c)/2 and (ii) if c is less than b. Furthermore, we study conditions for maximizing the productivity of strategy A, and we analyse the effect of mutations in both directions. Depending on the biological system, a mutation in an evolutionary game can be interpreted as a genetic alteration, a cellular differentiation, a change in gene expression, an accidental or deliberate modification in cultural transmission, or a learning error. In a cultural context, our results indicate that the equilibrium payoff of the population can be increased if players sometimes choose the strategy with lower payoff. In a genetic context, we have shown that for frequency-dependent selection mutation can enhance the average fitness of the population at equilibrium.     

Cross entropy minimization in uninvadable states of complex populations

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an order stabilizing effect.     

The good, the bad and the discriminator--errors in direct and indirect reciprocity

This paper presents, in a series of simple diagrams, concise results about the replicator dynamics of direct and indirect reciprocity. We consider repeated interactions between donors and recipients, and analyse the relationship between three basic strategies for the donor: unconditional cooperation, all-out defection, and conditional cooperation. In other words, we investigate the competition of discriminating and indiscriminating altruists with defectors. Discriminators and defectors form a bistable community, and hence a population of discriminators cannot be invaded by defectors. But unconditional altruists can invade a discriminating population and 'soften it up' for a subsequent invasion by defectors. The resulting dynamics exhibits various forms of rock-paper-scissors cycles and depends in subtle ways on noise, in the form of errors in implementation. The probability for another round (in the case of direct reciprocity), and information about the co-player (in the case of indirect reciprocity), add further elements to the ecology of reciprocation.     

Evolutionary dynamics for bimatrix games: A Hamiltonian system?

We review some properties of the evolutionary dynamics for asymmetric conflicts, give a simplified approach to them, and present some new results on the stability and bifurcations occurring in these conservative systems. In particular, we compare their dynamics to those of Hamiltonian systems     

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646-650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency , strategy α fixates in a β-population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.     

Game Dynamics with Learning and Evolution of Universal Grammar

We investigate a model of language evolution, based on population game dynamics with learning. First, we examine the case of two genetic variants of universal grammar (UG), the heart of the human language faculty, assuming each admits two possible grammars. The dynamics are driven by a communication game. We prove using dynamical systems techniques that if the payoff matrix obeys certain constraints, then the two UGs are stable against invasion by each other, that is, they are evolutionarily stable. Then, we prove a similar theorem for an arbitrary number of disjoint UGs. In both theorems, the constraints are independent of the learning process. Intuitively, if a mutation in UG results in grammars that are incompatible with the established languages, then the mutation will die out because mutants will be unable to communicate and therefore unable to realize any potential benefit of the mutation. An example for which these theorems do not apply shows that compatible mutations may or may not be able to invade, depending on the population's history and the learning process. These results suggest that the genetic history of language is constrained by the need for compatibility and that mutations in the language faculty may have died out or taken over due more to historical accident than to any straightforward notion of relative fitness. MSC 1991: 37N25 · 92D15 · 91F20     

Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

Persistence of predator-prey systems in an uncertain environment

Summary The time derivatives of prey and predator populations are assumed to satisfy a set of inequalities, instead of a precise differential equation, reflecting an uncertain environmental and/or lack of knowledge by the modeler. A system of differential equations is found whose solution gives the boundary of a persistent set, which is positive flow invariant for any system satisfying the inequalities. Conditions are given for the persistent set to be bounded away from both axes, which show that resonance effects cannot drive either predator or prey to extinction if that does not happen for an autonomous system satisfying the inequalities. In general predator-prey systems are more persistent when there is strong asymptotic stability, when there is correlation between prey and predator dynamics, when the effect of perturbations is density dependent, and are more persistent under perturbations of the prey than of the predator.     

How can we model selectively neutral density dependence in evolutionary games

The problem of density dependence appears in all approaches to the modelling of population dynamics. It is pertinent to classic models (i.e., Lotka-Volterra's), and also population genetics and game theoretical models related to the replicator dynamics. There is no density dependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the question arises: How is unlimited population growth suppressed in frequency-dependent models? Two categories of solutions can be found in the literature. In the first, replicator dynamics is independent of background fitness. In the second type of solution, a multiplicative suppression coefficient is used, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory and basic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reaches the maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitly consider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium value of population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectories is the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and then remains constant and dependent on the rate of turn-over of individuals.     

A stochastic model of evolutionary dynamics with deterministic large-population asymptotics

An evolutionary birth-death process is proposed as a model of evolutionary dynamics. Agents residing in a continuous spatial environment X, play a game G, with a continuous strategy set S, against other agents in the environment. The agents’ positions and strategies continuously change in response to other agents and to random effects. Agents spawn asexually at rates that depend on their current fitness, and agents die at rates that depend on their local population density. Agents’ individual evolutionary trajectories in X and S are governed by a system of stochastic ODEs. When the number of agents is large and distributed in a smooth density on (X,S), the collective dynamics of the entire population is governed by a certain (deterministic) PDE, which we call a fitness-diffusion equation.     

基于比率的具有反馈控制的捕食系统的持久性与全局渐近稳定性

研究了一类基于比率的具有反馈控制的非自治捕食模型,得到了系统一致持久和其周期系统存在唯一全局渐近稳定的周期解的充分性条件．     

An optimization framework of biological dynamical systems

Different biological dynamics are often described by different mathematical equations. On the other hand, some mathematical models describe many biological dynamics universally. Here, we focus on three biological dynamics: the Lotka-Volterra equation, the Hopfield neural networks, and the replicator equation. We describe these three dynamical models using a single optimization framework, which is constructed with employing the Riemannian geometry. Then, we show that the optimization structures of these dynamics are identical, and the differences among the three dynamics are only in the constraints of the optimization. From this perspective, we discuss the unified view for biological dynamics. We also discuss the plausible categorizations, the fundamental nature, and the efficient modeling of the biological dynamics, which arise from the optimization perspective of the dynamical systems.     

Reconnecting plants and pollinators: challenges in the restoration of pollination mutualisms

Ecological restoration of plant-pollinator interactions has received surprisingly little attention, despite animal-mediated pollination underpinning reproduction of the majority of higher plants. Here, we offer a conceptual and practical framework for the ecological restoration of pollination mutualisms. Through the use of targeted restoration plantings to attract and sustain pollinators and increased knowledge of the ecological requirements of pollinators, we propose that pollination could be successfully restored in degraded ecosystems. The challenge for pollination biologists is to integrate their findings with those of plant restoration ecologists to ensure sustainable pollination in restored ecosystems.     

Stochastic fluctuations in frequency-dependent selection: a one-locus, two-allele and two-phenotype model

Stochastic fluctuations in a simple frequency-dependent selection model with one-locus, two-alleles and two-phenotypes are investigated. The steady-state statistics of allele frequencies for an interior stable phenotypic equilibrium are shown to be similar to the stochastic fluctuations in standard evolutionary game dynamics [Tao, Y., Cressman, R., 2007. Stochastic fluctuations through intrinsic noise in evolutionary game dynamics. Bull. Math. Biol. 69, 1377-1399]. On the other hand, for an interior stable phenotypic or genotypic equilibrium, our main results show that the deterministic model cannot be used to predict the expectation of phenotypic frequency. The variance of phenotypic frequency for an interior stable genotypic equilibrium is more sensitive to the expected population size than for an interior stable phenotypic equilibrium. Furthermore, the stochastic fluctuations of allele frequency and phenotypic frequency can be considered approximately independent of each other for these genotypic equilibria, but not for phenotypic.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.     

The evolution of obligate pollination mutualisms: senita cactus and senita moth

We report a new obligate pollination mutualism involving the senita cactus, Lophocereus schottii (Cactaceae, Pachyceereae), and the senita moth, Upiga virescens (Pyralidae, Glaphyriinae) in the Sonoran Desert and discuss the evolution of specialized pollination mutualisms. L. schottii is a night-blooming, self-incompatible columnar cactus. Beginning at sunset, its flowers are visited by U. virescens females, which collect pollen on specialized abdominal scales, actively deposit pollen on flower stigmas, and oviposit a single egg on a flower petal. Larvae spend 6 days eating ovules before exiting the fruit and pupating in a cactus branch. Hand-pollination and pollinator exclusion experiments at our study site near Bahia Kino, Sonora, Mexico, revealed that fruit set in L. schottii is likely to be resource limited. About 50% of hand-outcrossed and open-pollinated senita flowers abort by day 6 after flower opening. Results of exclusion experiments indicated that senita moths accounted for 75% of open-pollinated fruit set in 1995 with two species of halictid bees accounting for the remaining fruit set. In 1996, flowers usually closed before sunrise, and senita moths accounted for at least 90% of open-pollinated fruit set. The net outcome of the senita/senita moth interaction is mutualistic, with senita larvae destroying about 30% of the seeds resulting from pollination by senita moths. Comparison of the senita system with the yucca/yucca moth mutualism reveals many similarities, including reduced nectar production, active pollination, and limited seed destruction. The independent evolution of many of the same features in the two systems suggests that a common pathway exists for the evolution of these highly specialized pollination mutualisms. Nocturnal flower opening, self-incompatible breeding systems, and resource-limited fruit production appear to be important during this evolution. Received: 19 August 1997 / Accepted: 24 November 1997