Evolutionarily stable payoff matrix in hawk-dove games

BACKGROUND

Classical matrix game models aim to find the endpoint of behavioural evolution for a set of fixed possible interaction outcomes. Here, we introduce an evolutionary model in which not only the players' strategies but also the payoff matrix evolves according to natural selection.

RESULTS

We start out from the hawk-dove matrix game and, in a way that is consistent with the monomorphic model setup of Maynard Smith and Price, introduce an evolving phenotypic trait that quantifies fighting ability and determines the probability of winning and the cost of losing escalated hawk-hawk fights. We define evolutionarily stable phenotypes as consisting of an evolutionarily stable strategy and an evolutionarily stable trait, which in turn describes a corresponding evolutionarily stable payoff matrix.

CONCLUSIONS

We find that the maximal possible cost of escalating fights remains constant during evolution assuming a separation in the time scales of fast behavioural and slow trait selection, despite the fact that the final evolutionarily stable phenotype maximizes the payoff of hawk-hawk fights. Our results mirror the dual nature of Darwinian evolution whereby the criteria of evolutionary success, as well as the successful phenotypes themselves, are a product of natural selection.

Do pathogens always evolve to be less virulent? The virulence–transmission trade-off in light of the COVID-19 pandemic

The direction the evolution of virulence takes in connection with any pathogen is a long-standing question. Formerly, it was theorized that pathogens should always evolve to be less virulent. As observations were not in line with this theoretical outcome, new theories emerged, chief among them the transmission–virulence trade-off hypotheses, which predicts an intermediate level of virulence as the endpoint of evolution. At the moment, we are very much interested in the future evolution of COVID-19’s virulence. Here, we show that the disease does not fulfill all the assumptions of the hypothesis. In the case of COVID-19, a higher viral load does not mean a higher risk of death; immunity is not long-lasting; other hosts can act as reservoirs for the virus; and death as a consequence of viral infection does not shorten the infectious period. Consequently, we cannot predict the short- or long-term evolution of the virulence of COVID-19.

Evolution of warfare by resource raiding favours polymorphism in belligerence and bravery

From protists to primates, intergroup aggression and warfare over resources have been observed in several taxa whose populations typically consist of groups connected by limited genetic mixing. Here, we model the coevolution between four traits relevant to this setting: (i) investment into common-pool resource production within groups (helping); (ii) proclivity to raid other groups to appropriate their resources (belligerence); and investments into (iii) defense and (iv) offense of group contests (defensive and offensive bravery). We show that when traits coevolve, the population often experiences disruptive selection favouring two morphs: 'Hawks', who express high levels of both belligerence and offensive bravery; and 'Doves', who express neither. This social polymorphism involves further among-traits associations when the fitness costs of helping and bravery interact. In particular, if helping is antagonistic with both forms of bravery, coevolution leads to the coexistence of individuals that either: (i) do not participate into common-pool resource production but only in its defense and appropriation (Scrounger Hawks) or (ii) only invest into common pool resource production (Producer Doves). Provided groups are not randomly mixed, these findings are robust to several modelling assumptions. This suggests that inter-group aggression is a potent mechanism in favouring within-group social diversity and behavioural syndromes. This article is part of the theme issue 'Intergroup conflict across taxa'.

Selection on age-specific survival: Constant versus fluctuating environment

According to a classic result in evolutionary biodemography, selection on age-specific survival invariably declines with reproductive age. The result assumes proportional changes in survival and a constant environment. Here, we look at selection on age-specific survival when changes are still proportional but the environment fluctuates. We find that selection may or may not decline with reproductive age depending on how exactly survival is proportionally altered by mutations. However, interpreted in neutral terms, the mathematics behind the classic result capture a general property that the genetics of populations with age structure possess both in a constant and in a fluctuating environment.

Trait positions for elevated invasiveness in adaptive ecological networks

Our ability to predict the outcome of invasion declines rapidly as non-native species progress through intertwined ecological barriers to establish and spread in recipient ecosystems. This is largely due to the lack of systemic knowledge on key processes at play as species establish self-sustaining populations within the invaded range. To address this knowledge gap, we present a mathematical model that captures the eco-evolutionary dynamics of native and non-native species interacting within an ecological network. The model is derived from continuous-trait evolutionary game theory (i.e., Adaptive Dynamics) and its associated concept of invasion fitness which depicts dynamic demographic performance that is both trait mediated and density dependent. Our approach allows us to explore how multiple resident and non-native species coevolve to reshape invasion performance, or more precisely invasiveness, over trait space. The model clarifies the role of specific traits in enabling non-native species to occupy realised opportunistic niches. It also elucidates the direction and speed of both ecological and evolutionary dynamics of residing species (natives or non-natives) in the recipient network under different levels of propagule pressure. The versatility of the model is demonstrated using four examples that correspond to the invasion of (i) a horizontal competitive community; (ii) a bipartite mutualistic network; (iii) a bipartite antagonistic network; and (iv) a multi-trophic food web. We identified a cohesive trait strategy that enables the success and establishment of non-native species to possess high invasiveness. Specifically, we find that a non-native species can achieve high levels of invasiveness by possessing traits that overlap with those of its facilitators (and mutualists), which enhances the benefits accrued from positive interactions, and by possessing traits outside the range of those of antagonists, which mitigates the costs accrued from negative interactions. This ‘central-to-reap, edge-to-elude’ trait strategy therefore describes the strategic trait positions of non-native species to invade an ecological network. This model provides a theoretical platform for exploring invasion strategies in complex adaptive ecological networks.

The evolution of site-selection strategy during dispersal

We propose a mathematical model that enables the evolutionary analysis of site-selection process of dispersing individuals that encounter sites of high or low quality. Since each site can be inhabited by at most one individual, all dispersers are not able to obtain a high-quality site. We study the evolutionary dynamics of the low-quality-site acceptance as a function of the time during the dispersal season using adaptive dynamics. We show that environmental changes affect the evolutionary dynamics in two ways: directly and indirectly via density-dependent factors. Direct evolutionary effects usually follow intuition, whereas indirect effects are often counter-intuitive and hence difficult to predict without mechanistic modeling. Therefore, the mechanistic derivation of the fitness function, with careful attention on density- and frequency dependence, is essential for predicting the consequences of environmental changes to site selection. For example, increasing fecundity in high-quality sites makes them more tempting for dispersers and hence the direct effect of this ecological change delays the acceptance of low-quality sites. However, increasing fecundity in high-quality sites also increases the population size, which makes the competition for sites more severe and thus, as an indirect effect, forces evolution to favor less picky individuals. Our results indicate that the indirect effects often dominate the intuitive effects, which emphasizes the need for mechanistic models of the immigration process.

Adaptive diversification in heterogeneous environments

The role of environmental heterogeneity in the evolution of biological diversity has been studied only for simple types of heterogeneities and dispersals. This article broadens previous results by considering heterogeneities and dispersals that are structured by several environmental factors. It studies the evolution of a metapopulation, living in a network of patches connected by dispersal, under the effects of mutation, selection and migration. First, it is assumed that patches are equally connected and that they carry habitats characterized by several factors exerting selection pressures on several individual traits. Habitat factors may vary in the environment independently or they may be correlated. It is shown that correlations between habitat factors promote adaptive diversification and that this effect may be modified by trait interactions on survival. Then, it is assumed that patches are structured by two crossed factors, called the row and column factors, such that patches are more connected when they occur in the same row or in the same column. Environmental patterns in which each habitat appears in each row the same number of times and appears in each column the same number of times are found to hinder adaptive diversification.

Adaptive dynamics of saturated polymorphisms

We study the joint adaptive dynamics of n scalar-valued strategies in ecosystems where n is the maximum number of coexisting strategies permitted by the (generalized) competitive exclusion principle. The adaptive dynamics of such saturated systems exhibits special characteristics, which we first demonstrate in a simple example of a host-pathogen-predator model. The main part of the paper characterizes the adaptive dynamics of saturated polymorphisms in general. In order to investigate convergence stability, we give a new sufficient condition for absolute stability of an arbitrary (not necessarily saturated) polymorphic singularity and show that saturated evolutionarily stable polymorphisms satisfy it. For the case [Formula: see text], we also introduce a method to construct different pairwise invasibility plots of the monomorphic population without changing the selection gradients of the saturated dimorphism.

Modeling evolutionary games in populations with demographic structure

Classic life history models are often based on optimization algorithms, focusing on the adaptation of survival and reproduction to the environment, while neglecting frequency dependent interactions in the population. Evolutionary game theory, on the other hand, studies frequency dependent strategy interactions, but usually omits life history and the demographic structure of the population. Here we show how an integration of both aspects can substantially alter the underlying evolutionary dynamics. We study the replicator dynamics of strategy interactions in life stage structured populations. Individuals have two basic strategic behaviours, interacting in pairwise games. A player may condition behaviour on the life stage of its own, or that of the opponent, or the matching of life stages between both players. A strategy is thus defined as the set of rules that determines a player׳s life stage dependent behaviours. We show that the diversity of life stage structures and life stage dependent strategies can promote each other, and the stable frequency of basic strategic behaviours can deviate from game equilibrium in populations with life stage structures.

Evolution of heritable behavioural differences in a model of social division of labour

The spectacular diversity of personality and behaviour of animals and humans has evoked many hypotheses intended to explain its developmental and evolutionary background. Although the list of the possible contributing mechanisms seems long, we propose that an underemphasised explanation is the division of labour creating negative frequency dependent selection. We use analytical and numerical models of social division of labour to show how selection can create consistent and heritable behavioural differences in a population, where randomly sampled individuals solve a collective task together. We assume that the collective task needs collaboration of individuals performing one of the two possible subtasks. The total benefit of the group is highest when the ratio of different subtasks is closest to 1. The probability of choosing one of the two costly subtasks and the costs assigned to them are under selection. By using adaptive dynamics we show that if a trade-off between the costs of the subtasks is strong enough, then evolution leads to coexistence of specialized individuals performing one of the subtasks with high probability and low cost. Our analytical results were verified and extended by numerical simulations.

Biological fitness and the fundamental theorem of natural selection

Fisher's fundamental theorem of natural selection is proved satisfactorily for the first time, resolving confusions in the literature about the nature of reproductive value and fitness. Reproductive value is defined following Fisher, without reference to genetic variation, and fitness is the proportional rate of increase in an individual's contribution to the demographic population size. The mean value of fitness is the same in each age class, and it also equals the population's Malthusian parameter. The statement and derivation are regarded as settled here, and so the general biological significance of the fundamental theorem can be debated. The main purpose of the theorem is to find a quantitative measure of the effect of natural selection in a Mendelian system, thus founding Darwinism on Mendelism and identifying the design criterion for biological adaptation, embodied in Fisher's ingenious definition of fitness. The relevance of the newly understood theorem to five current research areas is discussed.

Intrusive versus domiciliated triatomines and the challenge of adapting vector control practices against Chagas disease

Chagas disease prevention remains mostly based on triatomine vector control to reduce or eliminate house infestation with these bugs. The level of adaptation of triatomines to human housing is a key part of vector competence and needs to be precisely evaluated to allow for the design of effective vector control strategies. In this review, we examine how the domiciliation/intrusion level of different triatomine species/populations has been defined and measured and discuss how these concepts may be improved for a better understanding of their ecology and evolution, as well as for the design of more effective control strategies against a large variety of triatomine species. We suggest that a major limitation of current criteria for classifying triatomines into sylvatic, intrusive, domiciliary and domestic species is that these are essentially qualitative and do not rely on quantitative variables measuring population sustainability and fitness in their different habitats. However, such assessments may be derived from further analysis and modelling of field data. Such approaches can shed new light on the domiciliation process of triatomines and may represent a key tool for decision-making and the design of vector control interventions.

Cheating is evolutionarily assimilated with cooperation in the continuous snowdrift game

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We fully analyze continuous snowdrift games with quadratic payoff functions in diversified populations.

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It is well known that classical snowdrift games maintain the coexistence of cooperators and cheaters.

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We clarify that the continuous snowdrift games often lead to assimilation of cooperators and cheaters.

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Allowing the gradual evolution of cooperative behavior can facilitate social inequity aversion in joint ventures.

It is well known that in contrast to the Prisoner’s Dilemma, the snowdrift game can lead to a stable coexistence of cooperators and cheaters. Recent theoretical evidence on the snowdrift game suggests that gradual evolution for individuals choosing to contribute in continuous degrees can result in the social diversification to a 100% contribution and 0% contribution through so-called evolutionary branching. Until now, however, game-theoretical studies have shed little light on the evolutionary dynamics and consequences of the loss of diversity in strategy. Here, we analyze continuous snowdrift games with quadratic payoff functions in dimorphic populations. Subsequently, conditions are clarified under which gradual evolution can lead a population consisting of those with 100% contribution and those with 0% contribution to merge into one species with an intermediate contribution level. The key finding is that the continuous snowdrift game is more likely to lead to assimilation of different cooperation levels rather than maintenance of diversity. Importantly, this implies that allowing the gradual evolution of cooperative behavior can facilitate social inequity aversion in joint ventures that otherwise could cause conflicts that are based on commonly accepted notions of fairness.

Quantifying selection differentials caused by recreational fishing: development of modeling framework and application to reproductive investment in pike (Esox lucius)

Methods for quantifying selection pressures on adaptive traits affected by size-selective fishing are still scarce, and none have as yet been developed for recreational fishing. We present an ecologically realistic age-structured model specifically tailored to recreational fishing that allows estimating selection differentials on adaptive life-history traits. The model accounts for multiple ecological feedbacks, which result in density-dependent and frequency-dependent selection. We study selection differentials on annual reproductive investment under size-selective exploitation in a highly demanded freshwater recreational fish species, northern pike (Esox lucius L.). We find that recreational angling mortality exerts positive selection differentials on annual reproductive investment, in agreement with predictions from life-history theory. The strength of selection increases with the intensity of harvesting. We also find that selection on reproductive investment can be reduced by implementing simple harvest regulations such as minimum-size limits. The general, yet computationally simple, methods introduced here allow evaluating and comparing selection pressures on adaptive traits in other fish populations and species, and thus have the potential to become a tool for evolutionary impact assessment of harvesting.

Eco-evolutionary feedbacks, adaptive dynamics and evolutionary rescue theory

Adaptive dynamics theory has been devised to account for feedbacks between ecological and evolutionary processes. Doing so opens new dimensions to and raises new challenges about evolutionary rescue. Adaptive dynamics theory predicts that successive trait substitutions driven by eco-evolutionary feedbacks can gradually erode population size or growth rate, thus potentially raising the extinction risk. Even a single trait substitution can suffice to degrade population viability drastically at once and cause 'evolutionary suicide'. In a changing environment, a population may track a viable evolutionary attractor that leads to evolutionary suicide, a phenomenon called 'evolutionary trapping'. Evolutionary trapping and suicide are commonly observed in adaptive dynamics models in which the smooth variation of traits causes catastrophic changes in ecological state. In the face of trapping and suicide, evolutionary rescue requires that the population overcome evolutionary threats generated by the adaptive process itself. Evolutionary repellors play an important role in determining how variation in environmental conditions correlates with the occurrence of evolutionary trapping and suicide, and what evolutionary pathways rescue may follow. In contrast with standard predictions of evolutionary rescue theory, low genetic variation may attenuate the threat of evolutionary suicide and small population sizes may facilitate escape from evolutionary traps.

Function-valued adaptive dynamics and optimal control theory

In this article we further develop the theory of adaptive dynamics of function-valued traits. Previous work has concentrated on models for which invasion fitness can be written as an integral in which the integrand for each argument value is a function of the strategy value at that argument value only. For this type of models of direct effect, singular strategies can be found using the calculus of variations, with singular strategies needing to satisfy Euler's equation with environmental feedback. In a broader, more mechanistically oriented class of models, the function-valued strategy affects a process described by differential equations, and fitness can be expressed as an integral in which the integrand for each argument value depends both on the strategy and on process variables at that argument value. In general, the calculus of variations cannot help analyzing this much broader class of models. Here we explain how to find singular strategies in this class of process-mediated models using optimal control theory. In particular, we show that singular strategies need to satisfy Pontryagin's maximum principle with environmental feedback. We demonstrate the utility of this approach by studying the evolution of strategies determining seasonal flowering schedules.

The structure of mutations and the evolution of cooperation

Evolutionary game dynamics in finite populations assumes that all mutations are equally likely, i.e., if there are n strategies a single mutation can result in any strategy with probability 1/n. However, in biological systems it seems natural that not all mutations can arise from a given state. Certain mutations may be far away, or even be unreachable given the current composition of an evolving population. These distances between strategies (or genotypes) define a topology of mutations that so far has been neglected in evolutionary game theory. In this paper we re-evaluate classic results in the evolution of cooperation departing from the assumption of uniform mutations. We examine two cases: the evolution of reciprocal strategies in a repeated prisoner's dilemma, and the evolution of altruistic punishment in a public goods game. In both cases, alternative but reasonable mutation kernels shift known results in the direction of less cooperation. We therefore show that assuming uniform mutations has a substantial impact on the fate of an evolving population. Our results call for a reassessment of the "model-less" approach to mutations in evolutionary dynamics.

Sexual conflict and the tragedy of the commons

It is widely understood that the costs and benefits of mating can affect the fecundity and survival of individuals. Sexual conflict may have profound consequences for populations as a result of the negative effects it causes males and females to have on one another's fitness. Here we present a model describing the evolution of sexual conflict, in which males inflict a direct cost on female fitness. We show that these costs can drive the entire population to extinction. To males, females are an essential but finite resource over which they have to compete. Population extinction owing to sexual conflict can therefore be seen as an evolutionary tragedy of the commons. Our model shows that a positive feedback between harassment and the operational sex ratio is responsible for the demise of females and, thus, for population extinction. We further show that the evolution of female resistance to counter harassment can prevent a tragedy of the commons. Our findings not only demonstrate that sexual conflict can drive a population to extinction but also highlight how simple mechanisms, such as harassment costs to males and females and the coevolution between harassment and resistance, can help avert a tragedy of the commons caused by sexual conflict.

Ecological factors driving the long-term evolution of influenza's host range

The evolution of a pathogen's host range is shaped by the ecology of its hosts and by the physiological traits that determine host specificity. For many pathogen traits, there is a trade-off: a phenotype suitable for infecting one set of hosts poorly infects another. Introducing and analysing a simple evo-epidemiological model, here we study how such a trade-off is expected to affect evolution of the host ranges of influenza viruses. We examine a quantitative trait underlying host specificity, given by an influenza virus's degree of adaptation to certain conformations of sialic acid receptors, and investigate how this receptor preference evolves in a minimal network of host species, including humans, that differ in life history and receptor physiology. Using adaptive dynamics theory, we establish thresholds in interspecific transmission rates and host population sizes that govern the emergence and persistence of human-adapted viruses. These ecological thresholds turn out to be largely independent of the strength of the evolutionary trade-off, underscoring the importance of ecological conditions in determining a disease's host range.

CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces

Static continuously stable strategy (CSS) and neighborhood invader strategy (NIS) conditions are developed for two-species models of frequency-dependent behavioral evolution when individuals have traits in continuous strategy spaces. These are intuitive stability conditions that predict the eventual outcome of evolution from a dynamic perspective. It is shown how the CSS is related to convergence stability for the canonical equation of adaptive dynamics and the NIS to convergence to a monomorphism for the replicator equation of evolutionary game theory. The CSS and NIS are also shown to be special cases of neighborhood p(*)- superiority for p(*) equal to one half and zero, respectively. The theory is illustrated when each species has a one-dimensional trait space.

Live where you thrive: joint evolution of habitat choice and local adaptation facilitates specialization and promotes diversity

We derive a comprehensive overview of specialization evolution based on analytical results and numerical illustrations. We study the separate and joint evolution of two critical facets of specialization-local adaptation and habitat choice-under different life cycles, modes of density regulation, variance-covariance structures, and trade-off strengths. A particular feature of our analysis is the investigation of arbitrary trade-off functions. We find that local-adaptation evolution qualitatively changes the outcome of habitat-choice evolution under a wide range of conditions. In addition, habitat-choice evolution qualitatively and invariably changes the outcomes of local-adaptation evolution whenever trade-offs are weak. Even weak trade-offs, which favor generalists when habitat choice is fixed, select for specialists once local adaptation and habitat choice are both allowed to evolve. Unless trapped by maladaptive genetic constraints, joint evolution of local adaptation and habitat choice in the models analyzed here thus always leads to specialists, independent of life cycle, density regulation, and trade-off strength, thus raising the bar for evolutionarily sound explanations of generalism. Whether a single specialist or two specialists evolve depends on the life cycle and the mode of density regulation. Finally, we explain why the gradual evolutionary emergence of coexisting specialists requires more restrictive conditions than does their evolutionarily stable maintenance.

Adaptive dynamics of dormancy duration variability: evolutionary trade-off and priority effect lead to suboptimal adaptation

Many plants, insects, and crustaceans show within-population variability in dormancy length. The question of whether such variability corresponds to a genetic polymorphism of pure strategies or a mixed bet-hedging strategy, and how the level of phenotypic variability can evolve remain unknown for most species. Using an eco-genetic model rooted in a 25-year ecological field study of a Chestnut weevil, Curculio elephas, we show that its diapause-duration variability is more likely to have evolved by the spread of a bet-hedging strategy than by the establishment of a genetic polymorphism. Investigating further the adaptive dynamics of diapause-duration variability, we find two unanticipated patterns of general interest. First, there is a trade-off between the ability of bet-hedging strategies to persist on an ecological time scale and their ability to invade. The optimal strategy (in terms of persistence) cannot invade, whereas suboptimal bet-hedgers are good invaders. Second, we describe an original evolutionary dynamics where each bet-hedging strategy (defined by its rate of prolonged diapause) resists invasion by all others, so that the first type of bet-hedger to appear persists on an evolutionary time scale. Such "evolutionary priority effect" could drive the evolution of maladapted levels of diapause-duration variability.

The formal Darwinism project: a mid-term report

For 8 years I have been pursuing in print an ambitious and at times highly technical programme of work, the 'Formal Darwinism Project', whose essence is to underpin and formalize the fitness optimization ideas used by behavioural ecologists, using a new kind of argument linking the mathematics of motion and the mathematics of optimization. The value of the project is to give stronger support to current practices, and at the same time sharpening theoretical ideas and suggesting principled resolutions of some untidy areas, for example, how to define fitness. The aim is also to unify existing free-standing theoretical structures, such as inclusive fitness theory, Evolutionary Stable Strategy (ESS) theory and bet-hedging theory. The 40-year-old misunderstanding over the meaning of fitness optimization between mathematicians and biologists is explained. Most of the elements required for a general theory have now been implemented, but not together in the same framework, and 'general time' remains to be developed and integrated with the other elements to produce a final unified theory of neo-Darwinian natural selection.

Disparate maturation adaptations to size-dependent mortality

Body size is an important determinant of resource use, fecundity and mortality risk. Evolution of maturation size in response to size-dependent selection is thus a fundamental part of life-history theory. Increased mortality among small individuals has previously been predicted to cause larger maturation size, whereas increased mortality among large individuals is expected to have the opposite effect. Here we use a continuously size-structured model to demonstrate that, contrary to these widespread expectations, increased mortality among small individuals can have three alternative effects: maturation size may increase, decrease or become evolutionarily bistable. We show that such complex responses must be reckoned with whenever mortality is size-dependent, growth is indeterminate, reproduction impairs growth and fecundity increases with size. Predicting adaptive responses to altered size-dependent mortality is thus inherently difficult, since, as demonstrated here, such mortality cannot only reverse the direction of adaptation, but also cause abrupt shifts in evolutionarily stable maturation sizes.

Uninvadability in -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 suicide

The great majority of species that lived on this earth have gone extinct. These extinctions are often explained by invoking changes in the environment, to which the species has been unable to adapt. Evolutionary suicide is an alternative explanation to such extinctions. It is an evolutionary process in which a viable population adapts in such a way that it can no longer persist. In this paper different models, where evolutionary suicide occurs are discussed, and the theory behind the phenomenon is reviewed.

Stability of the replicator equation for a single species with a multi-dimensional continuous trait space

The replicator equation model for the evolution of individual behaviors in a single species with a multi-dimensional continuous trait space is developed as a dynamics on the set of probability measures. Stability of monomorphisms in this model using the weak topology is compared to more traditional methods of adaptive dynamics. For quadratic fitness functions and initial normal trait distributions, it is shown that the multi-dimensional continuously stable strategy (CSS) of adaptive dynamics is often relevant for predicting stability of the measure-theoretic model but may be too strong in general. For general fitness functions and trait distributions, the CSS is related to dominance solvability which can be used to characterize local stability for a large class of trait distributions that have no gaps in their supports whereas the stronger neighborhood invader strategy (NIS) concept is needed if the supports are arbitrary.

Emergence of asymmetry in evolution

We investigate symmetry-breaking bifurcation patterns in evolution in the framework of adaptive dynamics (AD). We define weak and strong symmetry. The former applies for populations where only the simultaneous reflection of all individuals is an invariant transformation. The symmetry is strong in populations where reflection of some, but not all, individuals leaves the situation unchanged. We show that in case of weak symmetry evolutionary branching can lead to the emergence of two asymmetric variants, which are mirror images of each other, and the loss of the symmetric ancestor. We also show that in case of strong symmetry, evolutionary branching can occur into a symmetric and an asymmetric variant, both of which survive. The latter, asymmetric branching differs from the generic branching patterns of AD, which is always symmetric. We discuss biological examples for weak and strong symmetries and a specific model producing the new kind of branching.

The adaptive dynamics of function-valued traits

This study extends the framework of adaptive dynamics to function-valued traits. Such adaptive traits naturally arise in a great variety of settings: variable or heterogeneous environments, age-structured populations, phenotypic plasticity, patterns of growth and form, resource gradients, and in many other areas of evolutionary ecology. Adaptive dynamics theory allows analysing the long-term evolution of such traits under the density-dependent and frequency-dependent selection pressures resulting from feedback between evolving populations and their ecological environment. Starting from individual-based considerations, we derive equations describing the expected dynamics of a function-valued trait in asexually reproducing populations under mutation-limited evolution, thus generalizing the canonical equation of adaptive dynamics to function-valued traits. We explain in detail how to account for various kinds of evolutionary constraints on the adaptive dynamics of function-valued traits. To illustrate the utility of our approach, we present applications to two specific examples that address, respectively, the evolution of metabolic investment strategies along resource gradients, and the evolution of seasonal flowering schedules in temporally varying environments.

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.

Fire, rain and the selection of seeder and resprouter life-histories in fire-recruiting, woody plants

Several Cape species of the genus Erica are known to present seeder and resprouter phenotypes, and this variation seems to have a genetic basis. Therefore, this genus provides ideal model systems for using to elucidate the evolution of nonsprouting or seeder and resprouter life-histories in woody, fire-recruiting plants. A simple simulation model was developed to identify, under life-history optimality, the ecological conditions (viz. rainfall conditions and fire frequency) that conferred a selective advantage to the seeder phenotype over the resprouter in a given Cape Erica species. The model illustrated that the seeder life-history was able to invade and replace a resprouter population only under a mild mediterranean climate, with short, moderate summer droughts. This simulation approach will contribute to a better understanding of the biogeographical pattern of seeder and resprouter lineages of one of the paradigmatic fynbos woody taxa throughout the Cape floristic region.

Function-valued adaptive dynamics and the calculus of variations

Adaptive dynamics has been widely used to study the evolution of scalar-valued, and occasionally vector-valued, strategies in ecologically realistic models. In many ecological situations, however, evolving strategies are best described as function-valued, and thus infinite-dimensional, traits. So far, such evolution has only been studied sporadically, mostly based on quantitative genetics models with limited ecological realism. In this article we show how to apply the calculus of variations to find evolutionarily singular strategies of function-valued adaptive dynamics: such a strategy has to satisfy Euler's equation with environmental feedback. We also demonstrate how second-order derivatives can be used to investigate whether or not a function-valued singular strategy is evolutionarily stable. We illustrate our approach by presenting several worked examples.

When do mixotrophs specialize? Adaptive dynamics theory applied to a dynamic energy budget model

In evolutionary history, several events have occurred at which mixotrophs specialized into pure autotrophs and heterotrophs. We studied the conditions under which such events take place, using the Dynamic Energy Budget (DEB) theory for physiological rules of the organisms' metabolism and Adaptive Dynamics (AD) theory for evolutionary behavior of parameter values. We modeled a population of mixotrophs that can take up dissolved inorganic nutrients by autotrophic assimilation and detritus by heterotrophic assimilation. The organisms have a certain affinity for both pathways; mutations that occur in the affinities enable the population to evolve. One of the possible evolutionary outcomes is a branching point which provides an opportunity for the mixotrophic population to split up and specialize into separate autotrophs and heterotrophs. Evolutionary branching is not a common feature of the studied system, but is found to occur only under specific conditions. These conditions depend on intrinsic properties such as the cost function, the level of the costs and the boundaries of the trait space: only at intermediate cost levels and when an explicit advantage exists to pure strategies over mixed ones may evolutionary branching occur. Usually, such an advantage (and hence evolutionary branching) can be induced by interference between the two affinities, but this result changes due to the constraints on the affinities. Now, only some of the more complicated cost functions give rise to a branching point. In contrast to the intrinsic properties, extrinsic properties such as the total nutrient content or light intensity were found to have no effect on the evolutionary outcomes at all.

Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics

The measure dynamics approach to modelling single-species coevolution with a one-dimensional trait space is developed and compared to more traditional methods of adaptive dynamics and the Maximum Principle. It is assumed that individual fitness results from pairwise interactions together with a background fitness that depends only on total population size. When fitness functions are quadratic in the real variables parameterizing the one-dimensional traits of interacting individuals, the following results are derived. It is shown that among monomorphisms (i.e. measures supported on a single trait value), the continuously stable strategy (CSS) characterize those that are Lyapunov stable and attract all initial measures supported in an interval containing this trait value. In the cases where adaptive dynamics predicts evolutionary branching, convergence to a dimorphism is established. Extensions of these results to general fitness functions and/or multi-dimensional trait space are discussed.

Trade-Off Geometries and Frequency-Dependent Selection

Life-history evolution is determined by the interplay between natural selection and adaptive constraints. The classical approach to studying constrained life-history evolution-Richard Levins's geometric comparison of fitness sets and adaptive functions-is applicable when selection pressures are frequency independent. Here we extend this widely used tool to frequency-dependent selection. Such selection pressures vary with a population's phenotypic composition and are increasingly recognized as ubiquitous. Under frequency dependence, two independent properties have to be distinguished: evolutionary stability (an evolutionarily stable strategy cannot be invaded once established) and convergence stability (only a convergence stable strategy can be attained through small, selectively advantageous steps). Combination of both properties results in four classes of possible evolutionary outcomes. We introduce a geometric mode of analysis that enables predicting, for any bivariate selection problem, evolutionary outcomes induced by trade-offs of given shape, shapes of trade-offs required for given evolutionary outcomes, the set of all evolutionary outcomes trade-offs can induce, and effects of ecological parameters on evolutionary outcomes independent of trade-off shape.

Evolutionarily stable transition rates in a stage-structured model. An application to the analysis of size distributions of badges of social status

This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete population model with a general transition matrix. In the model, individuals are categorized into n dominance classes, newborns lie in the subordinate class, and it is considered as evolutionary trait a vector eta of probabilities of transition among classes. For this trait, we obtain the evolutionary singular strategy and prove its neutral evolutionary stability. Finally, we obtain conditions for the invading potential of such a strategy, which is sufficient for the convergence stability of the latter. With the help of the previous results, we provide an explanation for the bimodal distribution of badges of status observed in the Siskin (Carduelis spinus). In the Siskin, as in several bird species, patches of pigmented plumage signal the dominance status of the bearer to opponents, and central to the discussion on the evolution of status signalling is the understanding of which should be the frequency distribution of badge sizes. Though some simple verbal models predicted a bimodal distribution, up to now most species display normal distributions and bimodality has only been described for the Siskin. In this paper, we give conditions leading to one of these two distributions in terms of the survival, fecundity and aggression rates in each dominance class.

Adaptive walks on changing landscapes: Levins' approach extended

The assumption that trade-offs exist is fundamental in evolutionary theory. Levins (Am. Nat. 96 (1962) 361-372) introduced a widely adopted graphical method for analyzing evolution towards an optimal combination of two quantitative traits, which are traded off. His approach explicitly excluded the possibility of density- and frequency-dependent selection. Here we extend Levins method towards models, which include these selection regimes and where therefore fitness landscapes change with population state. We employ the same kind of curves Levins used: trade-off curves and fitness contours. However, fitness contours are not fixed but a function of the resident traits and we only consider those that divide the trait space into potentially successful mutants and mutants which are not able to invade ('invasion boundaries'). The developed approach allows to make a priori predictions about evolutionary endpoints and about their bifurcations. This is illustrated by applying the approach to several examples from the recent literature.

Resident-invader dynamics and the coexistence of similar strategies

We study the resident-invader dynamics for a given class of models of unstructured populations of finite-dimensional strategies. We prove various results on the existence and uniqueness of omega-limit sets in the interior of the resident-invader population state space, and we classify the generically possible types of dynamics in terms of the invasion conditions when the resident and invader strategies are similar to one another.

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.

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 evolution of phenotypic plasticity in spatially structured environments: implications of intraspecific competition, plasticity costs and environmental characteristics

We model the evolution of reaction norms focusing on three aspects: frequency-dependent selection arising from resource competition, maintenance and production costs of phenotypic plasticity, and three characteristics of environmental heterogeneity (frequency of environments, their intrinsic carrying capacity and the sensitivity to phenotypic maladaptation in these environments). We show that (i) reaction norms evolve so as to trade adaptation for acquiring resources against cost avoidance; (ii) maintenance costs cause reaction norms to better adapt to frequent rather than to infrequent environments, whereas production costs do not; and (iii) evolved reaction norms confer better adaptation to environments with low rather than with high intrinsic carrying capacity. The two previous findings contradict earlier theoretical results and originate from two previously unexplored features that are included in our model. First, production costs of phenotypic plasticity are only incurred when a given phenotype is actually produced. Therefore, they are proportional to the frequency of environments, and these frequencies thus affect the selection pressure to avoid costs just as much as the selection pressure to improve adaptation. This prevents the frequency of environments from affecting the evolving reaction norm. Secondly, our model describes the evolution of plasticity for a phenotype determining an individual's capability to acquire resources, and thus its realized carrying capacity. When individuals are distributed randomly across environments, they cannot avoid experiencing environments with intrinsically low carrying capacity. As selection pressures arising from the need to improve adaptation are stronger under such extreme conditions than under mild ones, better adaptation to environments with low rather than with high intrinsic carrying capacity results.