Adaptive dynamics for physiologically structured population models

Differential Stage-Specific Mortality as a Mechanism for Diversification

AbstractIndividual variability in mortality is widespread in nature. The general rule is that larger organisms have a greater chance of survival than smaller conspecifics. There is growing evidence that differential mortality between developmental stages has important consequences for the ecology and evolution of populations and communities. However, we know little about how it can influence diversification. Using an eco-evolutionary model of diversification that considers individual variability in mortality, I show that commonly observed differences in mortality between juveniles and adults can facilitate adaptive diversification. In particular, diversification is expected to be less restricted when mortality is more biased toward juveniles. Additionally, I find stage-specific differences in metabolic cost and foraging capacity to further facilitate diversification when adults are slightly superior competitors, due to either a lower metabolic cost or a higher foraging capacity, than juveniles. This is because by altering the population composition, differential stage-specific mortality and competitive ability can modulate the strength of intraspecific competition, which in turn determines the outcome of diversification. These results demonstrate the strong influence that ecological differences between developmental stages have on diversification and highlight the need for integrating developmental processes into diversification theory.

Anisogamy Does Not Always Promote the Evolution of Mating Competition Traits in Males

AbstractAnisogamy has evolved in most sexually reproducing multicellular organisms allowing the definition of male and female sexes, producing small and large gametes. Anisogamy, as the initial sexual dimorphism, is a good starting point to understand the evolution of further sexual dimorphisms. For instance, it is generally accepted that anisogamy sets the stage for more intense mating competition in males than in females. We argue that this idea stems from a restrictive assumption on the conditions under which anisogamy evolved in the first place: the absence of sperm limitation (assuming that all female gametes are fertilized). Here, we relax this assumption and present a model that considers the coevolution of gamete size with a mating competition trait, starting in a population without dimorphism. We vary gamete density to produce different scenarios of gamete limitation. We show that while at high gamete density the evolution of anisogamy always results in male investment in competition, gamete limitation at intermediate gamete densities allows for either females or males to invest more into mating competition. Our results thus suggest that anisogamy does not always promote mating competition among males. The conditions under which anisogamy evolves matter, as does the competition trait.

A mathematical framework for evo-devo dynamics

Natural selection acts on phenotypes constructed over development, which raises the question of how development affects evolution. Classic evolutionary theory indicates that development affects evolution by modulating the genetic covariation upon which selection acts, thus affecting genetic constraints. However, whether genetic constraints are relative, thus diverting adaptation from the direction of steepest fitness ascent, or absolute, thus blocking adaptation in certain directions, remains uncertain. This limits understanding of long-term evolution of developmentally constructed phenotypes. Here we formulate a general, tractable mathematical framework that integrates age progression, explicit development (i.e., the construction of the phenotype across life subject to developmental constraints), and evolutionary dynamics, thus describing the evolutionary and developmental (evo-devo) dynamics. The framework yields simple equations that can be arranged in a layered structure that we call the evo-devo process, whereby five core elementary components generate all equations including those mechanistically describing genetic covariation and the evo-devo dynamics. The framework recovers evolutionary dynamic equations in gradient form and describes the evolution of genetic covariation from the evolution of genotype, phenotype, environment, and mutational covariation. This shows that genotypic and phenotypic evolution must be followed simultaneously to yield a dynamically sufficient description of long-term phenotypic evolution in gradient form, such that evolution described as the climbing of a fitness landscape occurs in "geno-phenotype" space. Genetic constraints in geno-phenotype space are necessarily absolute because the phenotype is related to the genotype by development. Thus, the long-term evolutionary dynamics of developed phenotypes is strongly non-standard: (1) evolutionary equilibria are either absent or infinite in number and depend on genetic covariation and hence on development; (2) developmental constraints determine the admissible evolutionary path and hence which evolutionary equilibria are admissible; and (3) evolutionary outcomes occur at admissible evolutionary equilibria, which do not generally occur at fitness landscape peaks in geno-phenotype space, but at peaks in the admissible evolutionary path where "total genotypic selection" vanishes if exogenous plastic response vanishes and mutational variation exists in all directions of genotype space. Hence, selection and development jointly define the evolutionary outcomes if absolute mutational constraints and exogenous plastic response are absent, rather than the outcomes being defined only by selection. Moreover, our framework provides formulas for the sensitivities of a recurrence and an alternative method to dynamic optimization (i.e., dynamic programming or optimal control) to identify evolutionary outcomes in models with developmentally dynamic traits. These results show that development has major evolutionary effects.

Marine reserves promote cycles in fish populations on ecological and evolutionary time scales

Marine reserves are considered essential for sustainable fisheries, although their effectiveness compared to traditional fisheries management is debated. The effect of marine reserves is mostly studied on short ecological time scales, whereas fisheries-induced evolution is a well-established consequence of harvesting. Using a size-structured population model for an exploited fish population of which individuals spend their early life stages in a nursery habitat, we show that marine reserves will shift the mode of population regulation from low size-selective survival late in life to low, early-life survival due to strong resource competition. This shift promotes the occurrence of rapid ecological cycles driven by density-dependent recruitment as well as much slower evolutionary cycles driven by selection for the optimal body to leave the nursery grounds, especially with larger marine reserves. The evolutionary changes increase harvesting yields in terms of total biomass but cause disproportionately large decreases in yields of larger, adult fish. Our findings highlight the importance of carefully considering the size of marine reserves and the individual life history of fish when managing eco-evolutionary marine systems to ensure both population persistence as well as stable fisheries yields.

Immunity-driven evolution of virulence and diversity in respiratory diseases

The time-honored paradigm in the theory of virulence evolution assumes a positive relation between infectivity and harmfulness. However, the etiology of respiratory diseases yields a negative relation, with diseases of the lower respiratory tract being less infective and more harmful. We explore the evolutionary consequences in a simple model incorporating cross-immunity between disease strains that diminishes with their distance in the respiratory tract, assuming that docking rate follows the match between the local mix of cell surface types and the pathogen's surface and cross-immunity the similarity of the pathogens' surfaces. The assumed relation between fitness components causes virulent strains infecting the lower airways to evolve to milder more transmissible variants. Limited cross-immunity, generally, causes a readiness to diversify that increases with host population density. In respiratory diseases that diversity will be highest in the upper respiratory tract. More tentatively, emerging respiratory diseases are likely to start low and virulent, to evolve up, and become milder. Our results extend to a panoply of realistic generalizations of the disease's ecology to including additional epitope axes. These extensions allow us to apply our results quantitatively to elucidate the differences in diversification between rhino- and coronavirus caused common colds.

A geometric process of evolutionary game dynamics

Many evolutionary processes occur in phenotype spaces which are continuous. It is therefore of interest to explore how selection operates in continuous spaces. One approach is adaptive dynamics, which assumes that mutants are local. Here we study a different process which also allows non-local mutants. We assume that a resident population is challenged by an invader who uses a strategy chosen from a random distribution on the space of all strategies. We study the repeated donation game of direct reciprocity. We consider reactive strategies given by two probabilities, denoting respectively the probability to cooperate after the co-player has cooperated or defected. The strategy space is the unit square. We derive analytic formulae for the stationary distribution of evolutionary dynamics and for the average cooperation rate as function of the cost-to-benefit ratio. For positive reactive strategies, we prove that cooperation is more abundant than defection if the area of the cooperative region is greater than 1/2 which is equivalent to benefit, b, divided by cost, c, exceeding [Formula: see text]. We introduce the concept of strategies that are stable with probability one. We also study an extended process and discuss other games.

How development affects evolution

Natural selection acts on developmentally constructed phenotypes, but how does development affect evolution? This question prompts a simultaneous consideration of development and evolution. However, there has been a lack of general mathematical frameworks mechanistically integrating the two, which may have inhibited progress on the question. Here, we use a new mathematical framework that mechanistically integrates development into evolution to analyse how development affects evolution. We show that, while selection pushes genotypic and phenotypic evolution up the fitness landscape, development determines the admissible evolutionary pathway, such that evolutionary outcomes occur at path peaks rather than landscape peaks. Changes in development can generate path peaks, triggering genotypic or phenotypic diversification, even on constant, single-peak landscapes. Phenotypic plasticity, niche construction, extra-genetic inheritance, and developmental bias alter the evolutionary path and hence the outcome. Thus, extra-genetic inheritance can have permanent evolutionary effects by changing the developmental constraints, even if extra-genetically acquired elements are not transmitted to future generations. Selective development, whereby phenotype construction points in the adaptive direction, may induce adaptive or maladaptive evolution depending on the developmental constraints. Moreover, developmental propagation of phenotypic effects over age enables the evolution of negative senescence. Overall, we find that development plays a major evolutionary role.

Eusociality through conflict dissolution

Eusociality, where largely unreproductive offspring help their mothers reproduce, is a major form of social organization. An increasingly documented feature of eusociality is that mothers induce their offspring to help by means of hormones, pheromones or behavioural displays, with evidence often indicating that offspring help voluntarily. The co-occurrence of maternal influence and offspring voluntary help may be explained by what we call the converted helping hypothesis, whereby maternally manipulated helping subsequently becomes voluntary. Such hypothesis requires that parent-offspring conflict is eventually dissolved-for instance, if the benefit of helping increases sufficiently over evolutionary time. We show that help provided by maternally manipulated offspring can enable the mother to sufficiently increase her fertility to transform parent-offspring conflict into parent-offspring agreement. This conflict-dissolution mechanism requires that helpers alleviate maternal life-history trade-offs, and results in reproductive division of labour, high queen fertility and honest queen signalling suppressing worker reproduction-thus exceptionally recovering diverse features of eusociality. As such trade-off alleviation seemingly holds widely across eusocial taxa, this mechanism offers a potentially general explanation for the origin of eusociality, the prevalence of maternal influence, and the offspring's willingness to help. Overall, our results explain how a major evolutionary transition can happen from ancestral conflict.

On the importance of evolving phenotype distributions on evolutionary diversification

Evolutionary branching occurs when a population with a unimodal phenotype distribution diversifies into a multimodally distributed population consisting of two or more strains. Branching results from frequency-dependent selection, which is caused by interactions between individuals. For example, a population performing a social task may diversify into a cooperator strain and a defector strain. Branching can also occur in multi-dimensional phenotype spaces, such as when two tasks are performed simultaneously. In such cases, the strains may diverge in different directions: possible outcomes include division of labor (with each population performing one of the tasks) or the diversification into a strain that performs both tasks and another that performs neither. Here we show that the shape of the population's phenotypic distribution plays a role in determining the direction of branching. Furthermore, we show that the shape of the distribution is, in turn, contingent on the direction of approach to the evolutionary branching point. This results in a distribution-selection feedback that is not captured in analytical models of evolutionary branching, which assume monomorphic populations. Finally, we show that this feedback can influence long-term evolutionary dynamics and promote the evolution of division of labor.

Resident-invader dynamics of similar strategies in fluctuating environments

We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers.

Habitat deterioration promotes the evolution of direct development in metamorphosing species

Although metamorphosis is widespread in the animal kingdom, several species have evolved life‐cycle modifications to avoid complete metamorphosis. Some species, for example, many salamanders and newts, have deleted the adult stage via a process called paedomorphosis. Others, for example, some frog species and marine invertebrates, no longer have a distinct larval stage and reach maturation via direct development. Here we study which ecological conditions can lead to the loss of metamorphosis via the evolution of direct development. To do so, we use size‐structured consumer‐resource models in conjunction with the adaptive‐dynamics approach. In case the larval habitat deteriorates, individuals will produce larger offspring and in concert accelerate metamorphosis. Although this leads to the evolutionary transition from metamorphosis to direct development when the adult habitat is highly favorable, the population will go extinct in case the adult habitat does not provide sufficient food to escape metamorphosis. With a phylogenetic approach we furthermore show that among amphibians the transition of metamorphosis to direct development is indeed, in line with model predictions, conditional on and preceded by the evolution of larger egg sizes.

Cooperation can promote rescue or lead to evolutionary suicide during environmental change

The adaptation of populations to changing conditions may be affected by interactions between individuals. For example, when cooperative interactions increase fecundity, they may allow populations to maintain high densities and thus keep track of moving environmental optima. Simultaneously, changes in population density alter the marginal benefits of cooperative investments, creating a feedback loop between population dynamics and the evolution of cooperation. Here we model how the evolution of cooperation interacts with adaptation to changing environments. We hypothesize that environmental change lowers population size and thus promotes the evolution of cooperation, and that this, in turn, helps the population keep up with the moving optimum. However, we find that the evolution of cooperation can have qualitatively different effects, depending on which fitness component is reduced by the costs of cooperation. If the costs decrease fecundity, cooperation indeed speeds adaptation by increasing population density; if, in contrast, the costs decrease viability, cooperation may instead slow adaptation by lowering the effective population size, leading to evolutionary suicide. Thus, cooperation can either promote or-counterintuitively-hinder adaptation to a changing environment. Finally, we show that our model can also be generalized to other social interactions by discussing the evolution of competition during environmental change.

Lotka-Volterra approximations for evolutionary trait-substitution processes

A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an appropriately chosen Lotka–Volterra model. To this end, the assumption is made that similar phenotypes in the community form clusters that are well-separated from each other, as is expected to be generally the case when evolution proceeds through small mutational steps. Each phenotypic cluster is represented by a single phenotype, which we call an approximate phenotype and assign the cluster’s total population density. We present our results in three steps. First, for a set of approximate phenotypes with arbitrary equilibrium population densities before the invasion, the Lotka–Volterra approximation is proved to apply if the changes of the population densities of these phenotypes are sufficiently small during the transient following the invasion. Second, quantitative conditions for such small changes of population densities are derived as a relationship between within-cluster differences and the leading eigenvalue of the community’s Jacobian matrix evaluated at the equilibrium population densities before the invasion. Third, to demonstrate the utility of our results, the ‘invasion implies substitution’ result for monomorphic populations is extended to arbitrarily polymorphic populations consisting of well-recognizable and -separated clusters.

Environmental dimensionality determines species coexistence

According to the competitive-exclusion principle, the number n of regulating variables describing a given community dynamics is an upper bound on the number of species (or types or morphs) that can coexist at equilibrium. On occasion, it is possible to reformulate a model with a lower number of regulating variables than appeared in the initial specification. We call the smallest number of such variables the dimension of the environmental feedback, or environmental dimension for short. For studying which species can invade a community, it is enough to know the sign of each species' long-term growth rate, i.e., invasion fitness. Therefore, different indicators of population growth - so-called fitness proxies, such as the basic reproduction number-are sometimes preferred. However, as we show, different fitness proxies may have different dimensions. Fundamental characteristics such as the environmental dimension should not depend on such arbitrary choices. Here, we resolve this difficulty by introducing a refined definition of environmental dimension that focuses on neutral fitness contours. On this basis, we show that this definition of environmental dimension is not only unambiguous, i.e., independent of the choice of fitness proxy, but also constructive, i.e., applicable without needing to assess an infinite number of possible fitness proxies. We then investigate how to determine environmental dimensions by analysing the two components of the environmental feedback: the impact map describing how a community's resident species affect the regulating variables and the sensitivity map describing how population growth depends on the regulating variables. The dimension of the impact map is lower than n when the set of feasible environments is of lower dimension than n, and the dimension of the sensitivity map is lower than n when not all n regulating variables affect the sign of population growth independently. While the minimum of the dimensions of the impact and sensitivity maps provides an upper bound on the environmental dimension, the combined effect of the two maps can result in an even lower environmental dimension, which happens when the sensitivity map is insensitive to some aspects of the impact map's image. To facilitate the applications of the framework introduced here, we illustrate all key concepts with detailed worked examples. In view of these results, we claim that the environmental dimension is the ultimate generalization of the traditional and widely used notions of the "number of regulating variables" or the "number of limiting factors", and is thus the sharpest generally applicable upper bound on the number of species that can robustly coexist in a community.

Ecological changes with minor effect initiate evolution to delayed regime shifts

Regime shifts have been documented in a variety of natural and social systems. These abrupt transitions produce dramatic shifts in the composition and functioning of socioecological systems. Existing theory on ecosystem resilience has only considered regime shifts to be caused by changes in external conditions beyond a tipping point and therefore lacks an evolutionary perspective. In this study, we show how a change in external conditions has little ecological effect and does not push the system beyond a tipping point. The change therefore does not cause an immediate regime shift but instead triggers an evolutionary process that drives a phenotypic trait beyond a tipping point, thereby resulting (after a substantial delay) in a selection-induced regime shift. Our finding draws attention to the fact that regime shifts observed in the present may result from changes in the distant past, and highlights the need for integrating evolutionary dynamics into the theoretical foundation for ecosystem resilience.

How Does Joint Evolution of Consumer Traits Affect Resource Specialization?

Consumers regularly experience trade-offs in their ability to find, handle, and digest different resources. Evolutionary ecologists recognized the significance of this observation for the evolution and maintenance of biological diversity long ago and continue to elaborate on the conditions under which to expect one or several specialists, generalists, or combinations thereof. Existing theory based on a single evolving trait predicts that specialization requires strong trade-offs such that generalists perform relatively poorly, while weak trade-offs favor a single generalist. Here, we show that this simple dichotomy does not hold true under joint evolution of two or more foraging traits. In this case, the boundary between trade-offs resulting in resource specialists and resource generalists is shifted toward weaker trade-off curvatures. In particular, weak trade-offs can result in evolutionary branching, leading to the evolution of two coexisting resource specialists, while the evolution of a single resource generalist requires particularly weak trade-offs. These findings are explained by performance benefits due to epistatic trait interactions enjoyed by phenotypes that are specialized in more than one trait for the same resource.

The Evolution of Immigration Strategies Facilitates Niche Expansion by Divergent Adaptation in a Structured Metapopulation Model

Local adaptation and habitat choice are two key factors that control the distribution and diversification of species. Here we model habitat choice mechanistically as the outcome of dispersal with nonrandom immigration. We consider a structured metapopulation with a continuous distribution of patch types and determine the evolutionarily stable immigration strategy as the function linking patch type to the probability of settling in the patch on encounter. We uncover a novel mechanism whereby coexisting strains that only slightly differ in their local adaptation trait can evolve substantially different immigration strategies. In turn, different habitat use selects for divergent adaptations in the two strains. We propose that the joint evolution of immigration and local adaptation can facilitate diversification and discuss our results in the light of niche conservatism versus niche expansion.

Joint evolution of dispersal and connectivity

Functional connectivity, the realized flow of individuals between the suitable sites of a heterogeneous landscape, is a prime determinant of the maintenance and evolution of populations in fragmented habitats. While a large body of literature examines the evolution of dispersal propensity, it is less known how evolution shapes functional connectivity via traits that influence the distribution of the dispersers. Here, we use a simple model to demonstrate that, in a heterogeneous environment with clustered and solitary sites (i.e., with variable structural connectivity), the evolutionarily stable population contains strains that are strongly differentiated in their pattern of connectivity (local vs. global dispersal), but not necessarily in the fraction of dispersed individuals. Also during evolutionary branching, selection is disruptive predominantly on the pattern of connectivity rather than on dispersal propensity itself. Our model predicts diversification along a hitherto neglected axis of dispersal strategies and highlights the role of the solitary sites-the more isolated and therefore seemingly less important patches of habitat-in maintaining global dispersal that keeps all sites connected.

Modelling optimal behavioural strategies in structured populations using a novel theoretical framework

Understanding complex behavioural patterns of organisms observed in nature can be facilitated using mathematical modelling. The conventional paradigm in animal behavior modelling consists of maximisation of some evolutionary fitness function. However, the definition of fitness of an organism or population is generally subjective, and using different criteria can lead us to contradictory model predictions regarding optimal behaviour. Moreover, structuring of natural populations in terms of individual size or developmental stage creates an extra challenge for theoretical modelling. Here we revisit and formalise the definition of evolutionary fitness to describe long-term selection of strategies in deterministic self-replicating systems for generic modelling settings which involve an arbitrary function space of inherited strategies. Then we show how optimal behavioural strategies can be obtained for different developmental stages in a generic von-Foerster stage-structured population model with an arbitrary mortality term. We implement our theoretical framework to explore patterns of optimal diel vertical migration (DVM) of two dominant zooplankton species in the north-eastern Black Sea. We parameterise the model using 7 years of empirical data from 2007-2014 and show that the observed DVM can be explained as the result of a trade-off between depth-dependent metabolic costs for grazers, anoxia zones, available food, and visual predation.

Understanding the emergence of bacterial pathogens in novel hosts

Our understanding of the ecological and evolutionary context of novel infections is largely based on viral diseases, even though bacterial pathogens may display key differences in the processes underlying their emergence. For instance, host-shift speciation, in which the jump of a pathogen into a novel host species is followed by the specialization on that host and the loss of infectivity of previous host(s), is commonly observed in viruses, but less often in bacteria. Here, we suggest that the extent to which pathogens evolve host generalism or specialism following a jump into a novel host will depend on their level of adaptation to dealing with different environments, their rates of molecular evolution and their ability to recombine. We then explore these hypotheses using a formal model and show that the high levels of phenotypic plasticity, low rates of evolution and the ability to recombine typical of bacterial pathogens should reduce their propensity to specialize on novel hosts. Novel bacterial infections may therefore be more likely to result in transient spillovers or increased host ranges than in host shifts. Finally, consistent with our predictions, we show that, in two unusual cases of contemporary bacterial host shifts, the bacterial pathogens both have small genomes and rapid rates of substitution. Further tests are required across a greater number of emerging pathogens to assess the validity of our hypotheses. This article is part of the theme issue 'Dynamic and integrative approaches to understanding pathogen spillover'.

Large-amplitude consumer-resource cycles allow for the evolution of ontogenetic niche shifts in consumer life history

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We analyze the evolution of ontogenetic niche shifts under non-equilibrium dynamics.

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We assume a secondary resource that is only available for large individuals.

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Specialization on this resource is hardly possible in case of small-amplitude cycles.

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Large-amplitude cycles allow for specialization on the secondary resource.

In many size-structured populations individuals change resources during the course of their ontogenetic development. Different resources often require different adaptations to be effectively exploited. This leads to a trade-off between small and large individuals in direct developing species. Specialization on the resource used later in life turns out to be hardly possible in case of equilibrium dynamics. However, size-structured populations often exhibit population cycles. Non-equilibrium dynamics can change evolutionary behavior when compared with equilibrium dynamics. Here, we study the evolution of specialization on a secondary resource that is available only to large individuals, using the framework of adaptive dynamics. We show that in case of small-amplitude cycles, specialization on a secondary resource is hardly possible. Specialization will either decrease the resource intake of large individuals or severely increase competition among small individuals such that they cannot mature. Specialization on a secondary resource is often possible in case the population exhibits large-amplitude cycles. Specialization in that case increases the resource intake of large individuals and therefore prevents starvation. While specialization on a secondary resource increases competition among small individuals, maturation is still possible in case of large-amplitude cycles. We furthermore show that there is ecological bistability where small- and large-amplitude cycles coexist, giving rise to evolutionary bistability.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Host-pathogen coevolution in the presence of predators: fluctuating selection and ecological feedbacks

Host-pathogen coevolution is central to shaping natural communities and is the focus of much experimental and theoretical study. For tractability, the vast majority of studies assume the host and pathogen interact in isolation, yet in reality, they will form one part of complex communities, with predation likely to be a particularly key interaction. Here, I present, to my knowledge, the first theoretical study to assess the impact of predation on the coevolution of costly host resistance and pathogen transmission. I show that fluctuating selection is most likely when predators selectively prey upon infected hosts, but that saturating predation, owing to large handling times, dramatically restricts the potential for fluctuations. I also show how host evolution may drive either enemy to extinction, and demonstrate that while predation selects for low host resistance and high pathogen infectivity, ecological feedbacks mean this results in lower infection rates when predators are present. I emphasize the importance of accounting for varying population sizes, and place the models in the context of recent experimental studies.

From the Price equation to the selection gradient in class-structured populations: a quasi-equilibrium route

Recent studies in theoretical evolutionary ecology have emphasised two approaches to modelling evolution. On the one hand, models based on a separation of time scales rely on the concept of invasion fitness. On the other hand, models based on the Price equation track the dynamics of a trait average, coupled with a description of ecological dynamics. The aim of this article is to show that, in class-structured populations, both approaches yield the same expression for the selection gradient under weak selection. Although the result is not new, I propose an alternative route to its derivation using the dynamics of scaled measures of between-class phenotypic differentiation. Under weak selection, these measures of phenotypic differentiation can be treated as fast variables compared to the trait mean, which allows for a quasi-equilibrium approximation. This suggests a different approach to calculating weak selection approximations of evolutionary dynamics, and clarifies the links between short- and long-term perspectives on evolution in structured populations.

Joint evolution of dispersal propensity and site selection in structured metapopulation models

We propose a novel mathematical model for a metapopulation in which dispersal occurs on two levels: juvenile dispersal from the natal site is mandatory but it may take place either locally within the natal patch or globally between patches. Within each patch, individuals live in sites. Each site can be inhabited by at most one individual at a time and it may be of high or low quality. A disperser immigrates into a high-quality site whenever it obtains one, but it immigrates into a low-quality site only with a certain probability that depends on the time within the dispersal season. The vector of these low-quality-site-acceptance probabilities is the site-selection strategy of an individual. We derive a proxy for the invasion fitness in this model and study the joint evolution of long-distance-dispersal propensity and site-selection strategy. We focus on the way different ecological changes affect the evolutionary dynamics and study the interplay between global patch-to-patch dispersal and local site-selection. We show that ecological changes affect site-selection mainly via the severeness of competition for sites, which often leads to effects that may appear counterintuitive. Moreover, the metapopulation structure may result in extremely complex site-selection strategies and even in evolutionary cycles. The propensity for long-distance dispersal is mainly determined by the metapopulation-level ecological factors. It is, however, also strongly affected by the winter-survival of the site-holders within patches, which results in surprising non-monotonous effects in the evolution of site-selection due to interplay with long-distance dispersal. Altogether, our results give new additional support to the recent general conclusion that evolution of site-selection is often dominated by the indirect factors that take place via density-dependence, which means that evolutionary responses can rarely be predicted by intuition.

Environmental dimensionality

The number of regulating variables n in a given system is an upper bound to the number of coexisting species at equilibrium according to the competitive exclusion principle. However, it may be possible to formulate the model with a lower number of regulating variables, the smallest number of which is the dimension of the environmental feedback. Here we investigate how that dimension can be determined by analysing the two parts of environmental feedback: The impact map describes how the extant species affect the regulating variables, and the sensitivity map describes how population growth depends on the regulating variables. For the equilibrium condition it is enough to know the sign of each population growth rate, and therefore as the sensitivity map, different measures of population growth can be chosen, such as the basic reproduction number. The dimension of the environmental feedback must not depend on that choice. Different sensitivity maps can have different global dimensions, on which the definition thus cannot be based. Here we show that the local sensitivity dimension is independent of the choice, so that the concept is well-defined. The impact dimension is lower than n when the feasible set of environments is of lower dimension than n, and sensitivity dimension is lower than n when not all environmental variables affect the sign of population growth independently. Their combined effect can result in even lower environmental dimension. We illustrate such situations with examples. In conclusion, the dimension of environmental feedback gives valuable information about the potential coexistence of species.

Theoretical Approaches in Evolutionary Ecology: Environmental Feedback as a Unifying Perspective

Evolutionary biology and ecology have a strong theoretical underpinning, and this has fostered a variety of modeling approaches. A major challenge of this theoretical work has been to unravel the tangled feedback loop between ecology and evolution. This has prompted the development of two main classes of models. While quantitative genetics models jointly consider the ecological and evolutionary dynamics of a focal population, a separation of timescales between ecology and evolution is assumed by evolutionary game theory, adaptive dynamics, and inclusive fitness theory. As a result, theoretical evolutionary ecology tends to be divided among different schools of thought, with different toolboxes and motivations. My aim in this synthesis is to highlight the connections between these different approaches and clarify the current state of theory in evolutionary ecology. Central to this approach is to make explicit the dependence on environmental dynamics of the population and evolutionary dynamics, thereby materializing the eco-evolutionary feedback loop. This perspective sheds light on the interplay between environmental feedback and the timescales of ecological and evolutionary processes. I conclude by discussing some potential extensions and challenges to our current theoretical understanding of eco-evolutionary dynamics.

Coevolution of patch-type dependent emigration and patch-type dependent immigration

The three phases of dispersal - emigration, transfer and immigration - are affecting each other and the former and latter decisions may depend on patch types. Despite the inevitable fact of the complexity of the dispersal process, patch-type dependencies of dispersal decisions modelled as emigration and immigration are usually missing in theoretical dispersal models. Here, I investigate the coevolution of patch-type dependent emigration and patch-type dependent immigration in an extended Hamilton-May model. The dispersing population inhabits a landscape structured into many patches of two types and disperses during a continuous-time season. The trait under consideration is a four dimensional vector consisting of two values for emigration probability from the patches and two values for immigration probability into the patches of each type. Using the adaptive dynamics approach I show that four qualitatively different dispersal strategies may evolve in different parameter regions, including a counterintuitive strategy, where patches of one type are fully dispersed from (emigration probability is one) but individuals nevertheless always immigrate into them during the dispersal season (immigration probability is one). I present examples of evolutionary branching in a wide parameter range, when the patches with high local death rate during the dispersal season guarantee a high expected disperser output. I find that two dispersal strategies can coexist after evolutionary branching: a strategy with full immigration only into the patches with high expected disperser output coexists with a strategy that immigrates into any patch. Stochastic simulations agree with the numerical predictions. Since evolutionary branching is also found when immigration evolves alone, the present study is adding coevolutionary constraints on the emigration traits and hence finds that the coevolution of a higher dimensional trait sometimes hinders evolutionary diversification.

Defining fitness in an uncertain world

The recently elucidated definition of fitness employed by Fisher in his fundamental theorem of natural selection is combined with reproductive values as appropriately defined in the context of both random environments and continuing fluctuations in the distribution over classes in a class-structured population. We obtain astonishingly simple results, generalisations of the Price Equation and the fundamental theorem, that show natural selection acting only through the arithmetic expectation of fitness over all uncertainties, in contrast to previous studies with fluctuating demography, in which natural selection looks rather complicated. Furthermore, our setting permits each class to have its characteristic ploidy, thus covering haploidy, diploidy and haplodiploidy at the same time; and allows arbitrary classes, including continuous variables such as condition. The simplicity is achieved by focussing just on the effects of natural selection on genotype frequencies: while other causes are present in the model, and the effect of natural selection is assessed in their presence, these causes will have their own further effects on genoytpe frequencies that are not assessed here. Also, Fisher’s uses of reproductive value are shown to have two ambivalences, and a new axiomatic foundation for reproductive value is endorsed. The results continue the formal darwinism project, and extend support for the individual-as-maximising-agent analogy to finite populations with random environments and fluctuating class-distributions. The model may also lead to improved ways to measure fitness in real populations.

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.

A Parent-Offspring Trade-Off Limits the Evolution of an Ontogenetic Niche Shift

Many free-living animal species, including the majority of fish, insects, and amphibians, change their food and habitat during their life. Even though these ontogenetic changes in niche are common, it is not well understood which ecological conditions have favored the evolution of these shifts. Using an adaptive dynamics approach, we show that it is evolutionarily advantageous to switch to an alternative food source in the course of ontogeny when this results in a higher intake rate for the switching consumers. Individuals are, however, not able to specialize on this new food source when this negatively affects the performance early in life on the original food source. Selection on these early life stages is so strong that in species with a complete diet shift, evolution results in large juveniles and adults that are maladapted to the alternative food source while their offspring are specialized on the original food source when young. These outcomes suggest strong selection to decouple the different life stages, such that they can maximize their performance on different food sources independently from each other. Metamorphosis could be a way to decouple the different life stages and therefore evolve in species that feed on multiple food sources during their life.

Evolution of dispersal under variable connectivity

The pattern of connectivity between local populations or between microsites supporting individuals within a population is a poorly understood factor affecting the evolution of dispersal. We modify the well-known Hamilton-May model of dispersal evolution to allow for variable connectivity between microsites. For simplicity, we assume that the microsites are either solitary, i.e., weakly connected through costly dispersal, or part of a well-connected cluster of sites with low-cost dispersal within the cluster. We use adaptive dynamics to investigate the evolution of dispersal, obtaining analytic results for monomorphic evolution and numerical results for the co-evolution of two dispersal strategies. A monomorphic population always evolves to a unique singular dispersal strategy, which may be an evolutionarily stable strategy or an evolutionary branching point. Evolutionary branching happens if the contrast between connectivities is sufficiently high and the solitary microsites are common. The dimorphic evolutionary singularity, when it exists, is always evolutionarily and convergence stable. The model exhibits both protected and unprotected dimorphisms of dispersal strategies, but the dimorphic singularity is always protected. Contrasting connectivities can thus maintain dispersal polymorphisms in temporally stable environments.

Frequency-dependent two-sex models: a new approach to sex ratio evolution with multiple maternal conditions

Mothers that experience different individual or environmental conditions may produce different proportions of male to female offspring. The Trivers-Willard hypothesis, for instance, suggests that mothers with different qualities (size, health, etc.) will use different sex ratios if maternal quality differentially affects sex-specific reproductive success. Condition-dependent, or facultative, sex ratio strategies like these allow multiple sex ratios to coexist within a population. They also create complex population structure due to the presence of multiple maternal conditions. As a result, modeling facultative sex ratio evolution requires not only sex ratio strategies with multiple components, but also two-sex population models with explicit stage structure. To this end, we combine nonlinear, frequency-dependent matrix models and multidimensional adaptive dynamics to create a new framework for studying sex ratio evolution. We illustrate the applications of this framework with two case studies where the sex ratios depend one of two possible maternal conditions (age or quality). In these cases, we identify evolutionarily singular sex ratio strategies, find instances where one maternal condition produces exclusively male or female offspring, and show that sex ratio biases depend on the relative reproductive value ratios for each sex.

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.

Why Darwin would have loved evolutionary game theory

Humans have marvelled at the fit of form and function, the way organisms' traits seem remarkably suited to their lifestyles and ecologies. While natural selection provides the scientific basis for the fit of form and function, Darwin found certain adaptations vexing or particularly intriguing: sex ratios, sexual selection and altruism. The logic behind these adaptations resides in frequency-dependent selection where the value of a given heritable phenotype (i.e. strategy) to an individual depends upon the strategies of others. Game theory is a branch of mathematics that is uniquely suited to solving such puzzles. While game theoretic thinking enters into Darwin's arguments and those of evolutionists through much of the twentieth century, the tools of evolutionary game theory were not available to Darwin or most evolutionists until the 1970s, and its full scope has only unfolded in the last three decades. As a consequence, game theory is applied and appreciated rather spottily. Game theory not only applies to matrix games and social games, it also applies to speciation, macroevolution and perhaps even to cancer. I assert that life and natural selection are a game, and that game theory is the appropriate logic for framing and understanding adaptations. Its scope can include behaviours within species, state-dependent strategies (such as male, female and so much more), speciation and coevolution, and expands beyond microevolution to macroevolution. Game theory clarifies aspects of ecological and evolutionary stability in ways useful to understanding eco-evolutionary dynamics, niche construction and ecosystem engineering. In short, I would like to think that Darwin would have found game theory uniquely useful for his theory of natural selection. Let us see why this is so.

The transition from evolutionary stability to branching: A catastrophic evolutionary shift

Evolutionary branching—resident-mutant coexistence under disruptive selection—is one of the main contributions of Adaptive Dynamics (AD), the mathematical framework introduced by S.A.H. Geritz, J.A.J. Metz, and coauthors to model the long-term evolution of coevolving multi-species communities. It has been shown to be the basic mechanism for sympatric and parapatric speciation, despite the essential asexual nature of AD. After 20 years from its introduction, we unfold the transition from evolutionary stability (ESS) to branching, along with gradual change in environmental, control, or exploitation parameters. The transition is a catastrophic evolutionary shift, the branching dynamics driving the system to a nonlocal evolutionary attractor that is viable before the transition, but unreachable from the ESS. Weak evolutionary stability hence qualifies as an early-warning signal for branching and a testable measure of the community’s resilience against biodiversity. We clarify a controversial theoretical question about the smoothness of the mutant invasion fitness at incipient branching. While a supposed nonsmoothness at third order long prevented the analysis of the ESS-branching transition, we argue that smoothness is generally expected and derive a local canonical model in terms of the geometry of the invasion fitness before branching. Any generic AD model undergoing the transition qualitatively behaves like our canonical model.

Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspective

The fitness concept and perforce the definition of frequency independent fitnesses from population genetics is closely tied to discrete time population models with non-overlapping generations. Evolutionary ecologists generally focus on trait evolution through repeated mutant substitutions in populations with complicated life histories. This goes with using the per capita invasion speed of mutants as their fitness. In this paper we develop a concept of frequency independence that attempts to capture the practical use of the term by ecologists, which although inspired by population genetics rarely fits its strict definition. We propose to call the invasion fitnesses of an eco-evolutionary model frequency independent when the phenotypes can be ranked by competitive strength, measured by who can invade whom. This is equivalent to the absence of weak priority effects, protected dimorphisms and rock-scissor-paper configurations. Our concept differs from that of Heino et al. (TREE 13:367-370, 1998) in that it is based only on the signs of the invasion fitnesses, whereas Heino et al. based their definitions on the structure of the feedback environment, summarising the effect of all direct and indirect interactions between individuals on fitness. As it turns out, according to our new definition an eco-evolutionary model has frequency independent fitnesses if and only if the effect of the feedback environment on the fitness signs can be summarised by a single scalar with monotonic effect. This may be compared with Heino et al.'s concept of trivial frequency dependence defined by the environmental feedback influencing fitness, and not just its sign, in a scalar manner, without any monotonicity restriction. As it turns out, absence of the latter restriction leaves room for rock-scissor-paper configurations. Since in 'realistic' (as opposed to toy) models frequency independence is exceedingly rare, we also define a concept of weak frequency dependence, which can be interpreted intuitively as almost frequency independence, and analyse in which sense and to what extent the restrictions on the potential model outcomes of the frequency independent case stay intact for models with weak frequency dependence.

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.

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program of the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set D in trait space squared harbouring protected dimorphisms. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to D. However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evolutionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to D, just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibility that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es.

The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin's maximum principle

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin’s maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.