The adaptive dynamics of function-valued traits

Evolutionary-developmental (evo-devo) dynamics of hominin brain size

Brain size tripled in the human lineage over four million years, but why this occurred remains uncertain. Here, to study what caused this brain expansion, I mathematically model the evolutionary and developmental (evo-devo) dynamics of hominin brain size. The model recovers (1) the evolution of brain and body sizes of seven hominin species starting from brain and body sizes of the australopithecine scale, (2) the evolution of the hominin brain-body allometry and (3) major patterns of human development and evolution. I show that the brain expansion recovered is not caused by direct selection for brain size but by its genetic correlation with developmentally late preovulatory ovarian follicles. This correlation is generated over development if individuals experience a challenging ecology and seemingly cumulative culture, among other conditions. These findings show that the evolution of exceptionally adaptive traits may not be primarily caused by selection for them but by developmental constraints that divert selection.

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.

Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact

Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

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.

Towards a replicator dynamics model of age structured populations

We present a new modelling framework combining replicator dynamics, the standard model of frequency dependent selection, with an age-structured population model. The new framework allows for the modelling of populations consisting of competing strategies carried by individuals who change across their life cycle. Firstly the discretization of the McKendrick von Foerster model is derived. We show that the Euler–Lotka equation is satisfied when the new model reaches a steady state (i.e. stable frequencies between the age classes). This discretization consists of unit age classes where the timescale is chosen so that only a fraction of individuals play a single game round. This implies a linear dynamics and individuals not killed during the round are moved to the next age class; linearity means that the system is equivalent to a large Bernadelli–Lewis–Leslie matrix. Then we use the methodology of multipopulation games to derive two, mutually equivalent systems of equations. The first contains equations describing the evolution of the strategy frequencies in the whole population, completed by subsystems of equations describing the evolution of the age structure for each strategy. The second contains equations describing the changes of the general population’s age structure, completed with subsystems of equations describing the selection of the strategies within each age class. We then present the obtained system of replicator dynamics in the form of the mixed ODE-PDE system which is independent of the chosen timescale, and much simpler. The obtained results are illustrated by the example of the sex ratio model which shows that when different mortalities of the sexes are assumed, the sex ratio of 0.5 is obtained but that Fisher’s mechanism, driven by the reproductive value of the different sexes, is not in equilibrium.

Hamilton's rule, gradual evolution, and the optimal (feedback) control of phenotypically plastic traits

Most traits expressed by organisms, such as gene expression profiles, developmental trajectories, behavioural sequences and reaction norms are function-valued traits (colloquially "phenotypically plastic traits"), since they vary across an individual's age and in response to various internal and/or external factors (state variables). Furthermore, most organisms live in populations subject to limited genetic mixing and are thus likely to interact with their relatives. We here formalise selection on genetically determined function-valued traits of individuals interacting in a group-structured population, by deriving the marginal version of Hamilton's rule for function-valued traits. This rule simultaneously gives a condition for the invasion of an initially rare mutant function-valued trait and its ultimate fixation in the population (invasion thus implies substitution). Hamilton's rule thus underlies the gradual evolution of function-valued traits and gives rise to necessary first-order conditions for their uninvadability (evolutionary stability). We develop a novel analysis using optimal control theory and differential game theory, to simultaneously characterise and compare the first-order conditions of (i) open-loop traits - functions of time (or age) only, and (ii) closed-loop (state-feedback) traits - functions of both time and state variables. We show that closed-loop traits can be represented as the simpler open-loop traits when individuals do not interact or when they interact with clonal relatives. Our analysis delineates the role of state-dependence and interdependence between individuals for trait evolution, which has implications to both life-history theory and social evolution.

Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods

Here, we propose a computational approach to explore evolutionary fitness in complex biological systems based on empirical data using artificial neural networks. The essence of our approach is the following. We first introduce a ranking order of inherited elements (behavioral strategies or/and life history traits) in considered self-reproducing systems: we use available empirical information on selective advantages of such elements. Next, we introduce evolutionary fitness, which is formally described as a certain function reflecting the introduced ranking order. Then, we approximate fitness in the space of key parameters using a Taylor expansion. To estimate the coefficients in the Taylor expansion, we utilize artificial neural networks: we construct a surface to separate the domains of superior and interior ranking of pair inherited elements in the space of parameters. Finally, we use the obtained approximation of the fitness surface to find the evolutionarily stable (optimal) strategy which maximizes fitness. As an ecologically important study case, we apply our approach to explore the evolutionarily stable diel vertical migration of zooplankton in marine and freshwater ecosystems. Using machine learning we reconstruct the fitness function of herbivorous zooplankton from empirical data and predict the daily trajectory of a dominant species in the northeastern Black Sea.

Replicator dynamics for the game theoretic selection models based on state

The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describeenergy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game,with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed.

A function-valued trait approach to estimating the genetic basis of size at age and its potential role in fisheries-induced evolution

Natural selection is inherently a multivariate phenomenon. The selection pressure on size (natural and artificial) and the age at which selection occurs is likely to induce evolutionary changes in growth rates across the entire life history. However, the covariance structure that will determine the path of evolution for size at age has been studied in only a few fish species. We therefore estimated the genetic covariance function for size throughout ontogeny using Atlantic silversides (Menidia menidia) as the model system. Over a 3-year period, a total of 542 families were used to estimate the genetic covariance in length at age from hatch through maturity. The function-valued trait approach was employed to estimate the genetic covariance functions. A Bayesian hierarchical model was used to account for the unbalanced design, unequal measurement intervals, unequal sample sizes, and family-aggregated data. To improve mixing, we developed a two-stage sampler using a Gibbs sampler to generate the posterior of a well-mixing approximate model followed by an importance sampler to draw samples from posterior of the completely specified model. We found that heritability of length is age-specific and there are strong genetic correlations in length across ages that last 30 days or more. We used these estimates in a hypothetical model predicting the evolutionary response to harvesting following a single generation of selection under both sigmoidal and unimodal patterns of gear selectivity to illustrate the potential outcomes of ignoring the genetic correlations. In these scenarios, genetic correlations were found to have a strong effect on both the direction and magnitude of the response to harvest selection.

Spatiotemporal variation in thermal niches suggests lability rather than conservatism of thermal physiology along an environmental gradient

Temperature variation throughout a species range can be extensive and exert divergent spatiotemporal patterns of selection. The estimation of phenotypic differences of populations along environmental gradients provides information regarding population-level responses to changing environments and evolutionary lability in climate-relevant traits. However, few studies have found physiological differentiation across environmental gradients attributable to behavioural thermoregulation buffering physiological evolution. Here, we compared thermal sensitivity of physiological performance among three populations of the ornate tree lizard (Urosaurus ornatus) along a 1100 m elevational gradient in southeastern Arizona across years in order to determine whether spatial differences in thermal environments are capable of driving local physiological differentiation. Lizards exhibited significant population-level differences in thermal physiology. The thermal traits of lizards at low elevations included warmer body temperatures and higher preferred and critical thermal temperatures. In contrast, lizards at higher elevations had cooler body temperatures and lower preferred and critical thermal temperatures. Populations also exhibited differences in the optimal temperature for performance and thermal performance breadth. The direction of population variation was consistent across years. Environmental gradients can provide model systems for studying the evolution of thermal physiology, and our study is one of the first to suggest that population differentiation in thermal physiology could be more prominent than previously thought.

Local adaptation in dispersal in multi-resource landscapes

The distribution of resources in space has important consequences for the evolution of dispersal-related traits. Dispersal moderates patterns of gene flow and, consequently, the potential for local adaptation to spatially differentiated resource types. We lack both models and experiments that evaluate how dispersal evolves in landscapes with multiple resources. Here, we investigate the evolution of dispersal in landscapes that contain two resource types that differ in their spatial autocorrelations. Individuals may possess ecological traits that give them a fitness advantage on one or the other resource. We find that resources differing in their spatial autocorrelation select for different optimal dispersal strategies and, further, that some multi-resource landscapes can support the stable coexistence of distinct dispersal strategies. Whether divergence in dispersal strategies between resource specialists occurs depends on the underlying structure of the resources and the degree of linkage between dispersal strategies and ecological specialization. This work indicates that the spatial autocorrelation of resources is an important factor in determining when evolutionary branching is likely to occur, and sheds light on when secondary isolating mechanisms should arise between locally adapted specialists.

Variability in life-history switch points across and within populations explained by Adaptive Dynamics

Understanding the factors that shape the timing of life-history switch points (SPs; e.g. hatching, metamorphosis and maturation) is a fundamental question in evolutionary ecology. Previous studies examining this question from a fitness optimization perspective have advanced our understanding of why the timing of life-history transitions may vary across populations and environments. However, in nature we also often observe variability among individuals within populations. Optimization theory, which typically predicts a single optimal SP under physiological and environmental constraints for a given environment, cannot explain this variability. Here, we re-examine the evolution of a single life-history SP between juvenile and adult stages from an Adaptive Dynamics (AD) perspective, which explicitly considers the feedback between the dynamics of population and the evolution of life-history strategy. The AD model, although simple in structure, exhibits a diverse range of evolutionary scenarios depending upon demographic and environmental conditions, including the loss of the juvenile stage, a single optimal SP, alternative optimal SPs depending on the initial phenotype, and sympatric coexistence of two SP phenotypes under disruptive selection. Such predictions are consistent with previous optimization approaches in predicting life-history SP variability across environments and between populations, and in addition they also explain within-population variability by sympatric disruptive selection. Thus, our model can be used as a theoretical tool for understanding life-history variability across environments and, especially, within species in the same environment.

Twelve fundamental life histories evolving through allocation-dependent fecundity and survival

An organism's life history is closely interlinked with its allocation of energy between growth and reproduction at different life stages. Theoretical models have established that diminishing returns from reproductive investment promote strategies with simultaneous investment into growth and reproduction (indeterminate growth) over strategies with distinct phases of growth and reproduction (determinate growth). We extend this traditional, binary classification by showing that allocation‐dependent fecundity and mortality rates allow for a large diversity of optimal allocation schedules. By analyzing a model of organisms that allocate energy between growth and reproduction, we find twelve types of optimal allocation schedules, differing qualitatively in how reproductive allocation increases with body mass. These twelve optimal allocation schedules include types with different combinations of continuous and discontinuous increase in reproduction allocation, in which phases of continuous increase can be decelerating or accelerating. We furthermore investigate how this variation influences growth curves and the expected maximum life span and body size. Our study thus reveals new links between eco‐physiological constraints and life‐history evolution and underscores how allocation‐dependent fitness components may underlie biological diversity.

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.

A model for brain life history evolution

Complex cognition and relatively large brains are distributed across various taxa, and many primarily verbal hypotheses exist to explain such diversity. Yet, mathematical approaches formalizing verbal hypotheses would help deepen the understanding of brain and cognition evolution. With this aim, we combine elements of life history and metabolic theories to formulate a metabolically explicit mathematical model for brain life history evolution. We assume that some of the brain's energetic expense is due to production (learning) and maintenance (memory) of energy-extraction skills (or cognitive abilities, knowledge, information, etc.). We also assume that individuals use such skills to extract energy from the environment, and can allocate this energy to grow and maintain the body, including brain and reproductive tissues. The model can be used to ask what fraction of growth energy should be allocated at each age, given natural selection, to growing brain and other tissues under various biological settings. We apply the model to find uninvadable allocation strategies under a baseline setting ("me vs nature"), namely when energy-extraction challenges are environmentally determined and are overcome individually but possibly with maternal help, and use modern-human data to estimate model's parameter values. The resulting uninvadable strategies yield predictions for brain and body mass throughout ontogeny and for the ages at maturity, adulthood, and brain growth arrest. We find that: (1) a me-vs-nature setting is enough to generate adult brain and body mass of ancient human scale and a sequence of childhood, adolescence, and adulthood stages; (2) large brains are favored by intermediately challenging environments, moderately effective skills, and metabolically expensive memory; and (3) adult skill is proportional to brain mass when metabolic costs of memory saturate the brain metabolic rate allocated to skills.

Adaptive significance of critical weight for metamorphosis in holometabolous insects

Holometabolous insect larvae become committed to metamorphosis when they reach a critical weight. Although the physiological mechanisms involved in this process have been well-studied, the adaptive significance of the critical weight remains unclear. Here, we developed a life history model for holometabolous insects and evaluated it from the viewpoint of optimal energy allocation. We found that, without a priori assumptions about critical weight, the optimal growth schedule is always biphasic: larval tissues grow predominately until they reach a certain threshold, after which the imaginal tissues begin rapid growth, suggesting that the emergence of a critical weight as a phase-transition point is a natural consequence of optimal growth scheduling. Our model predicts the optimal timing of critical-weight attainment, in agreement with observations in phylogenetically-distinct species. Furthermore, it also predicts the scaling of growth scheduling against environmental change, i.e., the relative value and timing of the critical weight should be constant, thus providing a general interpretation of observed phenotypic plasticity. This scaling relationship allows the classification of adaptive responses in critical weight into five possible types that reflect the ecological features of focal insects. In this manner, our theory and its consistency with experimental observations demonstrate the adaptive significance of critical weight.

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.

Towards developing a general framework for modelling vertical migration in zooplankton

Diel vertical migration (DVM) of zooplankton is a widespread phenomenon in both oceans and lakes, and is generally considered to be the largest synchronized movement of biomass on Earth. Most existing mathematical models of DVM are based on the assumption that animals maximize a certain criterion such as the expected reproductive value, the venturous revenue, the ratio of energy gain/mortality or some predator avoidance function when choosing their instantaneous depth. The major shortcoming of this general point of view is that the predicted DVM may be strongly affected by a subjective choice of a particular optimization criterion. Here we argue that the optimal strategy of DVM can be unambiguously obtained as an outcome of selection in the underlying equations of genotype/traits frequency dynamics. Using this general paradigm, we explore the optimal strategy for the migration across different depths by zooplankton grazers throughout the day. To illustrate our ideas we consider four generic DVM models, each making different assumptions on the population dynamics of zooplankton, and demonstrate that in each model we need to maximize a particular functional to find the optimal strategy. Surprisingly, patterns of DVM obtained for different models greatly differ in terms of their parameters dependence. We then show that the infinite dimensional trait space of different zooplankton trajectories can be projected onto a low dimensional space of generalized parameters and the genotype evolution dynamics can be easily followed using this low-dimensional space. Using this space of generalized parameters we explore the influence of mutagenesis on evolution of DVM, and we show that strong mutagenesis allows the coexistence of an infinitely large number of strategies whereas for weak mutagenesis the selection results in the extinction of most strategies, with the surviving strategies all staying close to the optimal strategy in the corresponding mutagenesis-free system.

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.

Evolution of dispersal distance: maternal investment leads to bimodal dispersal kernels

Since dispersal research has mainly focused on the evolutionary dynamics of dispersal rates, it remains unclear what shape evolutionarily stable dispersal kernels have. Yet, detailed knowledge about dispersal kernels, quantifying the statistical distribution of dispersal distances, is of pivotal importance for understanding biogeographic diversity, predicting species invasions, and explaining range shifts. We therefore examine the evolution of dispersal kernels in an individual-based model of a population of sessile organisms, such as trees or corals. Specifically, we analyze the influence of three potentially important factors on the shape of dispersal kernels: distance-dependent competition, distance-dependent dispersal costs, and maternal investment reducing an offspring׳s dispersal costs through a trade-off with maternal fecundity. We find that without maternal investment, competition and dispersal costs lead to unimodal kernels, with increasing dispersal costs reducing the kernel׳s width and tail weight. Unexpectedly, maternal investment inverts this effect: kernels become bimodal at high dispersal costs. This increases a kernel׳s width and tail weight, and thus the fraction of long-distance dispersers, at the expense of simultaneously increasing the fraction of non-dispersers. We demonstrate the qualitative robustness of our results against variations in the tested parameter combinations.

Fisheries-induced disruptive selection

Commercial harvesting is recognized to induce adaptive responses of life-history traits in fish populations, in particular by shifting the age and size at maturation through directional selection. In addition to such evolution of a target stock, the corresponding fishery itself may adapt, in terms of fishing policy, technological progress, fleet dynamics, and adaptive harvest. The aim of this study is to assess how the interplay between natural and artificial selection, in the simplest setting in which a fishery and a target stock coevolve, can lead to disruptive selection, which in turn may cause trait diversification. To this end, we build an eco-evolutionary model for a size-structured population, in which both the stock׳s maturation schedule and the fishery׳s harvest rate are adaptive, while fishing may be subject to a selective policy based on fish size and/or maturity stage. Using numerical bifurcation analysis, we study how the potential for disruptive selection changes with fishing policy, fishing mortality, harvest specialization, life-history tradeoffs associated with early maturation, and other demographic and environmental parameters. We report the following findings. First, fisheries-induced disruptive selection is readily caused by commonly used fishing policies, and occurs even for policies that are not specific for fish size or maturity, provided that the harvest is sufficiently adaptive and large individuals are targeted intensively. Second, disruptive selection is more likely in stocks in which the selective pressure for early maturation is naturally strong, provided life-history tradeoffs are sufficiently consequential. Third, when a fish stock is overexploited, fisheries targeting only large individuals might slightly increase sustainable yield by causing trait diversification (even though the resultant yield always remains lower than the maximum sustainable yield that could be obtained under low fishing mortality, without causing disruptive selection). We discuss the broader implications of our results and highlight how these can be taken into account for designing evolutionarily informed fisheries-management regimes.

Adaptive dynamics with interaction structure

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the canonical equation of adaptive dynamics with interaction structure, characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.

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.

Evolution of complex density-dependent dispersal strategies

The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, "triple-threshold" strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.

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.

Integrating mechanistic organism–environment interactions into the basic theory of community and evolutionary ecology

This paper presents an overview of how mechanistic knowledge of organism–environment interactions, including biomechanical interactions of heat, mass and momentum transfer, can be integrated into basic theoretical population biology through mechanistic functional responses that quantitatively describe how organisms respond to their physical environment. Integrating such functional responses into simple community and microevolutionary models allows scaling up of the organism-level understanding from biomechanics both ecologically and temporally. For community models, Holling-type functional responses for predator–prey interactions provide a classic example of the functional response affecting qualitative model dynamics, and recent efforts are expanding analogous models to incorporate environmental influences such as temperature. For evolutionary models, mechanistic functional responses dependent on the environment can serve as fitness functions in both quantitative genetic and game theoretic frameworks, especially those concerning function-valued traits. I present a novel comparison of a mechanistic fitness function based on thermal performance curves to a commonly used generic fitness function, which quantitatively differ in their predictions for response to environmental change. A variety of examples illustrate how mechanistic functional responses enhance model connections to biologically relevant traits and processes as well as environmental conditions and therefore have the potential to link theoretical and empirical studies. Sensitivity analysis of such models can provide biologically relevant insight into which parameters and processes are important to community and evolutionary responses to environmental change such as climate change, which can inform conservation management aimed at protecting response capacity. Overall, the distillation of detailed knowledge or organism–environment interactions into mechanistic functional responses in simple population biology models provides a framework for integrating biomechanics and ecology that allows both tractability and generality.

Impact of environmental covariation in growth and mortality on evolving maturation reaction norms

Maturation age and size have important fitness consequences through their effects on survival probabilities and body sizes. The evolution of maturation reaction norms in response to environmental covariation in growth and mortality is therefore a key subject of life-history theory. The eco-evolutionary model we present and analyze here incorporates critical features that earlier studies of evolving maturation reaction norms have often neglected: the trade-off between growth and reproduction, source-sink population structure, and population regulation through density-dependent growth and fecundity. We report the following findings. First, the evolutionarily optimal age at maturation can be decomposed into the sum of a density-dependent and a density-independent component. These components measure, respectively, the hypothetical negative age at which an individual's length would be 0 and the delay in maturation relative to this offset. Second, along any growth trajectory, individuals mature earlier when mortality is higher. This allows us to deduce, third, how the shapes of evolutionarily optimal maturation reaction norms depend on the covariation between growth and mortality (positive or negative, linear or curvilinear, and deterministic or probabilistic). Providing eco-evolutionary explanations for many alternative reaction-norm shapes, our results appear to be in good agreement with current empirical knowledge on maturation dynamics.

The evolution of individual variation in communication strategies

Communication is a process in which senders provide information via signals and receivers respond accordingly. This process relies on two coevolving conventions: a "sender code" that determines what kind of signal is to be sent given the sender's state; and a "receiver code" that determines the appropriate responses to different signal types. By means of a simple but generic model, we show that polymorphic sender and receiver strategies emerge naturally during the evolution of communication, and that the number of alternative strategies observed at equilibrium depends on the potential for error in signal production. Our model suggests that alternative communication strategies will evolve whenever senders possess imperfect information about their own quality or state, signals are costly, and genetic mechanisms allow for a correlation between sender and receiver behavior. These findings provide an explanation for recent reports of individual differences in communication strategies, and suggest that the amount of individual variation that can be expected in communication systems depends on the type of information being conveyed. Our model also suggests a link between communication and the evolution of animal personalities, which is that individual differences in the production and interpretation of signals can result in consistent differences in behavior.

Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner's dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.

Quantitative genetic variation in static allometry in the threespine stickleback

The common pattern of replicated evolution of a consistent shape-environment relationship might reflect selection acting in similar ways within each environment, but divergently among environments. However, phenotypic evolution depends on the availability of additive genetic variation as well as on the direction of selection, implicating a bias in the distribution of genetic variance as a potential contributor to replicated evolution. Allometry, the relationship between shape and size, is a potential source of genetic bias that is poorly understood. The threespine stickleback, Gasterosteus aculeatus, provides an ideal system for exploring the contribution of genetic variance in body shape allometry to evolutionary patterns. The stickleback system comprises marine populations that exhibit limited phenotypic variation, and young freshwater populations which, following independent colonization events, have often evolved similar phenotypes in similar environments. In particular, stickleback diversification has involved changes in both total body size and relative size of body regions (i.e., shape). In a laboratory-reared cohort derived from an oceanic Alaskan population that is phenotypically and genetically representative of the ancestor of the diverse freshwater populations in this region, we determined the phenotypic static allometry, and estimated the additive genetic variation about these population-level allometric functions. We detected significant allometry, with larger fish having relatively smaller heads, a longer base to their second dorsal fin, and longer, shallower caudal peduncles. There was additive genetic variance in body size and in size-independent body shape (i.e., allometric elevation), but typically not in allometric slopes. These results suggest that the parallel evolution of body shape in threespine stickleback is not likely to have been a correlated response to selection on body size, or vice versa. Although allometry is common in fishes, this study highlights the need for additional data on genetic variation in allometric functions to determine how allometry evolves and how it influences phenotypic evolution.

A differential game theoretical analysis of mechanistic models for territoriality

In this paper, elements of differential game theory are used to analyze a spatially explicit home range model for interacting wolf packs when movement behavior is uncertain. The model consists of a system of partial differential equations whose parameters reflect the movement behavior of individuals within each pack and whose steady-state solutions describe the patterns of space-use associated to each pack. By controlling the behavioral parameters in a spatially-dynamic fashion, packs adjust their patterns of movement so as to find a Nash-optimal balance between spreading their territory and avoiding conflict with hostile neighbors. On the mathematical side, we show that solving a nonzero-sum differential game corresponds to finding a non-invasible function-valued trait. From the ecological standpoint, when movement behavior is uncertain, the resulting evolutionarily stable equilibrium gives rise to a buffer-zone, or a no-wolf's land where deer are known to find refuge.

Evolution of age-dependent sex-reversal under adaptive dynamics

We investigate the evolution of the age (or size) at sex-reversal in a model of sequential hermaphroditism, by means of the function-valued adaptive dynamics. The trait is the probability law of the age at sex-reversal considered as a random variable. Our analysis starts with the ecological model which was first introduced and analyzed by Calsina and Ripoll (Math Biosci 208(2), 393-418, 2007). The structure of the population is extended to a genotype class and a new model for an invading/mutant population is introduced. The invasion fitness functional is derived from the ecological setting, and it turns out to be controlled by a formula of Shaw-Mohler type. The problem of finding evolutionarily stable strategies is solved by means of infinite-dimensional linear optimization. We have found that these strategies correspond to sex-reversal at a single particular age (or size) even if the set of feasible strategies is considerably broader and allows for a probabilistic sex-reversal. Several examples, including in addition the population-dynamical stability, are illustrated. For a special case, we can show that an unbeatable size at sex-reversal must be larger than 69.3% of the expected size at death.

Trait Substitution Sequence process and Canonical Equation for age-structured populations

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin-McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.

The evolution of function-valued traits for conditional cooperation

In this paper we study the evolution of function-valued traits for cooperation in environments that display varying degrees of population viscosity. Traits measure an individual's intrinsic propensity to cooperate in a standard bilateral Prisoner's dilemma and can be increasing, decreasing or constant functions of the probability to interact with individuals of ones own genotype. We first analyse adaptation to homogenous environments (with constant degree of viscosity). Comparing environments characterized by different degrees of viscosity, we find that the relation between viscosity and the equilibrium type distribution is not monotone. In fact, it is possible that in fluid populations (no viscosity) there is more cooperation in equilibrium than in populations with intermediate degrees of viscosity. In a second step we analyse heterogenous environments (with varying degrees of viscosity). We find that under very weak assumptions on the distribution of the viscosity parameter strictly increasing functions are always selected and under some parameter constellations they are uniquely so.