The evolutionary dynamics of hyperparasites

Evolutionary theory has typically focused on pairwise interactions, such as those between hosts and parasites, with relatively little work having been carried out on more complex interactions including hyperparasites: parasites of parasites. Hyperparasites are common in nature, with the chestnut blight fungus virus CHV-1 a well-known natural example, but also notably include the phages of important human bacterial diseases. We build a general modeling framework for the evolution of hyperparasites that highlights the central role that the ability of a hyperparasite to be transmitted with its parasite plays in their evolution. A key result is that hyperparasites which transmit with their parasite hosts (hitchhike) will be selected for lower virulence, trending towards hypermutualism or hypercommensalism. We examine the impact on the evolution of hyperparasite systems of a wide range of host and parasite traits showing, for example, that high parasite virulence selects for higher hyperparasite virulence resulting in reductions in parasite virulence when hyperparasitized. Furthermore, we show that acute parasite infection will also select for increased hyperparasite virulence. Our results have implications for hyperparasite research, both as biocontrol agents and for their role in shaping community ecology and evolution and moreover emphasize the importance of understanding evolution in the context of multitrophic interactions.

The impact of dormancy on evolutionary branching

In this paper, we investigate the consequences of dormancy in the 'rare mutation' and 'large population' regime of stochastic adaptive dynamics. Starting from an individual-based micro-model, we first derive the Polymorphic Evolution Sequence of the population, based on a previous work by Baar and Bovier (2018). After passing to a second 'small mutations' limit, we arrive at the Canonical Equation of Adaptive Dynamics, and state a corresponding criterion for evolutionary branching, extending a previous result of Champagnat and Méléard (2011). The criterion allows a quantitative and qualitative analysis of the effects of dormancy in the well-known model of Dieckmann and Doebeli (1999) for sympatric speciation. In fact, quite an intuitive picture emerges: Dormancy enlarges the parameter range for evolutionary branching, increases the carrying capacity and niche width of the post-branching sub-populations, and, depending on the model parameters, can either increase or decrease the 'speed of adaptation' of populations. Finally, dormancy increases diversity by increasing the genetic distance between subpopulations.

Separation of evolutionary timescales in coevolving species

Many coevolutionary processes, including host-parasite and host-symbiont interactions, involve one species or trait which evolves much faster than the other. Whether or not a coevolutionary trajectory converges depends on the relative rates of evolutionary change in the two species, and so current adaptive dynamics approaches generally either determine convergence stability by considering arbitrary (often comparable) rates of evolutionary change or else rely on necessary or sufficient conditions for convergence stability. We propose a method for determining convergence stability in the case where one species is expected to evolve much faster than the other. This requires a second separation of timescales, which assumes that the faster evolving species will reach its evolutionary equilibrium (if one exists) before a new mutation arises in the more slowly evolving species. This method, which is likely to be a reasonable approximation for many coevolving species, both provides straightforward conditions for convergence stability and is less computationally expensive than traditional analysis of coevolution models, as it reduces the trait space from a two-dimensional plane to a one-dimensional manifold. In this paper, we present the theory underlying this new separation of timescales and provide examples of how it could be used to determine coevolutionary outcomes from models.

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.

Life history and deleterious mutation rate coevolution

The cost of germline maintenance gives rise to a trade-off between lowering the deleterious mutation rate and investing in life history functions. Therefore, life history and the mutation rate coevolve, but this coevolution is not well understood. We develop a mathematical model to analyse the evolution of resource allocation traits, which simultaneously affect life history and the deleterious mutation rate. First, we show that the invasion fitness of such resource allocation traits can be approximated by the basic reproductive number of the least-loaded class; the expected lifetime production of offspring without deleterious mutations born to individuals without deleterious mutations. Second, we apply the model to investigate (i) the coevolution of reproductive effort and germline maintenance and (ii) the coevolution of age-at-maturity and germline maintenance. This analysis provides two resource allocation predictions when exposure to environmental mutagens is higher. First, selection favours higher allocation to germline maintenance, even if it comes at the expense of life history functions, and leads to a shift in allocation towards reproduction rather than survival. Second, life histories tend to be faster, characterised by individuals with shorter lifespans and smaller body sizes at maturity. Our results suggest that mutation accumulation via the cost of germline maintenance can be a major force shaping life-history traits.

Evolution of pathogens with cross-immunity in response to healthcare interventions

Cross-immunity, as an evolutionary driver, can contribute to pathogen evolution, particularly pathogen diversity. Healthcare interventions aimed at reducing disease severity or transmission are commonly used to control diseases and can also induce pathogen evolution. Understanding pathogen evolution in the context of cross-immunity and healthcare interventions is crucial for infection control. This study starts by modelling cross-immunity, the extent of which is determined by strain traits and host characteristics. Given that all hosts have the same characteristics, full cross-immunity between residents and mutants occurs when mutation step sizes are small enough. Cross-immunity can be partial when the step size is large. The presence of partial cross-immunity reduces pathogen load and shortens the infectious period inside hosts, reducing transmission between hosts and improving host population survival and recovery. This study focuses on how pathogens evolve through small and large mutational steps and how healthcare interventions affect pathogen evolution. Using the theory of adaptive dynamics, we found that when mutational steps are small (only full cross-immunity is present), pathogen diversity cannot occur because it maximises the basic reproduction number. This results in intermediate values for both pathogen growth and clearance rates. However, when large mutational steps are allowed (with full and partial cross-immunity present), pathogens can evolve into multiple strains and induce pathogen diversity. The study also shows that different healthcare interventions can have varying effects on pathogen evolution. Generally, low levels of intervention are more likely to induce strain diversity, while high levels are more likely to result in strain reduction.

Hamilton's rule, the evolution of behavior rules and the wizardry of control theory

This paper formalizes selection on a quantitative trait affecting the evolution of behavior (or development) rules through which individuals act and react with their surroundings. Combining Hamilton's marginal rule for selection on scalar traits and concepts from optimal control theory, a necessary first-order condition for the evolutionary stability of the trait in a group-structured population is derived. The model, which is of intermediate level of complexity, fills a gap between the formalization of selection on evolving traits that are directly conceived as actions (no phenotypic plasticity) and selection on evolving traits that are conceived as strategies or function valued actions (complete phenotypic plasticity). By conceptualizing individuals as open deterministic dynamical systems expressing incomplete phenotypic plasticity, the model captures selection on a large class of phenotypic expression mechanisms, including developmental pathways and learning under life-history trade-offs. As an illustration of the results, a first-order condition for the evolutionary stability of behavior response rules from the social evolution literature is re-derived, strengthened, and generalized. All results of the paper also generalize directly to selection on multidimensional quantitative traits affecting behavior rule evolution, thereby covering neural and gene network evolution.

Population regulation and adaptive dynamics of cross-feeding

The particular importance of evolutionary studies in microbial experimental systems is that starting from the level of the metabolism of individual cells, the adaptive dynamics can be followed step by step by biochemical, genetic, and population dynamical tools. Moreover, the coincidence of evolutionary and ecological time scales helps to clarify the mutual role of ecological and evolutionary principles in predicting adaptive dynamics in general. Ecological principles define the ecological conditions under which adaptive branching can occur. This paper overviews and interprets the results of empirical and modeling studies of the evolution of metabolic cross-feeding in glucose-limited E.coli chemostats and batch cultures in the context of theories of robust coexistence and adaptive dynamics. Empirical results consistently demonstrate that the interactions between cells are mediated by the changing metabolite concentrations in the cultures and modeling confirms that these changes may control the adaptive dynamics of the clones. In consequence, the potential results of evolution can be predicted at the functional level by evolutionary flux balance analysis (evoFBA), while the genetic changes are more contingent. evoFBA follows the scheme of adaptive dynamics theory by calculating the feedback environment that changes during the evolutionary process and provides a promising tool to further investigate adaptive divergence in small microbial communities. Three general conclusions close the paper.

Evolution of class-structured populations in periodic environments

What is the influence of periodic environmental fluctuations on life-history evolution? We present a general theoretical framework to understand and predict the long-term evolution of life-history traits under a broad range of ecological scenarios. Specifically, we investigate how periodic fluctuations affect selection when the population is also structured in distinct classes. This analysis yields time-varying selection gradients that clarify the influence of the fluctuations of the environment on the competitive ability of a specific life-history mutation. We use this framework to analyse the evolution of key life-history traits of pathogens. We examine three different epidemiological scenarios and we show how periodic fluctuations of the environment can affect the evolution of virulence and transmission as well as the preference for different hosts. These examples yield new and testable predictions on pathogen evolution, and illustrate how our approach can provide a better understanding of the evolutionary consequences of time-varying environmental fluctuations in a broad range of scenarios.

Life-History traits and the replicator equation

Due to the relevance for conservation biology, there is an increasing interest to extend evolutionary genomics models to plant, animal or microbial species. However, this requires to understand the effect of life-history traits absent in humans on genomic evolution. In this context, it is fundamentally of interest to generalize the replicator equation, which is at the heart of most population genomics models. However, as the inclusion of life-history traits generates models with a large state space, the analysis becomes involving. We focus, here, on quiescence and seed banks, two features common to many plant, invertebrate and microbial species. We develop a method to obtain a low-dimensional replicator equation in the context of evolutionary game theory, based on two assumptions: (1) the life-history traits are per se neutral, and (2) frequency-dependent selection is weak. We use the results to investigate the evolution and maintenance of cooperation based on the Prisoner's dilemma and the snowdrift game. We first consider the generalized replicator equation, and then refine the investigation using adaptive dynamics. It turns out that, depending on the structure and timing of the quiescence/dormancy life-history trait, cooperation in a homogeneous population can be stabilized. We finally discuss and highlight the relevance of these results for plant, invertebrate and microbial communities.

The importance of ecological dynamics in evolutionary processes: a host-bacteriophage model revisited

A recent study of adaptive dynamics of lysis propensity in temperate phages suggested that full lysogeny emerges as the outcome of bacteriophage evolution in a simple host-phage system. The conclusion is based on the premise that mutant strains necessarily appear in equilibrium host-phage environments. Revisiting the model, we show that the ecological system exhibits richer asymptotic dynamics and that, in a certain parameter regime, evolution may in fact drive lysis propensity towards an evolutionary singularity in which a non-zero proportion of phages initiate infection in a lytic cycle. These singularities act as points of evolutionary diversification, leading to periodic coexistence of two distinct phage strains on the evolutionary time-scale. One of the two strains in the dimorphic evolutionary singularity is fully lysogenic (in the sense that cell infection always leads to lysogeny), while the other is partially lytic. Our study thus highlights the importance of ecological interactions as a driver of evolution.

Evolutionary coexistence in a metacommunity: Competition-colonization trade-off, ownership effects, environmental fluctuations

We study the adaptive dynamics of the colonization rate of species living in a patchy habitat when there is a trade-off with the competitive strength for individual patches. To that end, we formulate a continuous-time competition-colonization model that also includes ownership effects as well as random disturbance affecting the mortality rate. We find that intermediate disturbance (as measured by the fluctuation intensity of the mortality rate), a strong competition-colonization trade-off, and a weak ownership effect are necessary conditions for evolutionary branching and hence for the emergence of polymorphisms (i.e., coexistence) by small evolutionary steps. Specifically, concerning ownership we find that with low-intermediate disturbance, a weak ownership advantage favours evolutionary branching while ownership disadvantage does not. This asymmetry disappears at the higher-intermediate disturbance. Moreover, at a low-intermediate disturbance, the effect of the strength of the competition-colonization trade-off on evolutionary branching is non-monotonic disappears because the possibility of branching disappears again when the trade-off is too strong. We also find that there can be multiple evolutionary attractors for polymorphic populations, each with its own basin of attraction. With small but non-zero random evolutionary steps and depending on the initial polymorphic condition just after branching, a coevolutionary trajectory may come arbitrarily close to the shared boundary of two such basins and may even jump from one side to the other, which can lead to various kinds of long-term evolutionary dynamics, including evolutionary branching-extinction cycles.

Virulence management: Closing the feedback loop between healthcare interventions and virulence evolution

We study the evolution of virulence of an endemic pathogen in response to healthcare interventions which affect host recovery and pathogen transmission. By anticipating the evolutionary response of the pathogen we may develop effective long-term management strategies for controlling the impact of the endemic on the society. To that end, we use standard Adaptive Dynamics techniques in an SIS model. The recovery rate and the transmission rate, both of which can be affected by healthcare interventions, are used as evolutionary control variables. The effect of interventions may be density-independent (self-help based on healthcare instructions) or density-dependent (when assistance of a healthcare worker is required). We consider the evolutionary response of the pathogen both to abrupt changes and to gradual changes in the level of healthcare intervention. Healthcare intervention is optimised for three alternative objectives: minimisation of virulence, minimisation of the probability that an infected individual dies of the disease, and total eradication of the endemic. We find that the optimal strategy may depend on the objective. High levels of healthcare intervention may eradicate the pathogen, but this option may not be available for budgetary reasons or otherwise. Counterintuitively, to minimise virulence, one should keep healthcare interventions at a minimum, while to minimise the probability for an infected individual to die of the disease, both low and high levels of healthcare intervention suffice. Changes in the level of healthcare intervention should be implemented fast (not gradually) in order to avoid sudden changes in pathogen evolution and the possible emergence of multiple simultaneously coexisting pathogen strains.

Coevolution of the reckless prey and the patient predator

The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator-prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.

Character displacement in the presence of multiple trait differences: Evolution of the storage effect in germination and growth

Ecological character displacement is a prominent hypothesis for the maintenance of ecological differences between species that are critical to stable coexistence. Models of character displacement often ascribe interspecific competitive interactions to a single character, but multiple characters contribute to competition, and their effects on selection can be nonadditive. Focusing on one character, we ask if other characters that affect competition alter evolutionary outcomes for the focal character. We address this question using the variable environment seed bank model for two species with two traits. The focal trait is the temporal pattern of germination, which is evolutionary labile. The other trait is the temporal pattern of plant growth, which is assumed fixed. We ask whether evolutionary divergence of germination patterns between species depends on species differences in plant growth. Patterns of growth can affect selection on germination patterns in two ways. First, cues present at germination can provide information about future growth. Second, germination and growth jointly determine the biomass of plants, which determines demand for resources. Germination and growth contribute to the selection gradient in distinct components, one density-independent and the other density-dependent. Importantly, the relative strengths of the components are key. When the density-dependent component is stronger, displacement in germination patterns between species is larger. Stronger cues at germination strengthen the density-independent component by increasing the benefits of germinating in years of favorable growth. But cues also affect the density-dependent component by boosting a species' biomass, and hence its competitive effect, in good years. Consequently, cues weaken character displacement when growth patterns are similar for two competitors, but favor displacement when growth patterns are species-specific. Understanding how these selection components change between contexts can help understand the origin and maintenance of species differences in germination patterns in temporally fluctuating environments.

Evolution and Adaptation of Anti-predation Response of Prey in a Two-Patchy Environment

When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. We formulate a patch model to investigate the aforementioned fear effect which is indirect, in contrast to the predation as a direct effect, of the predator on the prey population. We consider not only cost but also benefit of anti-predation response of the prey, and explore their trade-offs together as well as the impact of the fear effect mediated dispersals of the prey. In the case of constant response level, if there is no dispersal and for some given response functions, the model indicates the existence of an evolutionary stable strategy which is also a convergence stable strategy for the response level; and if there is dispersal, the analysis of the model shows that it will enhance the co-persistence of the prey on both patches. Considering the trait as another variable, we continue to study the evolution of anti-predation strategy for the model with dispersal, which leads to a three-dimensional system of ordinary differential equations. We perform some numerical simulations, which demonstrate global convergence to a positive equilibrium with the response level evolving towards a positive constant level, implying the existence of an optimal anti-predation response level.

The evolution of the irreversible transition from a free-swimming state to an immobile sessile state in aquatic invertebrates modelled in a chemostat

To better understand the environmental factors and ecological processes underlying the evolution of the irreversible transition from a free-swimming state to an immobile sessile state as seen in many aquatic invertebrates, we study the adaptive dynamics of the settling rate of a hypothetical microorganism onto the wall of a chemostat. The two states, floating or settled, differ in their nutrient ingestion, reproduction and death rate. We consider three different settling mechanisms involving competition for space on the wall: (i) purely exploitative competition where free-swimming individuals settle in vacant space only, (ii) mixed exploitative and interference competition where individuals attempt to settle in any place but fail and die if the space is already occupied, and (iii) mixed exploitative and interference competition, but now settling in occupied space is successful and the former occupant dies. In the simplified environment of the chemostat, the input concentration of nutrients and the dilution rate of the tank are the main environmental control variables. Using the theory of adaptive dynamics, we find that the settling mechanisms and environmental control variables have qualitatively different effects on the evolution of the settling rate in terms of the direction of evolution as well as on species diversity. In the case of purely exploitative competition a small change in the settings of the environmental control variables can lead to an abrupt reversal of the direction of evolution, while in the case of mixed exploitative and interference competition the effect is gradual. For all three settling mechanisms, periodic fluctuations in the nutrient input open the possibility of evolutionary branching leading to the long-term coexistence of an intermediate and an infinitely high settling rates (in the case of low-frequency fluctuations), and an intermediate and a zero settling rates (in the case of high-frequency fluctuations).

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.

Surprising spatiotemporal stability of a multi-peak fitness landscape revealed by independent field experiments measuring hybrid fitness

The effect of the environment on fitness in natural populations is a fundamental question in evolutionary biology. However, experimental manipulations of both environment and phenotype at the same time are rare. Thus, the relative importance of the competitive environment versus intrinsic organismal performance in shaping the location, height, and fluidity of fitness peaks and valleys remains largely unknown. Here, we experimentally tested the effect of competitor frequency on the complex fitness landscape driving adaptive radiation of a generalist and two trophic specialist pupfishes, a scale-eater and molluscivore, endemic to hypersaline lakes on San Salvador Island (SSI), Bahamas. We manipulated phenotypes, by generating 3407 F4/F5 lab-reared hybrids, and competitive environment, by altering the frequency of rare transgressive hybrids between field enclosures in two independent lake populations. We then tracked hybrid survival and growth rates across these four field enclosures for 3-11 months. In contrast to competitive speciation theory, we found no evidence that the frequency of hybrid phenotypes affected their survival. Instead, we observed a strikingly similar fitness landscape to a previous independent field experiment, each supporting multiple fitness peaks for generalist and molluscivore phenotypes and a large fitness valley isolating the divergent scale-eater phenotype. These features of the fitness landscape were stable across manipulated competitive environments, multivariate trait axes, and spatiotemporal heterogeneity. We suggest that absolute performance constraints and divergent gene regulatory networks shape macroevolutionary (interspecific) fitness landscapes in addition to microevolutionary (intraspecific) competitive dynamics. This interplay between organism and environment underlies static and dynamic features of the adaptive landscape.

Ecological and evolutionary consequences of predator-prey role reversal: Allee effect and catastrophic predator extinction

In many terrestrial, marine, and freshwater predator-prey communities, young predators can be vulnerable to attacks by large prey. Frequent prey counter-attacks may hinder the persistence of predators. Despite the commonness of such role reversals in nature, they have rarely been addressed in evolutionary modelling. To understand how role reversals affect ecological and evolutionary dynamics of a predator-prey community, we derived an ecological model from individual-level processes using ordinary differential equations. The model reveals complex ecological dynamics, with possible bistability between alternative coexistence states and an Allee effect for the predators. We find that when prey counter-attacks are frequent, cannibalism is necessary for predator persistence. Using numerical analysis, we also find that a sudden ecological shift from coexistence to predator extinction can occur through several catastrophic bifurcations, including 'saddle-node', 'homoclinic', and 'subcritical Hopf'. The analysis of single-species evolution reveals that predator selection towards increasing or decreasing cannibalism triggers a catastrophic shift towards an extinction state of the predators. Such an evolutionary extinction of the predators may also be caused by prey selection towards increasing foraging activity because it facilitates encounters with vulnerable, young predators. The analysis of predator-prey coevolution further demonstrates that predator's catastrophic extinction becomes an even more likely outcome than in single-species evolution. Our results suggest that when young predators are vulnerable to prey attacks, a sudden extinction of the predators may be more common than currently understood.

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.

How geographic productivity patterns affect food-web evolution

It is well recognized that spatial heterogeneity and overall productivity have important consequences for the diversity and community structure of food webs. Yet, few, if any, studies have considered the effects of heterogeneous spatial distributions of primary production. Here, we theoretically investigate how the variance and autocorrelation length of primary production affect properties of evolved food webs consisting of one autotroph and several heterotrophs. We report the following findings. (1) Diversity increases with landscape variance and is unimodal in autocorrelation length. (2) Trophic level increases with landscape variance and is unimodal in autocorrelation length. (3) The extent to which the spatial distribution of heterotrophs differ from that of the autotroph increases with landscape variance and decreases with autocorrelation length. (4) Components of initial disruptive selection experienced by the ancestral heterotroph predict properties of the final evolved communities. Prior to our study reported here, several authors had hypothesized that diversity increases with the landscape variance of productivity. Our results support their hypothesis and contribute new facets by providing quantitative predictions that also account for autocorrelation length and additional properties of the evolved communities.

A Mathematical Dissection of the Adaptation of Cell Populations to Fluctuating Oxygen Levels

The disordered network of blood vessels that arises from tumour angiogenesis results in variations in the delivery of oxygen into the tumour tissue. This brings about regions of chronic hypoxia (i.e. sustained low oxygen levels) and regions with alternating periods of low and relatively higher oxygen levels, and makes it necessary for cancer cells to adapt to fluctuating environmental conditions. We use a phenotype-structured model to dissect the evolutionary dynamics of cell populations exposed to fluctuating oxygen levels. In this model, the phenotypic state of every cell is described by a continuous variable that provides a simple representation of its metabolic phenotype, ranging from fully oxidative to fully glycolytic, and cells are grouped into two competing populations that undergo heritable, spontaneous phenotypic variations at different rates. Model simulations indicate that, depending on the rate at which oxygen is consumed by the cells, dynamic nonlinear interactions between cells and oxygen can stimulate chronic hypoxia and cycling hypoxia. Moreover, the model supports the idea that under chronic-hypoxic conditions lower rates of phenotypic variation lead to a competitive advantage, whereas higher rates of phenotypic variation can confer a competitive advantage under cycling-hypoxic conditions. In the latter case, the numerical results obtained show that bet-hedging evolutionary strategies, whereby cells switch between oxidative and glycolytic phenotypes, can spontaneously emerge. We explain how these results can shed light on the evolutionary process that may underpin the emergence of phenotypic heterogeneity in vascularised tumours.

How community adaptation affects biodiversity-ecosystem functioning relationships

Evidence is growing that evolutionary dynamics can impact biodiversity-ecosystem functioning (BEF) relationships. However the nature of such impacts remains poorly understood. Here we use a modelling approach to compare random communities, with no trait evolutionary fine-tuning, and co-adapted communities, where traits have co-evolved, in terms of emerging biodiversity-productivity, biodiversity-stability and biodiversity-invasion relationships. Community adaptation impacted most BEF relationships, sometimes inverting the slope of the relationship compared to random communities. Biodiversity-productivity relationships were generally less positive among co-adapted communities, with reduced contribution of sampling effects. The effect of community-adaptation, though modest regarding invasion resistance, was striking regarding invasion tolerance: co-adapted communities could remain very tolerant to invasions even at high diversity. BEF relationships are thus contingent on the history of ecosystems and their degree of community adaptation. Short-term experiments and observations following recent changes may not be safely extrapolated into the future, once eco-evolutionary feedbacks have taken place.

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.

Evolution of resource specialisation in competitive metacommunities

Spatial environmental heterogeneity coupled with dispersal can promote ecological persistence of diverse metacommunities. Does this premise hold when metacommunities evolve? Using a two-resource competition model, we studied the evolution of resource-uptake specialisation as a function of resource type (substitutable to essential) and shape of the trade-off between resource uptake affinities (generalist- to specialist-favouring). In spatially homogeneous environments, evolutionarily stable coexistence of consumers is only possible for sufficiently substitutable resources and specialist-favouring trade-offs. Remarkably, these same conditions yield comparatively low diversity in heterogeneous environments, because they promote sympatric evolution of two opposite resource specialists that, together, monopolise the two resources everywhere. Consumer diversity is instead maximised for intermediate trade-offs and clearly substitutable or clearly essential resources, where evolved metacommunities are characterised by contrasting selection regimes. Taken together, our results present new insights into resource-competition-mediated evolutionarily stable diversity in homogeneous and heterogeneous environments, which should be applicable to a wide range of systems.

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.

Is there a Trivers-Willard effect for parental investment? Modelling evolutionarily stable strategies using a matrix population model with nonlinear mating

The Trivers-Willard hypothesis (TWH) states that parents in good condition preferentially produce the sex with a higher variation in reproductive success, whereas parents in bad condition favour the opposite sex. Theorists distinguish two variants of the TWH: (a) a biased sex-ratio at birth and (b) biased parental investment after birth. It has been argued before that the conditions stated by Trivers and Willard (good condition is inherited and affects reproductive success more strongly for one of the sexes) are sufficient for the sex-ratio version but insufficient for the investment version of the TWH. However, it has not yet been investigated how these conditions affect parental investment in high and low quality parents, depending on the life-cycle of a species. The present study aims to fill this gap by introducing a multi-stage matrix population model with nonlinear mating to describe the effects of parental investment after birth on the reproductive values of male and female individuals. Using methods from adaptive dynamics and evolutionary invasion analysis, evolutionary trajectories and evolutionarily stable strategies are derived for different parameterizations of the model. Simulation results demonstrate that the conditions given by Trivers and Willard produce a general bias of parental investment towards the sex with higher variance in reproductive value. This bias is stronger for low-quality parents than for high-quality parents and matches the expected marginal offspring reproductive values for parental investment.

The effect of temporal fluctuations on the evolution of host tolerance to parasitism

There are many mechanisms that hosts can evolve to defend against parasites, two of which are resistance and tolerance. These defences often have different evolutionary behaviours, and it is important to consider how each individual mechanism may respond to changes in environment. In particular, host defence through tolerance is predicted to be unlikely to lead to variation, despite many observations of diversity in both animal and plant systems. Hence understanding the drivers of diversity in host defence and parasite virulence is vital for predicting future evolutionary changes in infectious disease dynamics. It has been suggested that heterogeneous environments might generally promote diversity, but the effect of temporal fluctuations has received little attention theoretically or empirically, and there has been no examination of how temporal fluctuations affects the evolution of host tolerance. In this study, we use a mathematical model to investigate the evolution of host tolerance in a temporally fluctuating environment. We show that investment in tolerance increases in more variable environments, giving qualitatively different evolutionary behaviours when compared to resistance. Once seasonality is introduced evolutionary branching though tolerance can occur and create diversity within the population, although potentially only temporarily. This branching behaviour arises due to the emergence of a negative feedback with the maximum infected density on a cycle, which is strongest when the infected population is large. This work reinforces the qualitative differences between tolerance and resistance evolution, but also provides theoretical evidence for the theory that heterogeneous environments promote host-parasite diversity, hence constant environment assumptions may omit important evolutionary outcomes.

From adaptive dynamics to adaptive walks

We consider an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modelled as a measure-valued Markov process. Multiple variations of this system have been studied in the simultaneous limit of large populations and rare mutations, where the regime is chosen such that mutations are separated. We consider the deterministic system, resulting from the large population limit, and then let the mutation probability tend to zero. This corresponds to a much higher frequency of mutations, where multiple microscopic types are present at the same time. The limiting process resembles an adaptive walk or flight and jumps between different equilibria of coexisting types. The graph structure on the type space, determined by the possibilities to mutate, plays an important role in defining this jump process. In a variation of the above model, where the radius in which mutants can be spread is limited, we study the possibility of crossing valleys in the fitness landscape and derive different kinds of limiting walks.

A birth-death model of ageing: from individual-based dynamics to evolutive differential inclusions

Ageing's sensitivity to natural selection has long been discussed because of its apparent negative effect on an individual's fitness. Thanks to the recently described (Smurf) 2-phase model of ageing (Tricoire and Rera in PLoS ONE 10(11):e0141920, 2015) we propose a fresh angle for modeling the evolution of ageing. Indeed, by coupling a dramatic loss of fertility with a high-risk of impending death-amongst other multiple so-called hallmarks of ageing-the Smurf phenotype allowed us to consider ageing as a couple of sharp transitions. The birth-death model (later called bd-model) we describe here is a simple life-history trait model where each asexual and haploid individual is described by its fertility period [Formula: see text] and survival period [Formula: see text]. We show that, thanks to the Lansing effect, the effect through which the "progeny of old parents do not live as long as those of young parents", [Formula: see text] and [Formula: see text] converge during evolution to configurations [Formula: see text] in finite time. To do so, we built an individual-based stochastic model which describes the age and trait distribution dynamics of such a finite population. Then we rigorously derive the adaptive dynamics models, which describe the trait dynamics at the evolutionary time-scale. We extend the Trait Substitution Sequence with age structure to take into account the Lansing effect. Finally, we study the limiting behaviour of this jump process when mutations are small. We show that the limiting behaviour is described by a differential inclusion whose solutions [Formula: see text] reach the diagonal [Formula: see text] in finite time and then remain on it. This differential inclusion is a natural way to extend the canonical equation of adaptive dynamics in order to take into account the lack of regularity of the invasion fitness function on the diagonal [Formula: see text].

The paradox of tolerance: Parasite extinction due to the evolution of host defence

Host defence against parasite infection can rely on two broad strategies: resistance and tolerance. The spread of resistance traits usually lowers parasite prevalence and decreases selection for higher defence. Conversely, tolerance mechanisms increase parasite prevalence and foster selection for more tolerance. Here we examine the potential for the host to drive parasites to extinction through the evolution of one or other defence mechanism. We analysed theoretical models of resistance and tolerance evolution in both the absence and the presence of a trade-off between defence and reproduction. In the absence of costs, resistance evolves towards maximisation and, consequently, parasite extinction. Tolerance also evolves towards maximisation but the positive feedback between tolerance and disease prevents the disappearance of the parasite. On the contrary, when defence comes with costs it is impossible for the host to eliminate the infection through resistance, because costly resistance is selected against when parasites are at low prevalence. We uncover that the only path to disease clearance in the presence of costs is through tolerance. Paradoxically, however, it is by lowering tolerance -and hence increasing disease-induced mortality- that extinction can occur. We also show that such extinction can occur even in the case of parasite counter-adaptation. Our results emphasise the importance of tolerance as a defence strategy, and identify key questions for future research.

The evolution of stage-specific virulence: Differential selection of parasites in juveniles

The impact of infectious disease is often very different in juveniles and adults, but theory has focused on the drivers of stage-dependent defense in hosts rather than the potential for stage-dependent virulence evolution in parasites. Stage structure has the potential to be important to the evolution of pathogens because it exposes parasites to heterogeneous environments in terms of both host characteristics and transmission pathways. We develop a stage-structured (juvenile-adult) epidemiological model and examine the evolutionary outcomes of stage-specific virulence under the classic assumption of a transmission-virulence trade-off. We show that selection on virulence against adults remains consistent with the classic theory. However, the evolution of juvenile virulence is sensitive to both demography and transmission pathway with higher virulence against juveniles being favored either when the transmission pathway is assortative (juveniles preferentially interact together) and the juvenile stage is long, or in contrast when the transmission pathway is disassortative and the juvenile stage is short. These results highlight the potentially profound effects of host stage structure on determining parasite virulence in nature. This new perspective may have broad implications for both understanding and managing disease severity.

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.

Alternative assembly processes from trait-mediated co-evolution in mutualistic communities

Ecological and evolutionary dynamics observed in mutualistic communities can be shaped by several mechanisms, including ecological interactions and their co-evolutionary consequences. Here we explore how intra and interspecific competition, together with mutualistic interactions, can affect community assembly through their effects on adaptive diversification and the emergence of biodiversity. To capture both ecological and evolutionary processes simultaneously, we used the adaptive dynamics approach based on a Lotka-Volterra framework and simulated the ecological dynamics of populations as well as the evolutionary dynamics of phenotypic traits. Depending on the initial trait values, two possible alternative evolutionary regimes emerged: traits evolve towards either optimal utilization of environmental resources or maximizing the benefits from mutualistic interactions. Diversification and overall biodiversity are mostly driven by frequency-dependent competition, while mutualism plays an important role in enhancing ecosystem productivity and evolutionary stability. Because different initial trait values in a community can lead to alternative evolutionary regimes, species loss and biological invasions could not only alter ecological dynamics but also push the system onto an alternative successional climax or evolutionary end point. It thus becomes essential to clarify the past evolutionary dynamics so as to draw conclusions on key community assembly processes.

Evolutionarily stable communities: a framework for understanding the role of trait evolution in the maintenance of diversity

Biological diversity depends on the interplay between evolutionary diversification and ecological mechanisms allowing species to coexist. Current research increasingly integrates ecology and evolution over a range of timescales, but our common conceptual framework for understanding species coexistence requires better incorporation of evolutionary processes. Here, we focus on the idea of evolutionarily stable communities (ESCs), which are theoretical endpoints of evolution in a community context. We use ESCs as a unifying framework to highlight some important but under-appreciated theoretical results, and we review empirical research relevant to these theoretical predictions. We explain how, in addition to generating diversity, evolution can also limit diversity by reducing the effectiveness of coexistence mechanisms. The coevolving traits of competing species may either diverge or converge, depending on whether the number of species in the community is low (undersaturated) or high (oversaturated) relative to the ESC. Competition in oversaturated communities can lead to extinction or neutrally coexisting, ecologically equivalent species. It is critical to consider trait evolution when investigating fundamental ecological questions like the strength of different coexistence mechanisms, the feasibility of ecologically equivalent species, and the interpretation of different patterns of trait dispersion.

Evolution of host preference in anthropophilic mosquitoes

Background

Insecticide-treated bed nets (ITNs) have played a large role in reducing the burden of malaria. There is concern however regarding the potential of the mass distributions and use of ITNs to select for insecticide and behavioural resistance in mosquito populations. A key feature of the vectorial capacity of the major sub-Saharan African malaria vector Anopheles gambiae sensu stricto (s.s.) is its tendency to feed almost exclusively on humans. Here, an evolutionary model is used to investigate the potential for ITNs to select for increased zoophily in this highly anthropophilic species and how this is influenced by ecological and operational conditions.

Results

The evolution of a single trait, namely the tendency to accept cattle as hosts, is modelled in mosquito populations which initially only bite humans. Thus, the conditions under which a resource specialist would broaden its diet and become a generalist are investigated. The results indicate that in the absence of insecticide-treated nets, host specialization in mosquitoes is either driven toward human specialization (when humans are more abundant than alternative hosts), or displays evolutionary bistability. The latter implies that the evolutionary endpoint relies on the initial trait value of the population. Bed nets select for increased zoophily while in use. When ITNs are removed, whether or not the population reverts to anthropophagic or zoophagic behaviour depends on whether the intervention had been maintained sufficiently long to drive the population past the evolutionarily unstable point.

Conclusions

The use of ITNs is likely to select for an increase in the biting preference for cattle. Bed nets may thus alter the population composition of major vector species in a manner that has positive epidemiological ramifications. Whether populations are set on a trajectory toward increased zoophily following the cessation of intense bed net usage in an area depends on the composition of host communities as well as operational conditions. This has potential implications for bed net campaigns, particularly with an eye toward scaling down interventions following interruption of transmission. Further research on malaria mosquito feeding behaviour is warranted to explore the conditions under which such adaptive shifts may actually occur in the field.

Why does invasion imply substitution? Beyond the paradigm of invasion fitness

Adaptive dynamics combines deterministic population dynamics of groups having different trait values and random process describing mutation and tries to predict the course of evolution of a species of interest. One of basic interests is to know which group survives, residents or mutants. By using invasion fitness as the primary tool, "invasion implies substitution" principle, IIS principle for short, has been established under the existence of a generating function in the sense of Brown and Vincent (Theor Popul Biol 31(1):140-166, 1987) and Vincent and Brown (Evolutionary game theory, natural selection, and darwinian dynamics. Cambridge University Press, Cambridge, 2005). This principle essentially says that the local gradient of invasion fitness ultimately determines the outcome of the competition. However, as we will see in this paper, even if a system is within the scope of IIS principle, its neighborhood always contains systems which are beyond this scope. In this paper, in order to overcome such a limitation, we establish a wider class of systems which is still reasonable as a model of evolution of a species. For our wider class, the notion of raw invasion fitness is introduced. In terms of raw invasion fitness, an explicit criterion for the existence of generating function and a counterpart of IIS principle are obtained. This enables us to discuss small perturbations of a system within or without the scope of generating functions/IIS principle. Eventually, we understand why invasion implies substitution, i.e. why the method using invasion fitness works well with the existence of generating function, from our broader point of view.

The recovery of a recessive allele in a Mendelian diploid model

We study the large population limit of a stochastic individual-based model which describes the time evolution of a diploid hermaphroditic population reproducing according to Mendelian rules. Neukirch and Bovier (J Math Biol 75:145-198, 2017) proved that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they would in populations reproducing asexually. In the present paper we prove that this indeed opens the possibility that individuals with a pure genotype can reinvade in the population after the appearance of further mutations. We thus expose a rigorous description of a mechanism by which a recessive allele can re-emerge in a population. This can be seen as a statement of genetic robustness exhibited by diploid populations performing sexual reproduction.

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.

An Alternative Formulation for a Distributed Delayed Logistic Equation

We study an alternative single species logistic distributed delay differential equation (DDE) with decay-consistent delay in growth. Population oscillation is rarely observed in nature, in contrast to the outcomes of the classical logistic DDE. In the alternative discrete delay model proposed by Arino et al. (J Theor Biol 241(1):109-119, 2006), oscillatory behavior is excluded. This study adapts their idea of the decay-consistent delay and generalizes their model. We establish a threshold for survival and extinction: In the former case, it is confirmed using Lyapunov functionals that the population approaches the delay modified carrying capacity; in the later case the extinction is proved by the fluctuation lemma. We further use adaptive dynamics to conclude that the evolutionary trend is to make the mean delay in growth as short as possible. This confirms Hutchinson's conjecture (Hutchinson in Ann N Y Acad Sci 50(4):221-246, 1948) and fits biological evidence.

Evolution of mate-finding Allee effect in prey

The search for mates is often accompanied with conspicuous behaviour or morphology that can be exploited by predators. Here we explore the evolutionary consequences of a trade-off that arises naturally between mate acquisition and risk of predation and study evolution of the rate at which male prey search for mates in a population subject to a mate-finding Allee effect and exposed to either generalist or specialist predators. Since we show that the mate search rate determines the strength of the mate-finding Allee effect, we can alternatively view this as evolution of the mate-finding Allee effect in prey. We contrast two different life histories and find that, predominantly, male prey either evolve towards the maximal mate search rate yielding the weakest possible mate-finding Allee effect (thus showing no adaptive response in mating behaviour to predation risk) or evolutionary bi-stability occurs. In the latter case, males evolve a relatively low mate search rate (hence a relatively strong mate-finding Allee effect, interpreted as an adaptive response of male prey to predation) when initially slow or the maximal mate search rate when initially fast. Disruptive selection does not occur in populations exposed to generalist predators but is possible when predators are specialists. The dimorphic phase, in which fast and conspicuous male prey coexist with slow and cryptic ones, is however but a transient in evolutionary dynamics as one branch goes extinct while the other evolves towards the maximal mate search rate.

The evolution of host defence to parasitism in fluctuating environments

Given rapidly changing environments, it is important for us to understand how the evolution of host defence responds to fluctuating environments. Here we present the first theoretical study of evolution of host resistance to parasitism in a classic epidemiological model where the host birth rate varies seasonally. We show that this form of seasonality has clear qualitative and quantitative impacts on the evolution of resistance. When the host can recover from infection, it evolves a lower level of defence when the amplitude is high. However, when recovery is absent, the host increases its defence for higher amplitudes. Between these different behaviours we find a region of parameter space that allows evolutionary bistability. When this occurs, the level of defence the host evolves depends on initial conditions, and in some cases a switch between attractors can lead to different periods in the population dynamics at each of the evolutionary stable strategies. Crucially, we find that evolutionary behaviour found in a constant environment for this model doesn't always hold for hosts with highly variable birth rates. Hence we argue that seasonality must be taken into account if we want to make predictions about evolutionary trends in real-world host-parasite systems.

Rethinking the evolution of specialization: A model for the evolution of phenotypic heterogeneity

Phenotypic heterogeneity refers to genetically identical individuals that express different phenotypes, even when in the same environment. Traditionally, "bet-hedging" in fluctuating environments is offered as the explanation for the evolution of phenotypic heterogeneity. However, there are an increasing number of examples of microbial populations that display phenotypic heterogeneity in stable environments. Here we present an evolutionary model of phenotypic heterogeneity of microbial metabolism and a resultant theory for the evolution of phenotypic versus genetic specialization. We use two-dimensional adaptive dynamics to track the evolution of the population phenotype distribution of the expression of two metabolic processes with a concave trade-off. Rather than assume a Gaussian phenotype distribution, we use a Beta distribution that is capable of describing genotypes that manifest as individuals with two distinct phenotypes. Doing so, we find that environmental variation is not a necessary condition for the evolution of phenotypic heterogeneity, which can evolve as a form of specialization in a stable environment. There are two competing pressures driving the evolution of specialization: directional selection toward the evolution of phenotypic heterogeneity and disruptive selection toward genetically determined specialists. Because of the lack of a singular point in the two-dimensional adaptive dynamics and the fact that directional selection is a first order process, while disruptive selection is of second order, the evolution of phenotypic heterogeneity dominates and often precludes speciation. We find that branching, and therefore genetic specialization, occurs mainly under two conditions: the presence of a cost to maintaining a high phenotypic variance or when the effect of mutations is large. A cost to high phenotypic variance dampens the strength of selection toward phenotypic heterogeneity and, when sufficiently large, introduces a singular point into the evolutionary dynamics, effectively guaranteeing eventual branching. Large mutations allow the second order disruptive selection to dominate the first order selection toward phenotypic heterogeneity.

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.

The evolution of site-selection strategy during dispersal

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

ADAPTIVE DYNAMICS IN ALLELE SPACE: EVOLUTION OF GENETIC POLYMORPHISM BY SMALL MUTATIONS IN A HETEROGENEOUS ENVIRONMENT

We demonstrate how a genetic polymorphism of distinctly different alleles can develop during long-term frequency-dependent evolution in an initially monomorphic diploid population, if mutations have only small phenotypic effect. As a specific example, we use a version of Levene's (1953) soft selection model, where stabilizing selection acts on a continuous trait within each of two habitats. If the optimal phenotypes within the habitats are sufficiently different, then two distinctly different alleles evolve gradually from a single ancestral allele. In a wide range of parameter values, the two locally optimal phenotypes will be realized by one of the homozygotes and the heterozygote, rather than by the two homozygotes. Unlike in the haploid analogue of the model, there can be multiple polymorphic evolutionary attractors with different probabilities of convergence. Our results differ from the population genetic models of short-term evolution in two aspects: (1) a polymorphism that is population genetically stable may be invaded by a new mutant allele and, as a consequence, the population may fall back to monomorphism, (2) long-term evolution by allele substitutions may lead from a population where polymorphism is not possible into one where polymorphism is possible.

Outline of a unified Darwinian evolutionary theory for physical and biological systems

The scheme of a unified Darwinian evolutionary theory for physical and biological systems is described. Every physical system is methodologically endowed with a classical information processor, which turns every system into an agent being also susceptible to evolution. Biological systems retain this structure as natural extensions of physical systems from which they are built up. Optimization of information flows turns out to be the key element to study the possible emergence of quantum behavior and the unified Darwinian description of physical and biological systems. The Darwinian natural selection scheme is completed by the Lamarckian component in the form of the anticipation of states of surrounding bio-physical systems.

A numerical approach to determine mutant invasion fitness and evolutionary singular strategies

We propose a numerical approach to study the invasion fitness of a mutant and to determine evolutionary singular strategies in evolutionary structured models in which the competitive exclusion principle holds. Our approach is based on a dual representation, which consists of the modeling of the small size mutant population by a stochastic model and the computation of its corresponding deterministic model. The use of the deterministic model greatly facilitates the numerical determination of the feasibility of invasion as well as the convergence-stability of the evolutionary singular strategy. Our approach combines standard adaptive dynamics with the link between the mutant survival criterion in the stochastic model and the sign of the eigenvalue in the corresponding deterministic model. We present our method in the context of a mass-structured individual-based chemostat model. We exploit a previously derived mathematical relationship between stochastic and deterministic representations of the mutant population in the chemostat model to derive a general numerical method for analyzing the invasion fitness in the stochastic models. Our method can be applied to the broad class of evolutionary models for which a link between the stochastic and deterministic invasion fitnesses can be established.

The evolution of parasitic and mutualistic plant-virus symbioses through transmission-virulence trade-offs

Virus-plant interactions range from parasitism to mutualism. Viruses have been shown to increase fecundity of infected plants in comparison with uninfected plants under certain environmental conditions. Increased fecundity of infected plants may benefit both the plant and the virus as seed transmission is one of the main virus transmission pathways, in addition to vector transmission. Trade-offs between vertical (seed) and horizontal (vector) transmission pathways may involve virulence, defined here as decreased fecundity in infected plants. To better understand plant-virus symbiosis evolution, we explore the ecological and evolutionary interplay of virus transmission modes when infection can lead to an increase in plant fecundity. We consider two possible trade-offs: vertical seed transmission vs infected plant fecundity, and horizontal vector transmission vs infected plant fecundity (virulence). Through mathematical models and numerical simulations, we show (1) that a trade-off between virulence and vertical transmission can lead to virus extinction during the course of evolution, (2) that evolutionary branching can occur with subsequent coexistence of mutualistic and parasitic virus strains, and (3) that mutualism can out-compete parasitism in the long-run. In passing, we show that ecological bi-stability is possible in a very simple discrete-time epidemic model. Possible extensions of this study include the evolution of conditional (environment-dependent) mutualism in plant viruses.