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.