Evolution of dispersal under spatio-temporal heterogeneity

Theoretical studies over the past decades have revealed various factors that favor or disfavor the evolution of dispersal. Among these, environmental heterogeneity is one driving force that can impact dispersal traits, because dispersing individuals can obtain a fitness benefit through finding better environments. Despite this potential benefit, some previous works have shown that the existence of spatial heterogeneity hinders evolution of dispersal. On the other hand, temporal heterogeneity has been shown to promote dispersal through a bet-hedging mechanism. When they are combined in a patch-structured population in which the quality of each patch varies over time independently of the others, it has been shown that spatiotemporal heterogeneity can favor evolution of dispersal. When individuals can use patch quality information so that dispersal decision is conditional, the evolutionary outcome can be different since individuals have options to disperse more/less offspring from bad/good patches. In this paper, we generalize the model and results of previous studies. We find richer dynamics including bistable evolutionary dynamics when there is arrival bias towards high-productivity patches. Then we study the evolution of conditional dispersal strategy in this generalized model. We find a surprising result that no offspring will disperse from a patch whose productivity was low when these offspring were born. In addition to mathematical proofs, we also provide intuition behind this initially counter-intuitive result based on reproductive-value arguments. Dispersal from high-productivity patches can evolve, and its parameter dependence behaves similarly, but not identically, to the case of unconditional dispersal. Our results unveil an importance of whether or not individuals can use patch quality information in dispersal evolution.

Evolution of Reproduction Periods in Seasonal Environments

AbstractMany species are subject to seasonal cycles in resource availability, affecting the timing of their reproduction. Using a stage-structured consumer-resource model in which juvenile development and maturation are resource dependent, we study how a species' reproductive schedule evolves, dependent on the seasonality of its resource. We find three qualitatively different reproduction modes. First, continuous income breeding (with adults reproducing throughout the year) evolves in the absence of significant seasonality. Second, seasonal income breeding (with adults reproducing unless they are starving) evolves when resource availability is sufficiently seasonal and juveniles are more efficient resource foragers. Third, seasonal capital breeding (with adults reproducing partly through the use of energy reserves) evolves when resource availability is sufficiently seasonal and adults are more efficient resource foragers. Such capital breeders start reproduction already while their offspring are still experiencing starvation. Changes in seasonality lead to continuous transitions between continuous and seasonal income breeding, but the change between income and capital breeding involves a hysteresis pattern, such that a population's evolutionarily stable reproduction pattern depends on its initial one. Taken together, our findings show how adaptation to seasonal environments can result in a rich array of outcomes, exhibiting seasonal or continuous reproduction with or without energy reserves.

Environmental dimensionality determines species coexistence

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

Modelling of killer T-cell and cancer cell subpopulation dynamics under immuno- and chemotherapies

Each patient's cancer has a unique molecular makeup, often comprised of distinct cancer cell subpopulations. Improved understanding of dynamic processes between cancer cell populations is therefore critical for making treatment more effective and personalized. It has been shown that immunotherapy increases the survival of melanoma patients. However, there remain critical open questions, such as timing and duration of immunotherapy and its added benefits when combined with other types of treatments. We introduce a model for the dynamics of active killer T-cells and cancer cell subpopulations. Rather than defining the cancer cell populations based on their genetic makeup alone, we consider also other, non-genetic differences that make the cell populations either sensitive or resistant to a therapy. Using the model, we make predictions of possible outcomes of the various treatment strategies in virtual melanoma patients, providing hypotheses regarding therapeutic efficacy and side-effects. It is shown, for instance, that starting immunotherapy with a denser treatment schedule may enable changing to a sparser schedule later during the treatment. Furthermore, combination of targeted and immunotherapy results in a better treatment effect, compared to mono-immunotherapy, and a stable disease can be reached with a patient-tailored combination. These results offer better understanding of the competition between T-cells and cancer cells, toward personalized immunotherapy regimens.

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

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

Environmental dimensionality

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

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.

Evolution of density-dependent cooperation

Cooperation is surprisingly common in life despite of its vulnerability to selfish cheating, i.e. defecting. Defectors do not contribute to common resources but take the advantage of cooperators' investments. Therefore, the emergence and maintenance of cooperation have been considered irrational phenomena. In this study, we focus on plastic, quantitative cooperation behaviour, especially on its evolution. We assume that individuals are capable to sense the population density in their neighbourhood and adjust their real-valued investments on public goods based on that information. The ecological setting is described with stochastic demographic events, e.g. birth and death, occurring at individual level. Individuals form small populations, which further constitute a structured metapopulation. For evolutionary investigations, we apply the adaptive dynamics framework. The cost of cooperative investment is incorporated into the model in two ways, by decreasing the birth rate or by increasing the death rate. In the first case, density-dependent cooperation evolves to be a decreasing function of population size as expected. In the latter case, however, the density-dependent cooperative investment can have a qualitatively different form as it may evolve to be highest in intermediate-sized populations. Indeed, we emphasize that some details in modelling may have a significant impact on the results obtained.

Consequences of asymmetric competition between resident and invasive defoliators: a novel empirically based modelling approach

Invasive species can have profound effects on a resident community via indirect interactions among community members. While long periodic cycles in population dynamics can make the experimental observation of the indirect effects difficult, modelling the possible effects on an evolutionary time scale may provide the much needed information on the potential threats of the invasive species on the ecosystem. Using empirical data from a recent invasion in northernmost Fennoscandia, we applied adaptive dynamics theory and modelled the long term consequences of the invasion by the winter moth into the resident community. Specifically, we investigated the outcome of the observed short-term asymmetric preferences of generalist predators and specialist parasitoids on the long term population dynamics of the invasive winter moth and resident autumnal moth sharing these natural enemies. Our results indicate that coexistence after the invasion is possible. However, the outcome of the indirect interaction on the population dynamics of the moth species was variable and the dynamics might not be persistent on an evolutionary time scale. In addition, the indirect interactions between the two moth species via shared natural enemies were able to cause asynchrony in the population cycles corresponding to field observations from previous sympatric outbreak areas. Therefore, the invasion may cause drastic changes in the resident community, for example by prolonging outbreak periods of birch-feeding moths, increasing the average population densities of the moths or, alternatively, leading to extinction of the resident moth species or to equilibrium densities of the two, formerly cyclic, herbivores.

Self-extinction through optimizing selection

Evolutionary suicide is a process in which selection drives a viable population to extinction. So far, such selection-driven self-extinction has been demonstrated in models with frequency-dependent selection. This is not surprising, since frequency-dependent selection can disconnect individual-level and population-level interests through environmental feedback. Hence it can lead to situations akin to the tragedy of the commons, with adaptations that serve the selfish interests of individuals ultimately ruining a population. For frequency-dependent selection to play such a role, it must not be optimizing. Together, all published studies of evolutionary suicide have created the impression that evolutionary suicide is not possible with optimizing selection. Here we disprove this misconception by presenting and analyzing an example in which optimizing selection causes self-extinction. We then take this line of argument one step further by showing, in a further example, that selection-driven self-extinction can occur even under frequency-independent selection.

Evolution of specialization under non-equilibrium population dynamics

We analyze the evolution of specialization in resource utilization in a mechanistically underpinned discrete-time model using the adaptive dynamics approach. We assume two nutritionally equivalent resources that in the absence of consumers grow sigmoidally towards a resource-specific carrying capacity. The consumers use resources according to the law of mass-action with rates involving trade-off. The resulting discrete-time model for the consumer population has over-compensatory dynamics. We illuminate the way non-equilibrium population dynamics affect the evolutionary dynamics of the resource consumption rates, and show that evolution to the trimorphic coexistence of a generalist and two specialists is possible due to asynchronous non-equilibrium population dynamics of the specialists. In addition, various forms of cyclic evolutionary dynamics are possible. Furthermore, evolutionary suicide may occur even without Allee effects and demographic stochasticity.

Evolutionary suicide as a consequence of runaway selection for greater aggregation tendency

Aggregation of individuals is a common phenomenon in nature. By aggregating, individuals can reap benefits but may also be subject to associated costs from increased competition. The benefits of aggregation can depend on population density, which in turn can be affected by aggregation when it determines reproductive success of individuals. The Allee effect is often considered to be one of the factors that can explain the evolution of aggregation behavior. We investigated this hypothesis with a mathematical model which integrates population dynamics and evolution. Individuals gain synergistically from aggregation but suffer from scramble competition with aggregation tendency as an evolving trait. We found that aggregation behavior can stabilize the population dynamics and reduce population growth. The results show that the Allee effect alone is not sufficient for aggregative behavior to evolve as an evolutionarily stable strategy. We also found that weak local competition does not promote aggregation due to feedback from the population level: under low competition, the population can achieve high density such that aggregation becomes costly rather than beneficial. Our model instead exhibits an escalation of aggregation tendency, leading to the extinction of the population in a process known as evolutionary suicide. We conclude that for aggregation to evolve as an evolutionarily stable strategy we need to consider other factors such as inter-patch dispersal to new patches and avoidance of excessively large groups.

Evolution of complex density-dependent dispersal strategies

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

Function-valued adaptive dynamics and optimal control theory

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

Adaptive dynamics of altruistic cooperation in a metapopulation: evolutionary emergence of cooperators and defectors or evolutionary suicide?

We investigate the evolution of public goods cooperation in a metapopulation model with small local populations, where altruistic cooperation can evolve due to assortment and kin selection, and the evolutionary emergence of cooperators and defectors via evolutionary branching is possible. Although evolutionary branching of cooperation has recently been demonstrated in the continuous snowdrift game and in another model of public goods cooperation, the required conditions on the cost and benefit functions are rather restrictive, e.g., altruistic cooperation cannot evolve in a defector population. We also observe selection for too low cooperation, such that the whole metapopulation goes extinct and evolutionary suicide occurs. We observed intuitive effects of various parameters on the numerical value of the monomorphic singular strategy. Their effect on the final coexisting cooperator-defector pair is more complex: changes expected to increase cooperation decrease the strategy value of the cooperator. However, at the same time the population size of the cooperator increases enough such that the average strategy does increase. We also extend the theory of structured metapopulation models by presenting a method to calculate the fitness gradient in a general class of metapopulation models, and try to make a connection with the kin selection approach.

Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction

It has recently been demonstrated that ecological feedback mechanisms can facilitate the emergence and maintenance of cooperation in public goods interactions: the replicator dynamics of defectors and cooperators can result, for example, in the ecological coexistence of cooperators and defectors. Here we show that these results change dramatically if cooperation strategy is not fixed but instead is a continuously varying trait under natural selection. For low values of the factor with which the value of resources is multiplied before they are shared among all participants, evolution will always favour lower cooperation strategies until the population falls below an Allee threshold and goes extinct, thus evolutionary suicide occurs. For higher values of the factor, there exists a unique evolutionarily singular strategy, which is convergence stable. Because the fitness function is linear with respect to the strategy of the mutant, this singular strategy is neutral against mutant invasions. This neutrality disappears if a nonlinear functional response in receiving benefits is assumed. For strictly concave functional responses, singular strategies become uninvadable. Evolutionary branching, which could result in the evolutionary emergence of cooperators and defectors, can occur only with locally convex functional responses, but we illustrate that it can also result in coevolutionary extinction.

The Allee effect in mechanistic models based on inter-individual interaction processes

Recently, Eskola and Geritz (Bull. Math. Biol. 69:329-346, 2007) showed that several discrete-time population models can be derived mechanistically within a single ecological framework by varying the within-season patterns of reproduction and inter-individual aggression. However, these models do not have the Allee effect. In this paper, we modify the original modelling framework by adding different mate finding processes, and thus derive mechanistically several population models with the Allee effect.

Evolution of specialization in resource utilization in structured metapopulations

We study the evolution of resource utilization in a structured discrete-time metapopulation model with an infinite number of patches, prone to local catastrophes. The consumer faces a trade-off in the abilities to consume two resources available in different amounts in each patch. We analyse how the evolution of specialization in the utilization of the resources is affected by different ecological factors: migration, local growth, local catastrophes, forms of the trade-off and distribution of the resources in the patches. Our modelling approach offers a natural way to include more than two patch types into the models. This has not been usually possible in the previous spatially heterogeneous models focusing on the evolution of specialization.

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261-269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.

Evolutionary suicide

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

Evolution of Dispersal in a Structured Metapopulation Model in Discrete Time

In this article, a structured metapopulation model in discrete time with catastrophes and density-dependent local growth is introduced. The fitness of a rare mutant in an environment set by the resident is defined, and an efficient method to calculate fitness is presented. With this fitness measure evolutionary analysis of this model becomes feasible. This article concentrates on the evolution of dispersal. The effect of catastrophes, dispersal cost, and local dynamics on the evolution of dispersal is investigated. It is proved that without catastrophes, if all population-dynamical attractors are fixed points, there will be selection for no dispersal. A new mechanism for evolutionary branching is also found: Even though local population sizes approach fixed points, catastrophes can cause enough temporal variability, so that evolutionary branching becomes possible.

The adaptive dynamics of function-valued traits

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

Function-valued adaptive dynamics and the calculus of variations

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

Dispersal and the evolution of specialisation in a two-habitat type metapopulation

Metapopulation theory for the evolution of specialisation is virtually absent. In this article, therefore, we study a metapopulation model for consumers with a fitness trade-off between two habitats. We focus on effects of habitat abundance, dispersal rate and trade-off strength on the evolution of specialisation under two types of trade-off. Adaptation affects either the intrinsic growth rates r or the carrying capacities K. Depending on dispersal rate and trade-off strength, evolution can result in one generalist, one specialist or two specialist types. Higher dispersal rate and a weaker trade-off favour the evolution of a generalist, for both trade-off structures. However, we also find differences between the two trade-off structures. Our results are qualitatively similar to analyses of two-patch models, suggesting that insights from such simpler models can be extrapolated to metapopulation models. Additional effects, however, occur because in classical metapopulations patch lifetime depends on extinction rate. Counterintuitively, this favours the evolution of specialisation when the trade-off affects r.

Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity

In this paper, we predict the outcome of dispersal evolution in metapopulations based on the following assumptions: (i) population dynamics within patches are density-regulated by realistic growth functions; (ii) demographic stochasticity resulting from finite population sizes within patches is accounted for; and (iii) the transition of individuals between patches is explicitly modelled by a disperser pool. We show, first, that evolutionarily stable dispersal rates do not necessarily increase with rates for the local extinction of populations due to external disturbances in habitable patches. Second, we describe how demographic stochasticity affects the evolution of dispersal rates: evolutionarily stable dispersal rates remain high even when disturbance-related rates of local extinction are low, and a variety of qualitatively different responses of adapted dispersal rates to varied levels of disturbance become possible. This paper shows, for the first time, that evolution of dispersal rates may give rise to monotonically increasing or decreasing responses, as well as to intermediate maxima or minima.

Evolutionary branching of dispersal strategies in structured metapopulations

Dispersal polymorphism and evolutionary branching of dispersal strategies has been found in several metapopulation models. The mechanism behind those findings has been temporal variation caused by cyclic or chaotic local dynamics, or temporally and spatially varying carrying capacities. We present a new mechanism: spatial heterogeneity in the sense of different patch types with sufficient proportions, and temporal variation caused by catastrophes. The model where this occurs is a generalization of the model by Gyllenberg and Metz (2001). Their model is a size-structured metapopulation model with infinitely many identical patches. We present a generalized version of their metapopulation model allowing for different types of patches. In structured population models, defining and computing fitness in polymorphic situations is, in general, difficult. We present an efficient method, which can be applied also to other structured population or metapopulation models.

Evolutionary suicide and evolution of dispersal in structured metapopulations

We study the evolution of dispersal in a structured metapopulation model. The metapopulation consists of a large (infinite) number of local populations living in patches of habitable environment. Dispersal between patches is modelled by a disperser pool and individuals in transit between patches are exposed to a risk of mortality. Occasionally, local catastrophes eradicate a local population: all individuals in the affected patch die, yet the patch remains habitable. We prove that, in the absence of catastrophes, the strategy not to migrate is evolutionarily stable. Under a given set of environmental conditions, a metapopulation may be viable and yet selection may favor dispersal rates that drive the metapopulation to extinction. This phenomenon is known as evolutionary suicide. We show that in our model evolutionary suicide can occur for catastrophe rates that increase with decreasing local population size. Evolutionary suicide can also happen for constant catastrophe rates, if local growth within patches shows an Allee effect. We study the evolutionary bifurcation towards evolutionary suicide and show that a discontinuous transition to extinction is a necessary condition for evolutionary suicide to occur. In other words, if population size smoothly approaches zero at a boundary of viability in parameter space, this boundary is evolutionarily repelling and no suicide can occur.

Invasion dynamics and attractor inheritance

We study the dynamics of a population of residents that is being invaded by an initially rare mutant. We show that under relatively mild conditions the sum of the mutant and resident population sizes stays arbitrarily close to the initial attractor of the monomorphic resident population whenever the mutant has a strategy sufficiently similar to that of the resident. For stochastic systems we show that the probability density of the sum of the mutant and resident population sizes stays arbitrarily close to the stationary probability density of the monomorphic resident population. Attractor switching, evolutionary suicide as well as most cases of "the resident strikes back" in systems with multiple attractors are possible only near a bifurcation point in the strategy space where the resident attractor undergoes a discontinuous change. Away from such points, when the mutant takes over the population from the resident and hence becomes the new resident itself, the population stays on the same attractor. In other words, the new resident "inherits" the attractor from its predecessor, the former resident.

Necessary and sufficient conditions for evolutionary suicide

Evolutionary suicide is an evolutionary process where a viable population adapts in such a way that it can no longer persist. It has already been found that a discontinuous transition to extinction is a necessary condition for suicide. Here we present necessary and sufficient conditions, concerning the bifurcation point, for suicide to occur. Evolutionary suicide has been found in structured metapopulation models. Here we show that suicide can occur also in unstructured population models. Moreover, a structured model does not guarantee the possibility of suicide: we show that suicide cannot occur in age-structured population models of the Gurtin-MacCamy type. The point is that the mutant's fitness must explicitly depend not only on the environmental interaction variable, but also on the resident strategy.

Evolution of Migration in a Metapopulation

In this paper a general deterministic discrete-time metapopulation model with a finite number of habitat patches is analysed within the framework of adaptive dynamics. We study a general model and prove analytically that (i) if the resident populations state is a fixed point, then the resident strategy with no migration is an evolutionarily stable strategy, (ii) a mutant population with no migration can invade any resident population in a fixed point state, (iii) in the uniform migration case the strategy not to migrate is attractive under small mutational steps so that selection favours low migration. Some of these results have been previously observed in simulations, but here they are proved analytically in a general case. If the resident population is in a two-cyclic orbit, then the situation is different. In the uniform migration case the invasion behaviour depends both on the type of the residents attractor and the survival probability during migration. If the survival probability during migration is low, then the system evolves towards low migration. If the survival probability is high enough, then evolutionary branching can happen and the system evolves to a situation with several coexisting types. In the case of out-of-phase attractor, evolutionary branching can happen with significantly lower survival probabilities than in the in-phase attractor case. Most results in the two-cyclic case are obtained by numerical simulations. Also, when migration is not uniform we observe in numerical simulations in the two-cyclic orbit case selection for low migration or evolutionary branching depending on the survival probability during migration.