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 dispersal in a spatially heterogeneous population with finite patch sizes.

Dispersal is one of the fundamental life-history strategies of organisms, so understanding the selective forces shaping the dispersal traits is important. In the Wright's island model, dispersal evolves due to kin competition even when dispersal is costly, and it has traditionally been assumed that the living conditions are the same everywhere. To study the effect of spatial heterogeneity, we extend the model so that patches may receive different amounts of immigrants, foster different numbers of individuals, and give different reproduction efficiency to individuals therein. We obtain an analytical expression for the fitness gradient, which shows that directional selection consists of three components: As in the homogeneous case, the direct cost of dispersal selects against dispersal and kin competition promotes dispersal. The additional component, spatial heterogeneity, more precisely the variance of so-called relative reproductive potential, tends to select against dispersal. We also obtain an expression for the second derivative of fitness, which can be used to determine whether there is disruptive selection: Unlike the homogeneous case, we found that divergence of traits through evolutionary branching is possible in the heterogeneous case. Our numerical explorations suggest that evolutionary branching is promoted more by differences in patch size than by reproduction efficiency. Our results show the importance of the existing spatial heterogeneity in the real world as a key determinant in dispersal evolution.

Tumor microenvironment as a metapopulation model: The effects of angiogenesis, emigration and treatment modalities.

Tumors consist of heterogeneous cell subpopulations that may develop differing phenotypes, such as increased cell growth, metastatic potential and treatment sensitivity or resistance. To study the dynamics of cancer development at a single-cell level, we model the tumor microenvironment as a metapopulation, in which habitat patches correspond to possible sites for cell subpopulations. Cancer cells may emigrate into dispersal pool (e.g. circulation system) and spread to new sites (i.e. metastatic disease). In the patches, cells divide and new variants may arise, possibly leading into an invasion provided the aberration promotes the cell growth. To study such adaptive landscape of cancer ecosystem, we consider various evolutionary strategies (phenotypes), such as emigration and angiogenesis, which are important determinants during early stages of tumor development. We use the metapopulation fitness of new variants to investigate how these strategies evolve through natural selection and disease progression. We further study various treatment effects and investigate how different therapy regimens affect the evolution of the cell populations. These aspects are relevant, for example, when examining the dynamic process of a benign tumor becoming cancerous, and what is the best treatment strategy during the early stages of cancer development. It is shown that positive angiogenesis promotes cancer cell growth in the absence of anti-angiogenic treatment, and that the anti-angiogenic treatment reduces the need of cytotoxic treatment when used in a combination. Interestingly, the model predicts that treatment resistance might become a favorable quality to cancer cells when the anti-angiogenic treatment is intensive enough. Thus, the optimal treatment dosage should remain below a patient-specific level to avoid treatment resistance.

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.

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.

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.

The components of directional and disruptive selection in heterogeneous group-structured populations.

We derive how directional and disruptive selection operate on scalar traits in a heterogeneous group-structured population for a general class of models. In particular, we assume that each group in the population can be in one of a finite number of states, where states can affect group size and/or other environmental variables, at a given time. Using up to second-order perturbation expansions of the invasion fitness of a mutant allele, we derive expressions for the directional and disruptive selection coefficients, which are sufficient to classify the singular strategies of adaptive dynamics. These expressions include first- and second-order perturbations of individual fitness (expected number of settled offspring produced by an individual, possibly including self through survival); the first-order perturbation of the stationary distribution of mutants (derived here explicitly for the first time); the first-order perturbation of pairwise relatedness; and reproductive values, pairwise and three-way relatedness, and stationary distribution of mutants, each evaluated under neutrality. We introduce the concept of individual k-fitness (defined as the expected number of settled offspring of an individual for which k-1 randomly chosen neighbors are lineage members) and show its usefulness for calculating relatedness and its perturbation. We then demonstrate that the directional and disruptive selection coefficients can be expressed in terms individual k-fitnesses with k=1,2,3 only. This representation has two important benefits. First, it allows for a significant reduction in the dimensions of the system of equations describing the mutant dynamics that needs to be solved to evaluate explicitly the two selection coefficients. Second, it leads to a biologically meaningful interpretation of their components. As an application of our methodology, we analyze directional and disruptive selection in a lottery model with either hard or soft selection and show that many previous results about selection in group-structured populations can be reproduced as special cases of our model.

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.

Spatial heterogeneity and evolution of fecundity-affecting traits.

It is widely recognized that spatial structure in a population has some, and occasionally great, impacts on ecological and evolutionary dynamics. However, it has been observed that in the homogeneous Wright's island model with a certain standard demographic assumption, spatial structure does not affect the fitness gradient of a fecundity-affecting trait. The location and convergence stability of singular strategies thus remain unchanged. Furthermore, evolutionary branching is impossible for small dispersal rates, and for a wide class of fecundity functions, evolutionary branching is impossible for any dispersal rate if branching does not occur in the corresponding well-mixed model. Spatially homogeneous structure thus often inhibits evolutionary branching. Here we study the impact of spatial heterogeneity on evolutionary dynamics. We consider an infinite Wright's island model, where different islands have different capacity and fecundity consequences, and therefore the population is spatially heterogeneous. Through the analysis of metapopulation fitness, we derive its first-order and second-order derivatives with respect to mutant's trait, which are explicitly represented in terms of fecundity derivatives. The selection gradient turns out to be a biased average of local selection pressures in different patch types. We find that evolutionary branching is generally favored in the presence of spatial heterogeneity. We also find a simple condition under which evolutionary branching is particularly favored. Applications to public-goods cooperation and emergent evolutionary branching to cooperators and defectors are discussed.

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.

The effect of fecundity derivatives on the condition of evolutionary branching in spatial models.

By investigating metapopulation fitness, we present analytical expressions for the selection gradient and conditions for convergence stability and evolutionary stability in Wright's island model in terms of fecundity function. Coefficients of each derivative of fecundity function appearing in these conditions have fixed signs. This illustrates which kind of interaction promotes or inhibits evolutionary branching in spatial models. We observe that Taylor's cancellation result holds for any fecundity function: Not only singular strategies but also their convergence stability is identical to that in the corresponding well-mixed model. We show that evolutionary branching never occurs when the dispersal rate is close to zero. Furthermore, for a wide class of fecundity functions (including those determined by any pairwise game), evolutionary branching is impossible for any dispersal rate if branching does not occur in the corresponding well-mixed model. Spatial structure thus often inhibits evolutionary branching, although we can construct a fecundity function for which evolutionary branching only occurs for intermediate dispersal rates.

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 Site-Selection Stabilizes Population Dynamics, Promotes Even Distribution of Individuals, and Occasionally Causes Evolutionary Suicide.

Species that compete for access to or use of sites, such as parasitic mites attaching to honey bees or apple maggots laying eggs in fruits, can potentially increase their fitness by carefully selecting sites at which they face little or no competition. Here, we systematically investigate the evolution of site-selection strategies among animals competing for discrete sites. By developing and analyzing a mechanistic and population-dynamical model of site selection in which searching individuals encounter sites sequentially and can choose to accept or continue to search based on how many conspecifics are already there, we give a complete characterization of the different site-selection strategies that can evolve. We find that evolution of site-selection stabilizes population dynamics, promotes even distribution of individuals among sites, and occasionally causes evolutionary suicide. We also discuss the broader implications of our findings and propose how they can be reconciled with an earlier study (Nonaka et al. in J Theor Biol 317:96-104, 2013) that reported selection toward ever higher levels of aggregation among sites as a consequence of site-selection.

On fitness in metapopulations that are both size- and stage-structured.

A proxy for the invasion fitness in structured metapopulation models has been defined as a metapopulation reproduction ratio, which is the expected number of surviving dispersers produced by a mutant immigrant and a colony of its descendants. When a size-structured metapopulation model involves also individual stages (such as juveniles and adults), there exists a generalized definition for the invasion fitness proxy. The idea is to calculate the expected numbers of dispersers of all different possible types produced by a mutant clan initiated with a single mutant, and to collect these values into a matrix. The metapopulation reproduction ratio is then the dominant eigenvalue of this matrix. The calculation method has been published in detail in the case of small local populations. However, in case of large patches the previously published numerical calculation method to obtain the expected number of dispersers does not generalize as such, which gives us one aim of this article. Here, we thus derive a generalized method to calculate the invasion fitness in a metapopulation, which consists of large local populations, and is both size- and stage-structured. We also prove that the metapopulation reproduction ratio is well-defined, i.e., it is equal to 1 for a mutant with a strategy equal to the strategy of a resident. Such a proof has not been previously published even for the case with only one type of individuals.

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.

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.

Joint evolution of altruistic cooperation and dispersal in a metapopulation of small local populations.

We investigate the joint evolution of public goods cooperation and dispersal in a metapopulation model with small local populations. Altruistic cooperation can evolve due to assortment and kin selection, and dispersal can evolve because of demographic stochasticity, catastrophes and kin selection. Metapopulation structures resulting in assortment have been shown to make selection for cooperation possible. But how does dispersal affect cooperation and vice versa, when both are allowed to evolve as continuous traits? We found four qualitatively different evolutionary outcomes. (1) Monomorphic evolution to full defection with positive dispersal. (2) Monomorphic evolution to an evolutionarily stable state with positive cooperation and dispersal. In this case, parameter changes selecting for increased cooperation typically also select for increased dispersal. (3) Evolutionary branching can result in the evolutionarily stable coexistence of defectors and cooperators. Although defectors could be expected to disperse more than cooperators, here we show that the opposite case is also possible: Defectors tend to disperse less than cooperators when the total amount of cooperation in the dimorphic population is low enough. (4) Selection for too low cooperation can cause the extinction of the evolving population. For moderate catastrophe rates dispersal needs to be initially very frequent for evolutionary suicide to occur. Although selection for less dispersal in principle could prevent such evolutionary suicide, in most cases this rescuing effect is not sufficient, because selection in the cooperation trait is typically much stronger. If the catastrophe rate is large enough, a part of the boundary of viability can be evolutionarily attracting with respect to both strategy components, in which case evolutionary suicide is expected from all initial conditions.

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.

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.

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.