Advantage of storage in a fluctuating environment

We will elaborate the evolutionary course of an ecosystem consisting of a population in a chemostat environment with periodically fluctuating nutrient supply. The organisms that make up the population consist of structural biomass and energy storage compartments. In a constant chemostat environment a species without energy storage always out-competes a species with energy reserves. This hinders evolution of species with storage from those without storage. Using the adaptive dynamics approach for non-equilibrium ecological systems we will show that in a fluctuating environment there are multiple stable evolutionary singular strategies (ss's): one for a species without, and one for a species with energy storage. The evolutionary end-point depends on the initial evolutionary state. We will formulate the invasion fitness in terms of Floquet multipliers for the oscillating non-autonomous system. Bifurcation theory is used to study points where due to evolutionary development by mutational steps, the long-term dynamics of the ecological system changes qualitatively. To that end, at the ecological time scale, the trait value at which invasion of a mutant into a resident population becomes possible can be calculated using numerical bifurcation analysis where the trait is used as the free parameter, because it is just a bifurcation point. In a constant environment there is a unique stable equilibrium for one species following the “competitive exclusion” principle. In contrast, due to the oscillatory dynamics on the ecological time scale two species may coexist. That is, non-equilibrium dynamics enhances biodiversity. However, we will show that this coexistence is not stable on the evolutionary time scale and always one single species survives.     

Environmental fluctuations restrict eco-evolutionary dynamics in predator–prey system

Environmental fluctuations, species interactions and rapid evolution are all predicted to affect community structure and their temporal dynamics. Although the effects of the abiotic environment and prey evolution on ecological community dynamics have been studied separately, these factors can also have interactive effects. Here we used bacteria–ciliate microcosm experiments to test for eco-evolutionary dynamics in fluctuating environments. Specifically, we followed population dynamics and a prey defence trait over time when populations were exposed to regular changes of bottom-up or top-down stressors, or combinations of these. We found that the rate of evolution of a defence trait was significantly lower in fluctuating compared with stable environments, and that the defence trait evolved to lower levels when two environmental stressors changed recurrently. The latter suggests that top-down and bottom-up changes can have additive effects constraining evolutionary response within populations. The differences in evolutionary trajectories are explained by fluctuations in population sizes of the prey and the predator, which continuously alter the supply of mutations in the prey and strength of selection through predation. Thus, it may be necessary to adopt an eco-evolutionary perspective on studies concerning the evolution of traits mediating species interactions.     

Bifurcation theory, adaptive dynamics and dynamic energy budget-structured populations of iteroparous species

In this paper, we describe a technique to evaluate the evolutionary dynamics of the timing of spawning for iteroparous species. The life cycle of the species consists of three life stages, embryonic, juvenile and adult whereby the transitions of life stages (gametogenesis, birth and maturation) occur at species-specific sizes. The dynamics of the population is studied in a semi-chemostat environment where the inflowing food concentration is periodic (annual). A dynamic energy budget-based continuous-time model is used to describe the uptake of the food, storage in reserves and allocation of the energy to growth, maintenance, development (embryos, juveniles) and reproduction (adults). A discrete-event process is used for modelling reproduction. At a fixed spawning date of the year, the reproduction buffer is emptied and a new cohort is formed by eggs with a fixed size and energy content. The population consists of cohorts: for each year one consisting of individuals with the same age which die after their last reproduction event. The resulting mathematical model is a finite-dimensional set of ordinary differential equations with fixed 1-year periodic boundary conditions yielding a stroboscopic map. We will study the evolutionary development of the population using the adaptive dynamics approach. The trait is the timing of spawning. Pairwise and mutual invasibility plots are calculated using bifurcation analysis of the stroboscopic map. The evolutionary singular strategy value belonging to the evolutionary endpoint for the trait allows for an interpretation of the reproduction strategy of the population. In a case study, parameter values from the literature for the bivalve Macoma balthica are used.     

Adaptive evolution of foraging-related traits in a predator-prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.     

Eco-evolutionary maintenance of diversity in fluctuating environments

Growing evidence suggests that temporally fluctuating environments are important in maintaining variation both within and between species. To date, however, studies of genetic variation within a population have been largely conducted by evolutionary biologists (particularly population geneticists), while population and community ecologists have concentrated more on diversity at the species level. Despite considerable conceptual overlap, the commonalities and differences of these two alternative paradigms have yet to come under close scrutiny. Here, we review theoretical and empirical studies in population genetics and community ecology focusing on the ‘temporal storage effect’ and synthesise theories of diversity maintenance across different levels of biological organisation. Drawing on Chesson's coexistence theory, we explain how temporally fluctuating environments promote the maintenance of genetic variation and species diversity. We propose a further synthesis of the two disciplines by comparing models employing traditional frequency-dependent dynamics and those adopting density-dependent dynamics. We then address how temporal fluctuations promote genetic and species diversity simultaneously via rapid evolution and eco-evolutionary dynamics. Comparing and synthesising ecological and evolutionary approaches will accelerate our understanding of diversity maintenance in nature.     

Energy storage and the evolution of population dynamics

We explore the mutual dependence of life history evolution and population dynamics by modeling a structured rotifer population that preys on a dynamic food supply. We focus on the ecological role of energy storage. A physiologically based submodel describes how individual predators allocate assimilated energy among growth, reproduction, and storage. We use invasibility analyses to predict evolutionary stable strategies for energy allocation. Various proxy measures of fitness based on measurable biological quantities, such as average population size or average per-capita fecundity, fail to predict evolutionary stable strategies. The predicted strategies indicate that selection strongly favors storage allocation for juveniles, but only for adults when prey densities are high. With the evolution of energy storage, population dynamics can shift from aperiodic to stable cycles without any need to invoke group selection.     

EVOLUTION OF DISPERSAL RATES IN METAPOPULATION MODELS: BRANCHING AND CYCLIC DYNAMICS IN PHENOTYPE SPACE

We study the evolution of dispersal rates in a two patch metapopulation model. The local dynamics in each patch are given by difference equations, which, together with the rate of dispersal between the patches, determine the ecological dynamics of the metapopulation. We assume that phenotypes are given by their dispersal rate. The evolutionary dynamics in phenotype space are determined by invasion exponents, which describe whether a mutant can invade a given resident population. If the resident metapopulation is at a stable equilibrium, then selection on dispersal rates is neutral if the population sizes in the two patches are the same, while selection drives dispersal rates to zero if the local abundances are different. With non-equilibrium metapopulation dynamics, non-zero dispersal rates can be maintained by selection. In this case, and if the patches are ecologically identical, dispersal rates always evolve to values which induce synchronized metapopulation dynamics. If the patches are ecologically different, evolutionary branching into two coexisting dispersal phenotypes can be observed. Such branching can happen repeatedly, leading to polymorphisms with more than two phenotypes. If there is a cost to dispersal, evolutionary cycling in phenotype space can occur due to the dependence of selection pressures on the ecological attractor of the resident population, or because phenotypic branching alternates with the extinction of one of the branches. Our results extend those of Holt and McPeek (1996), and suggest that phenotypic branching is an important evolutionary process. This process may be relevant for sympatric speciation.     

Fluctuation Induces Evolutionary Branching in a Mathematical Model of Ecosystems

Ecological systems are always subjected to various environmental fluctuations. They evolve under these fluctuations and the resulting systems are robust against them. The diversity in ecological systems is also acquired through the evolution. How do the fluctuations affect the evolutionary processes? Do the fluctuations have direct impact on the species diversity in ecological systems? In the present paper, we investigate the relation between the environmental fluctuation and the evolution of species diversity with a mathematical model of evolutionary ecology. In the model, individual organisms compete for a single restricted resource and the temporal fluctuation in the resource supply is introduced as the environmental fluctuation. The evolutionary process is represented by the mutational change of genotypes which determines their resource utilization strategies. We found that when the environmental state is switched form static to fluctuating conditions, the initial closely related population distributed around the genotype adapted for the static environment is destabilized and divided into two groups in the genotype space; i.e., the evolutionary branching is induced by the environmental fluctuation. The consequent multiple species structures is evolutionary stable at the presence of the fluctuation. We perform the evolutionary invasion analysis for the phenomena and illustrate the mechanisms of the branchings. The results indicate a novel process of increasing the species diversity via evolutionary branching, and the analysis reveals the mechanisims of the branching preocess as the response to the environmental fluctuation. The robustness of the evolutionary process is also discussed.     

Evolution of cannibalistic traits: scenarios derived from adaptive dynamics

The evolution of cannibalistic traits in consumer populations is studied in this paper with the approach of adaptive dynamics theory. The model is kept at its minimum complexity by eliminating some environmental characteristics, like heterogeneity and seasonalities, and by hiding the size-structure of the population. Evolutionary dynamics are identified through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the canonical equation of adaptive dynamics. The result is a rich catalog of evolutionary scenarios involving evolutionary stable strategies and branching points both in the monomorphic and dimorphic dynamics. The possibility of evolutionary extinction of highly cannibalistic populations is also ascertained. This allows one to explain why cannibalism can be a transient stage of evolution.     

Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods

In recent years, three related methods have been used to model the phenotypic dynamics of traits under the influence of natural selection. The first is based on an approximation to quantitative genetic recursion equations for sexual populations. The second is based on evolution in asexual lineages with mutation-generated variation. The third method finds an evolutionarily stable set of phenotypes for species characterized by a given set of fitness functions, assuming that the mode of reproduction places no constraints on the number of distinct types that can be maintained in the population. The three methods share the property that the rate of change of a trait within a homogeneous population is approximately proportional to the individual fitness gradient. The methods differ in assumptions about the potential magnitude of phenotypic differences in mutant forms, and in their assumptions about the probability that invasion or speciation occurs when a species has a stable, yet invadable phenotype. Determining the range of applicability of the different methods is important for assessing the validity of optimization methods in predicting the evolutionary outcome of ecological interactions. Methods based on quantitative genetic models predict that fitness minimizing traits will often be evolutionarily stable over significant time periods, while other approaches suggest this is likely to be rare. A more detailed study of cases of disruptive selection might reveal whether fitness-minimizing traits occur frequently in natural communities.     

Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis

On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.     

Evolutionary coexistence in a fluctuating environment by specialization on resource level

Microbial communities in fluctuating environments, such as oceans or the human gut, contain a wealth of diversity. This diversity contributes to the stability of communities and the functions they have in their hosts and ecosystems. To improve stability and increase production of beneficial compounds, we need to understand the underlying mechanisms causing this diversity. When nutrient levels fluctuate over time, one possibly relevant mechanism is coexistence between specialists on low and specialists on high nutrient levels. The relevance of this process is supported by the observations of coexistence in the laboratory, and by simple models, which show that negative frequency dependence of two such specialists can stabilize coexistence. However, as microbial populations are often large and fast growing, they evolve rapidly. Our aim is to determine what happens when species can evolve; whether evolutionary branching can create diversity or whether evolution will destabilize coexistence. We derive an analytical expression of the invasion fitness in fluctuating environments and use adaptive dynamics techniques to find that evolutionarily stable coexistence requires a special type of trade-off between growth at low and high nutrients. We do not find support for the necessary evolutionary trade-off in data available for the bacterium Escherichia coli and the yeast Saccharomyces cerevisiae on glucose. However, this type of data is scarce and might exist for other species or in different conditions. Moreover, we do find evidence for evolutionarily stable coexistence of the two species together. Since we find this coexistence in the scarce data that are available, we predict that specialization on resource level is a relevant mechanism for species diversity in microbial communities in fluctuating environments in natural settings.     

Eco-evolutionary feedbacks,adaptive dynamics and evolutionary rescue theory

Adaptive dynamics theory has been devised to account for feedbacks between ecological and evolutionary processes. Doing so opens new dimensions to and raises new challenges about evolutionary rescue. Adaptive dynamics theory predicts that successive trait substitutions driven by eco-evolutionary feedbacks can gradually erode population size or growth rate, thus potentially raising the extinction risk. Even a single trait substitution can suffice to degrade population viability drastically at once and cause ‘evolutionary suicide’. In a changing environment, a population may track a viable evolutionary attractor that leads to evolutionary suicide, a phenomenon called ‘evolutionary trapping’. Evolutionary trapping and suicide are commonly observed in adaptive dynamics models in which the smooth variation of traits causes catastrophic changes in ecological state. In the face of trapping and suicide, evolutionary rescue requires that the population overcome evolutionary threats generated by the adaptive process itself. Evolutionary repellors play an important role in determining how variation in environmental conditions correlates with the occurrence of evolutionary trapping and suicide, and what evolutionary pathways rescue may follow. In contrast with standard predictions of evolutionary rescue theory, low genetic variation may attenuate the threat of evolutionary suicide and small population sizes may facilitate escape from evolutionary traps.     

Evolution of reproductive effort in viscous populations: the importance of population dynamics

I present two ecological models for the evolution of reproductive effort in viscous populations with empty sites. In contrast with previous studies, I show that limited dispersal needs not have a positive effect on the evolutionarily stable allocation of resources to fecundity versus survival. Rather, depending on the feedback between the trait and the population dynamics, population viscosity may have no effect or even lead to a decrease in the evolutionarily stable reproductive effort when individuals can degrade their environment during their lifetime. I show that the different evolutionary outcomes can be explained by the asymmetry in the level of kin competition resulting from investing into juveniles or into adults.     

Coevolution can stabilize a mutualistic interaction

Interspecific mutualisms are ubiquitous in nature, despite their ecological and evolutionary instability. Recent studies have developed coevolutionary theory of mutualisms, which coupled population and evolutionary dynamics, to resolve the longstanding puzzle. However, earlier studies assumed a time-scale separation between these dynamics, leaving an unanswered question of how a relaxation in the time-scale separation affects the coevolutionary dynamics of mutualism. Here I relax the strong assumption to theoretically show that ecological and evolutionary dynamics occurring in a similar time scale can stabilize an otherwise unstable mutualism. I show that the coevolutionary dynamics can cause a stable limit cycle or stable equilibrium in the population sizes, even if the population sizes increase unbounded in the absence of evolutionary adaptation. In contrast, coevolution can also cause stable limit cycle even if the population dynamics is stable in the absence of evolutionary adaptation. Furthermore, the model predicts that the population dynamics is likely to converge to equilibrium when the evolutionary speed of two species is similar and fast or highly dissimilar. The results suggest that the ease of the evolutionary ‘arms race’ is of crucial importance to maintain mutualism.     

MUTANT INVASIONS AND ADAPTIVE DYNAMICS IN VARIABLE ENVIRONMENTS

The evolution of natural organisms is ultimately driven by the invasion and possible fixation of mutant alleles. The invasion process is highly stochastic, however, and the probability of success is generally low, even for advantageous alleles. Additionally, all organisms live in a stochastic environment, which may have a large influence on what alleles are favorable, but also contributes to the uncertainty of the invasion process. We calculate the invasion probability of a beneficial, mutant allele in a monomorphic, large population subject to stochastic environmental fluctuations, taking into account density‐ and frequency‐dependent selection, stochastic population dynamics and temporal autocorrelation of the environment. We treat both discrete and continuous time population dynamics, and allow for overlapping generations in the continuous time case. The results can be generalized to diploid, sexually reproducing organisms embedded in communities of interacting species. We further use these results to derive an extended canonical equation of adaptive dynamics, predicting the rate of evolutionary change of a heritable trait on long evolutionary time scales.     

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.     

Delayed maturation in temporally structured populations with non-equilibrium dynamics

In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple — cyclic and/or chaotic — attractors, thus allowing us to illustrate how (i) evolutionary stability is primarily a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. Our approach is based on numerically estimating invasion exponents or “mutant fitnesses”. The invasion exponent is defined as the theoretical long-term average relative growth rate of a population of mutants in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values, we have to consider multi-valued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multi-valuedness occurs (i) when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or in odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. Non-equilibrium population dynamics or random environmental noise selects for strategists with a non-zero probability to delay maturation. When there is an evolutionarily attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, that is an Evolutionarily Unbeatable Strategy which is also Stable in a long term evolutionary sense. Population dynamical coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.     

Ecological bistability and evolutionary reversals under asymmetrical competition

Abstract How does the process of life‐history evolution interplay with population dynamics? Almost all models that have addressed this question assume that any combination of phenotypic traits uniquely determine the ecological population state. Here we show that if multiple ecological equilibria can exist, the evolution of a trait that relates to competitive performance can undergo adaptive reversals that drive cyclic alternation between population equilibria. The occurrence of evolutionary reversals requires neither environmentally driven changes in selective forces nor the coevolution of interactions with other species. The mechanism inducing evolutionary reversals is twofold. First, there exist phenotypes near which mutants can invade and yet fail to become fixed; although these mutants are eventually eliminated, their transitory growth causes the resident population to switch to an alternative ecological equilibrium. Second, asymmetrical competition causes the direction of selection to revert between high and low density. When ecological conditions for evolutionary reversals are not satisfied, the population evolves toward a steady state of either low or high abundance, depending on the degree of competitive asymmetry and environmental parameters. A sharp evolutionary transition between evolutionary stasis and evolutionary reversals and cycling can occur in response to a smooth change in ecological parameters, and this may have implications for our understanding of size‐abundance patterns.