Evolutionary Branching in a Finite Population: Deterministic Branching vs. Stochastic Branching

Adaptive dynamics formalism demonstrates that, in a constant environment, a continuous trait may first converge to a singular point followed by spontaneous transition from a unimodal trait distribution into a bimodal one, which is called “evolutionary branching.” Most previous analyses of evolutionary branching have been conducted in an infinitely large population. Here, we study the effect of stochasticity caused by the finiteness of the population size on evolutionary branching. By analyzing the dynamics of trait variance, we obtain the condition for evolutionary branching as the one under which trait variance explodes. Genetic drift reduces the trait variance and causes stochastic fluctuation. In a very small population, evolutionary branching does not occur. In larger populations, evolutionary branching may occur, but it occurs in two different manners: in deterministic branching, branching occurs quickly when the population reaches the singular point, while in stochastic branching, the population stays at singularity for a period before branching out. The conditions for these cases and the mean branching-out times are calculated in terms of population size, mutational effects, and selection intensity and are confirmed by direct computer simulations of the individual-based model.     

Can possible evolutionary outcomes be determined directly from the population dynamics?

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria — one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.     

Superinfections can induce evolutionarily stable coexistence of pathogens

Parasites reproduce and are subject to natural selection at several different, but intertwined, levels. In the recent paper, Gilchrist and Coombs (Theor. Popul. Biol. 69:145–153, 2006) relate the between-host transmission in the context of an SI model to the dynamics within a host. They demonstrate that within-host selection may lead to an outcome that differs from the outcome of selection at the host population level. In this paper we combine the two levels of reproduction by considering the possibility of superinfection and study the evolution of the pathogen’s within-host reproduction rate p. We introduce a superinfection function φ = φ(p,q), giving the probability with which pathogens with trait q, upon transmission to a host that is already infected by pathogens with trait p, “take over” the host. We consider three cases according to whether the function q → φ(p,q) (i) has a discontinuity, (ii) is continuous, but not differentiable, or (iii) is differentiable in q = p. We find that in case (i) the within-host selection dominates in the sense that the outcome of evolution at the host population level coincides with the outcome of evolution in a single infected host. In case (iii), it is the transmission to susceptible hosts that dominates the evolution to the extent that the singular strategies are the same as when the possibility of superinfections is ignored. In the biologically most relevant case (ii), both forms of reproduction contribute to the value of a singular trait. We show that when φ is derived from a branching process variant of the submodel for the within-host interaction of pathogens and target cells, the superinfection functions fall under case (ii). We furthermore demonstrate that the superinfection model allows for steady coexistence of pathogen traits at the host population level, both on the ecological, as well as on the evolutionary time scale.      

Can possible evolutionary outcomes be determined directly from the population dynamics?

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria — one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.     

Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor,II

In our previous report [Aita, T., Morinaga, S., Hosimi, Y., 2004. Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor I. Bull. Math. Biol. 66, 1371–1403], an analogy between thermodynamics and adaptive walks on a Mt. Fuji-type fitness landscape in an artificial selection system was presented. Introducing the ‘free fitness’ as the sum of a fitness term and an entropy term and ‘evolutionary force’ as the gradient of free fitness on a fitness coordinate, we demonstrated that the adaptive walk (=evolution) is driven by the evolutionary force in the direction in which free fitness increases. In this report, we examine the effect of various modifications of the original model on the properties of the adaptive walk. The modifications were as follows: first, mutation distance d was distributed obeying binomial distribution; second, the selection process obeyed the natural selection protocol; third, ruggedness was introduced to the landscape according to the NK model; fourth, a noise was included in the fitness measurement. The effect of each modification was described in the same theoretical framework as the original model by introducing ‘effective’ quantities such as the effective mutation distance or the effective screening size.     

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.     

Evolutionary ecology of inducible morphological plasticity in predator–prey interaction: toward the practical links with population ecology

The outcome of species interactions is often strongly influenced by variation in the functional traits of the individuals participating. A rather large body of work demonstrates that inducible morphological plasticity in predators and prey can both influence and be influenced by species interaction strength, with important consequences for individual fitness. Much of the past research in this area has focused on the ecological and evolutionary significance of trait plasticity by studying single predator–prey pairs and testing the performance of individuals having induced and noninduced phenotypes. This research has thus been critical in improving our understanding of the adaptive value of trait plasticity and its widespread occurrence across species and community types. More recently, researchers have expanded this foundation by examining how the complexity of organismal design and community-level properties can shape plasticity in functional traits. In addition, researchers have begun to merge evolutionary and ecological perspectives by linking trait plasticity to community dynamics, with particular attention on trait-mediated indirect interactions. Here, we review recent studies on inducible morphological plasticity in predators and their prey with an emphasis on internal and external constraints and how the nature of predator–prey interactions influences the expression of inducible phenotypes. In particular, we focus on multiple-trait plasticity, flexibility and modification of inducible plasticity, and reciprocal plasticity between predator and prey. Based on our arguments on these issues, we propose future research directions that should better integrate evolutionary and population studies and thus improve our understanding of the role of phenotypic plasticity in predator–prey population and community dynamics.     

The evolution of resource use

 The evolution of a consumer exploiting two resources is investigated. The strategy x under selection represents the fraction of time or energy an individual invests into extracting the first resource. In the model, a dimensionless parameter α quantifies how simultaneous consumption of both resources influences consumer growth; α<0 corresponds to hemi-essential resources, 0<α<1 corresponds to complementary resources, α=1 corresponds to perfectly substitutable resources, and α>1 corresponds to antagonistic resources. An analysis of the ecological and evolutionary dynamics leads to five conclusions. First, when α≤1, there is a unique singular strategy x ^* for the adaptive dynamics and it is evolutionarily stable and globally convergent stable. Second, when α=1, the singular strategy x ^* corresponds to the populations exhibiting an ideal free distribution and a population playing this strategy can invade and displace populations playing any other strategy. Third, when α>1, the strategies x=0 and x=1 are evolutionarily stable and convergent stable. Hence, if the populations initially specialize on one resource, evolution amplifies this specialization. Fourth, when α is slightly larger than one (i.e. the resources are slightly antagonistic), there is a convergent stable singular strategy whose basin of attraction is almost the entire strategy space (0,1). This singular strategy is evolutionarily unstable and serves as an evolutionary branching point. Following evolutionary branching, our analysis and numerical simulations suggest that evolutionary dynamics are driven toward an end state consisting of two populations specializing on different resources. Fifth, when α>>1, there is only one singular strategy and it is convergent unstable and evolutionarily unstable. Hence, if resources are overly antagonistic, evolutionary branching does not occur and ultimately only one resource is exploited. Received: 8 June 2002 / Revised version: 28 November 2002 / Published online: 23 April 2003 This work was supported by NSF Grant DMS-0077986 Key words or phrases: Consumer-resource interactions – Adaptive dynamics – Evolutionary branching     

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.     

Quantification and decomposition of environment‐selection relationships

In nature, selection varies across time in most environments, but we lack an understanding of how specific ecological changes drive this variation. Ecological factors can alter phenotypic selection coefficients through changes in trait distributions or individual mean fitness, even when the trait‐absolute fitness relationship remains constant. We apply and extend a regression‐based approach in a population of Soay sheep (Ovis aries) and suggest metrics of environment‐selection relationships that can be compared across studies. We then introduce a novel method that constructs an environmentally structured fitness function. This allows calculation of full (as in existing approaches) and partial (acting separately through the absolute fitness function slope, mean fitness, and phenotype distribution) sensitivities of selection to an ecological variable. Both approaches show positive overall effects of density on viability selection of lamb mass. However, the second approach demonstrates that this relationship is largely driven by effects of density on mean fitness, rather than on the trait‐fitness relationship slope. If such mechanisms of environmental dependence of selection are common, this could have important implications regarding the frequency of fluctuating selection, and how previous selection inferences relate to longer term evolutionary dynamics.     

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.     

Darwinian adaptation,population genetics and the streetcar theory of evolution

 This paper investigates the problem of how to conceive a robust theory of phenotypic adaptation in non-trivial models of evolutionary biology. A particular effort is made to develop a foundation of this theory in the context of n-locus population genetics. Therefore, the evolution of phenotypic traits is considered that are coded for by more than one gene. The potential for epistatic gene interactions is not a priori excluded. Furthermore, emphasis is laid on the intricacies of frequency-dependent selection. It is first discussed how strongly the scope for phenotypic adaptation is restricted by the complex nature of ‘reproduction mechanics’ in sexually reproducing diploid populations. This discussion shows that one can easily lose the traces of Darwinism in n-locus models of population genetics. In order to retrieve these traces, the outline of a new theory is given that I call ‘streetcar theory of evolution’. This theory is based on the same models that geneticists have used in order to demonstrate substantial problems with the ‘adaptationist programme’. However, these models are now analyzed differently by including thoughts about the evolutionary removal of genetic constraints. This requires consideration of a sufficiently wide range of potential mutant alleles and careful examination of what to consider as a stable state of the evolutionary process. A particular notion of stability is introduced in order to describe population states that are phenotypically stable against the effects of all mutant alleles that are to be expected in the long-run. Surprisingly, a long-term stable state can be characterized at the phenotypic level as a fitness maximum, a Nash equilibrium or an ESS. The paper presents these mathematical results and discusses – at unusual length for a mathematical journal – their fundamental role in our current understanding of evolution. Received 22 April 1994; received in revised form 10 July 1995     

Evolutionary dynamics as a component of stage-structured matrix models: an example using Trillium grandiflorum

Evolution by natural selection improves fitness and may therefore influence population trajectories. Demographic matrix models are often employed in conservation studies to project population dynamics, but such analyses have not incorporated evolutionary dynamics. We project evolutionarily informed population trajectories for a population of the perennial plant Trillium grandiflorum, which is declining due to high levels of herbivory by white-tailed deer. Individuals with later flowering times are less often consumed, so there is selection on this trait. We first incorporated selection analyses into a deterministic matrix model in three ways (corresponding to different methods that have been used for analyzing evolution in structured populations). Because it is not clear which of these methods works best for stage-structured models, we compared each with a more realistic, individual-based model. Deterministic models using fitness averaged over the phenotypic distribution gave trajectories that were similar to those of the individual-based model, whereas the deterministic model using fitness at the mean phenotype gave a much faster rate of evolution than that which was observed. This illustrates that subtle differences in the way in which one splices evolution into demographic models can have a large effect on expected outcomes. This study demonstrates that, by combining demographic and selection analyses, one can gauge the potential relevance of evolution to population dynamics and persistence.     

Population and evolutionary dynamics in spatially structured seasonally varying environments

Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life‐history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco‐evolutionary theory and models have not yet fully encompassed within‐individual and among‐individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life‐history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal‐tracking technologies are increasingly demonstrating substantial within‐population variation in the occurrence and form of migration versus year‐round residence, generating diverse forms of ‘partial migration’ spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year‐round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio‐temporal population dynamics, we define a ‘partially migratory meta‐population’ system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within‐individual and among‐individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco‐evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta‐populations given diverse forms of seasonal environmental variation and change, and to forecast system‐specific dynamics. To demonstrate one such approach, we use an evolutionary individual‐based model to illustrate that multiple forms of partial migration can readily co‐exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco‐evolutionary responses to spatio‐temporal seasonal environmental variation and change.     

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.     

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.     

What life cycle graphs can tell about the evolution of life histories

We analyze long-term evolutionary dynamics in a large class of life history models. The model family is characterized by discrete-time population dynamics and a finite number of individual states such that the life cycle can be described in terms of a population projection matrix. We allow an arbitrary number of demographic parameters to be subject to density-dependent population regulation and two or more demographic parameters to be subject to evolutionary change. Our aim is to identify structural features of life cycles and modes of population regulation that correspond to specific evolutionary dynamics. Our derivations are based on a fitness proxy that is an algebraically simple function of loops within the life cycle. This allows us to phrase the results in terms of properties of such loops which are readily interpreted biologically. The following results could be obtained. First, we give sufficient conditions for the existence of optimisation principles in models with an arbitrary number of evolving traits. These models are then classified with respect to their appropriate optimisation principle. Second, under the assumption of just two evolving traits we identify structural features of the life cycle that determine whether equilibria of the monomorphic adaptive dynamics (evolutionarily singular points) correspond to fitness minima or maxima. Third, for one class of frequency-dependent models, where optimisation is not possible, we present sufficient conditions that allow classifying singular points in terms of the curvature of the trade-off curve. Throughout the article we illustrate the utility of our framework with a variety of examples.     

Evolutionary tradeoff and equilibrium in an aquatic predator-prey system

Due to the conventional distinction between ecological (rapid) and evolutionary (slow) timescales, ecological and population models have typically ignored the effects of evolution. Yet the potential for rapid evolutionary change has been recently established and may be critical to understanding how populations persist in changing environments. In this paper we examine the relationship between ecological and evolutionary dynamics, focusing on a well-studied experimental aquatic predator-prey system (Fussmann et al., 2000, Science, 290, 1358–1360; Shertzer et al., 2002, J. Anim. Ecol., 71, 802–815; Yoshida et al., 2003, Nature, 424, 303–306). Major properties of predator-prey cycles in this system are determined by ongoing evolutionary dynamics in the prey population. Under some conditions, however, the populations tend to apparently stable steady-state densities. These are the subject of the present paper. We examine a previously developed model for the system, to determine how evolution shapes properties of the equilibria, in particular the number and identity of coexisting prey genotypes. We then apply these results to explore how evolutionary dynamics can shape the responses of the system to ‘management’: externally imposed alterations in conditions. Specifically, we compare the behavior of the system including evolutionary dynamics, with predictions that would be made if the potential for rapid evolutionary change is neglected. Finally, we posit some simple experiments to verify our prediction that evolution can have significant qualitative effects on observed population-level responses to changing conditions.     

Defining fitness in evolutionary models

The analysis of evolutionary models requires an appropriate definition for fitness. In this paper, I review such definitions in relation to the five major dimensions by which models may be described, namely (i) finite versus infinite (or very large) population size, (ii) type of environment (constant, fixed length, temporally stochastic, temporally predictable, spatially stochastic, spatially predictable and social environment), (iii) density-independent or density-dependent, (iv) inherent population dynamics (equilibrium, cyclical and chaotic), and (v) frequency-dependent or independent. In simple models, the Malthusian parameter ‘r’ or the net reproductive rate R ₀ may be satisfactory, but once density-dependence or complex population dynamics is introduced the invasion exponent should be used. Defining fitness in a social environment or when there is frequency-dependence requires special consideration.