Class-structured Populations Research Articles

On the Interpretation of the Operation of Natural Selection in Class-Structured Populations.

AbstractBiological adaptation is the outcome of allele-frequency change by natural selection. At the same time, populations are usually class structured as individuals occupy different states, such as age, sex, or stage. This is known to result in the differential transmission of alleles through nonheritable fitness differences called class transmission, which also affects allele-frequency change even in the absence of selection. How does one then isolate allele-frequency change due to selection from that due to class transmission? We decompose one-generational allele-frequency change in terms of effects of selection and class transmission and show how reproductive values can be used to reach a decomposition between any two distant generations of the evolutionary process. This provides a missing relationship between multigenerational allele-frequency change and the operation of selection. It also allows a measure of fitness to be defined summarizing the effect of selection in a multigenerational evolutionary process, which connects asymptotically to invasion fitness.

How to make an inclusive-fitness model.

Social behaviours are typically modelled using neighbour-modulated fitness, which focuses on individuals having their fitness altered by neighbours. However, these models are either interpreted using inclusive fitness, which focuses on individuals altering the fitness of neighbours, or not interpreted at all. This disconnect leads to interpretational mistakes and obscures the adaptive significance of behaviour. We bridge this gap by presenting a systematic methodology for constructing inclusive-fitness models. We find a behaviour's 'inclusive-fitness effect' by summing primary and secondary deviations in reproductive value. Primary deviations are the immediate result of a social interaction; for example, the cost and benefit of an altruistic act. Secondary deviations are compensatory effects that arise because the total reproductive value of the population is fixed; for example, the increased competition that follows an altruistic act. Compared to neighbour-modulated fitness methodologies, our approach is often simpler and reveals the model's inclusive-fitness narrative clearly. We implement our methodology first in a homogeneous population, with supplementary examples of help under synergy, help in a viscous population and Creel's paradox. We then implement our methodology in a class-structured population, where the advantages of our approach are most evident, with supplementary examples of altruism between age classes, and sex-ratio evolution.

Multimorph Eco-Evolutionary Dynamics in Structured Populations.

AbstractOur understanding of the evolution of quantitative traits in nature is still limited by the challenge of including realistic trait distributions in the context of frequency-dependent selection and ecological feedbacks. We extend to class-structured populations a recently introduced "oligomorphic approximation," which bridges the gap between adaptive dynamics and quantitative genetics approaches and allows for the joint description of the dynamics of ecological variables and of the moments of multimodal trait distributions. Our theoretical framework allows us to analyze the dynamics of populations composed of several morphs and structured into distinct classes (e.g., age, size, habitats, infection status, and species). We also introduce a new approximation to simplify the eco-evolutionary dynamics using reproductive values. We illustrate the effectiveness of this approach by applying it to the important conceptual case of two-habitat migration-selection models. In particular, we show that our approach allows us to predict both the long-term evolutionary end points and the short-term transient dynamics of the eco-evolutionary process, including fast evolution regimes. We discuss the theoretical and practical implications of our results and sketch perspectives for future work.

Code for "Evolution of class-structured populations in periodic environments"

Code for "Evolution of class-structured populations in periodic environments"

Code for "Evolution of class-structured populations in periodic environments"

Code for "Evolution of class-structured populations in periodic environments"

Epidemiological and evolutionary consequences of periodicity in treatment coverage.

Host heterogeneity is a key driver of host–pathogen dynamics. In particular, the use of treatments against infectious diseases creates variation in quality among hosts, which can have both epidemiological and evolutionary consequences. We present a general theoretical model to highlight the consequences of different imperfect treatments on pathogen prevalence and evolution. These treatments differ in their action on host and pathogen traits. In contrast with previous studies, we assume that treatment coverage can vary in time, as in seasonal or pulsed treatment strategies. We show that periodic treatment strategies can limit both disease spread and virulence evolution, depending on the type of treatment. We also introduce a new method to analytically calculate the selection gradient in periodic environments, which allows our predictions to be interpreted using the concept of reproductive value, and can be applied more generally to analyse eco-evolutionary dynamics in class-structured populations and fluctuating environments.

Invasion implies substitution in ecological communities with class-structured populations

Long-term evolution of quantitative traits is classically and usefully described as the directional change in phenotype due to the recurrent fixation of new mutations. A formal justification for such continual evolution ultimately relies on the “invasion implies substitution”-principle. Here, whenever a mutant allele causing a small phenotypic change can successfully invade a population, the ancestral (or wild-type) allele will be replaced, whereby fostering gradual phenotypic change if the process is repeated. It has been argued that this principle holds in a broad range of situations, including spatially and demographically structured populations experiencing frequency- and density-dependent selection under demographic and environmental fluctuations. However, prior studies have not been able to account for all aspects of population structure, leaving unsettled the conditions under which the “invasion implies substitution”-principle really holds. In this paper, we start by laying out a program to explore and clarify the generality of the “invasion implies substitution”-principle. Particular focus is given on finding an explicit and functionally constant representation of the selection gradient on a quantitative trait. Using geometric singular perturbation methods, we then show that the “invasion implies substitution”-principle generalizes to well-mixed and scalar-valued polymorphic multispecies ecological communities that are structured into finitely many demographic (or physiological) classes. The selection gradient is shown to be constant over the evolutionary timescale and that it depends only on the resident phenotype, individual growth-rates, population steady states and reproductive values, all of which are calculated from the resident dynamics. Our work contributes to the theoretical foundations of evolutionary ecology.

When Do Individuals Maximize Their Inclusive Fitness?

Adaptation is often described in behavioral ecology as individuals maximizing their inclusive fitness. Under what conditions does this hold, and how does this relate to the gene-centered perspective of adaptation? We unify and extend the literature on these questions to class-structured populations. We demonstrate that the maximization (in the best-response sense) of class-specific inclusive fitness obtains in uninvadable population states (meaning that all deviating mutants become extinct). This defines a genuine actor-centered perspective on adaptation. But this inclusive fitness is assigned to all bearers of a mutant allele in a given class and depends on distributions of demographic and genetic contexts. These distributions, in turn, usually depend on events in previous generations and are thus not under individual control. This prevents, in general, envisioning individuals themselves as autonomous fitness maximizers, each with its own inclusive fitness. For weak selection, however, the dependence on earlier events can be neglected. We then show that each individual in each class appears to maximize its own inclusive fitness when all other individuals exhibit inclusive fitness-maximizing behavior. This defines a genuine individual-centered perspective of adaptation and justifies formally, as a first-order approximation, the long-heralded view of individuals appearing to maximize their own inclusive fitness.

From the Price equation to the selection gradient in class-structured populations: a quasi-equilibrium route

Recent studies in theoretical evolutionary ecology have emphasised two approaches to modelling evolution. On the one hand, models based on a separation of time scales rely on the concept of invasion fitness. On the other hand, models based on the Price equation track the dynamics of a trait average, coupled with a description of ecological dynamics. The aim of this article is to show that, in class-structured populations, both approaches yield the same expression for the selection gradient under weak selection. Although the result is not new, I propose an alternative route to its derivation using the dynamics of scaled measures of between-class phenotypic differentiation. Under weak selection, these measures of phenotypic differentiation can be treated as fast variables compared to the trait mean, which allows for a quasi-equilibrium approximation. This suggests a different approach to calculating weak selection approximations of evolutionary dynamics, and clarifies the links between short- and long-term perspectives on evolution in structured populations.

Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection.

In this paper we consider class-structured populations in discrete time in the limit of weak selection and with the inverse of the intensity of selection as unit of time. The aim is to establish a continuous model that approximates the discrete model. More precisely, we study frequency-dependent growth in an infinite haploid population structured into a finite number of classes such that individuals in each class contribute to a given subset of classes from one time step to the next. These contributions take the form of generalized fecundity parameters with perturbations of order 1/N that depends on the class frequencies of each type and the type frequencies. Moreover, they satisfy some mild conditions that ensure mixing in the long run. The dynamics in the limit as [Formula: see text] with N time steps as unit of time is considered first in the case of a single type, and second in the case of multiple types. The main result is that the type frequencies as [Formula: see text] obey the replicator equation with instantaneous growth rates for the different types that depend only on instantaneous equilibrium class frequencies and reproductive values. An application to evolutionary game theory complemented by simulation results is presented.

Defining fitness in an uncertain world

The recently elucidated definition of fitness employed by Fisher in his fundamental theorem of natural selection is combined with reproductive values as appropriately defined in the context of both random environments and continuing fluctuations in the distribution over classes in a class-structured population. We obtain astonishingly simple results, generalisations of the Price Equation and the fundamental theorem, that show natural selection acting only through the arithmetic expectation of fitness over all uncertainties, in contrast to previous studies with fluctuating demography, in which natural selection looks rather complicated. Furthermore, our setting permits each class to have its characteristic ploidy, thus covering haploidy, diploidy and haplodiploidy at the same time; and allows arbitrary classes, including continuous variables such as condition. The simplicity is achieved by focussing just on the effects of natural selection on genotype frequencies: while other causes are present in the model, and the effect of natural selection is assessed in their presence, these causes will have their own further effects on genoytpe frequencies that are not assessed here. Also, Fisher’s uses of reproductive value are shown to have two ambivalences, and a new axiomatic foundation for reproductive value is endorsed. The results continue the formal darwinism project, and extend support for the individual-as-maximising-agent analogy to finite populations with random environments and fluctuating class-distributions. The model may also lead to improved ways to measure fitness in real populations.

Quantifying injury to common bottlenose dolphins from the Deepwater Horizon oil spill using an age-, sex- and class-structured population model

Quantifying injury to common bottlenose dolphins from the Deepwater Horizon oil spill using an age-, sex- and class-structured population model

Invasion fitness, inclusive fitness, and reproductive numbers in heterogeneous populations.

How should fitness be measured to determine which phenotype or "strategy" is uninvadable when evolution occurs in a group-structured population subject to local demographic and environmental heterogeneity? Several fitness measures, such as basic reproductive number, lifetime dispersal success of a local lineage, or inclusive fitness have been proposed to address this question, but the relationships between them and their generality remains unclear. Here, we ascertain uninvadability (all mutant strategies always go extinct) in terms of the asymptotic per capita number of mutant copies produced by a mutant lineage arising as a single copy in a resident population ("invasion fitness"). We show that from invasion fitness uninvadability is equivalently characterized by at least three conceptually distinct fitness measures: (i) lineage fitness, giving the average individual fitness of a randomly sampled mutant lineage member; (ii) inclusive fitness, giving a reproductive value weighted average of the direct fitness costs and relatedness weighted indirect fitness benefits accruing to a randomly sampled mutant lineage member; and (iii) basic reproductive number (and variations thereof) giving lifetime success of a lineage in a single group, and which is an invasion fitness proxy. Our analysis connects approaches that have been deemed different, generalizes the exact version of inclusive fitness to class-structured populations, and provides a biological interpretation of natural selection on a mutant allele under arbitrary strength of selection.

Biological fitness and the Price Equation in class-structured populations.

Price׳s extended covariance selection mathematics is applied to class-structured populations with additional assumptions, to derive the 'genetic Price Equation with class structure'. Each individual belongs to a class, and there may be overlapping generations; the equation is genetic because the trait is restricted to an arbitrary weighted sum of allele frequencies. Two special cases are then considered, a demography-like case corresponding to Fisher's fundamental theorem of natural selection, and a sex-ratio-like case corresponding to Fisher's sex ratio argument: these differ in whether it is natural to assume that the per-capita or the total reproductive values of each class are maintained from the parental to the descendant population. These cases also match the two existing attempts to eliminate from the effects of natural selection those passive changes in allele frequencies that are caused by the class structure, and suggest improvements in one of them. In each case a more specialised Price Equation, and a 'fundamental theorem of natural selection', are proved, which hold out of class-structure equilibrium, showing that passive changes can be eliminated in more than one way, and hinting at the possibility of a more general formulation. Previous class-structured Price Equations and a 'fundamental theorem' are linked to these results. The power of Price's formal approach is vividly illustrated by this lucid conspectus of otherwise self-standing theories with confusing interconnections.

Stochastic Demography and the Neutral Substitution Rate in Class-Structured Populations

The neutral rate of allelic substitution is analyzed for a class-structured population subject to a stationary stochastic demographic process. The substitution rate is shown to be generally equal to the effective mutation rate, and under overlapping generations it can be expressed as the effective mutation rate in newborns when measured in units of average generation time. With uniform mutation rate across classes the substitution rate reduces to the mutation rate.

Foundations of a mathematical theory of darwinism

This paper pursues the 'formal darwinism' project of Grafen, whose aim is to construct formal links between dynamics of gene frequencies and optimization programmes, in very abstract settings with general implications for biologically relevant situations. A major outcome is the definition, within wide assumptions, of the ubiquitous but problematic concept of 'fitness'. This paper is the first to present the project for mathematicians. Within the framework of overlapping generations in discrete time and no social interactions, the current model shows links between fitness maximization and gene frequency change in a class-structured population, with individual-level uncertainty but no uncertainty in the class projection operator, where individuals are permitted to observe and condition their behaviour on arbitrary parts of the uncertainty. The results hold with arbitrary numbers of loci and alleles, arbitrary dominance and epistasis, and make no assumptions about linkage, linkage disequilibrium or mating system. An explicit derivation is given of Fisher's Fundamental Theorem of Natural Selection in its full generality.

The stationary distribution of a continuously varying strategy in a class‐structured population under mutation–selection–drift balance

Many traits and/or strategies expressed by organisms are quantitative phenotypes. Because populations are of finite size and genomes are subject to mutations, these continuously varying phenotypes are under the joint pressure of mutation, natural selection and random genetic drift. This article derives the stationary distribution for such a phenotype under a mutation-selection-drift balance in a class-structured population allowing for demographically varying class sizes and/or changing environmental conditions. The salient feature of the stationary distribution is that it can be entirely characterized in terms of the average size of the gene pool and Hamilton's inclusive fitness effect. The exploration of the phenotypic space varies exponentially with the cumulative inclusive fitness effect over state space, which determines an adaptive landscape. The peaks of the landscapes are those phenotypes that are candidate evolutionary stable strategies and can be determined by standard phenotypic selection gradient methods (e.g. evolutionary game theory, kin selection theory, adaptive dynamics). The curvature of the stationary distribution provides a measure of the stability by convergence of candidate evolutionary stable strategies, and it is evaluated explicitly for two biological scenarios: first, a coordination game, which illustrates that, for a multipeaked adaptive landscape, stochastically stable strategies can be singled out by letting the size of the gene pool grow large; second, a sex-allocation game for diploids and haplo-diploids, which suggests that the equilibrium sex ratio follows a Beta distribution with parameters depending on the features of the genetic system.

The evolution of juvenile-adult interactions in populations structured in age and space

We study the evolution of a spatially structured population with two age classes using spatial moment equations. In the model, adults can either help juveniles by increasing their survival, or adopt a cannibalistic behaviour and consume juveniles. While cannibalism is the sole evolutionary outcome when the population is well-mixed, both cannibalism and parental care can be evolutionarily stable if the population is viscous. Our analysis allows us to make two main technical points. First, we present a method to define invasion fitness in class-structured viscous populations, which allows us to apply adaptive dynamics methodology. Second, we show that ordinary pair approximation introduces an important quantitative bias in the evolutionary model, even on random networks. We propose a correction to the ordinary pair approximation that yields quantitative accuracy, and discuss how the bias associated with this approach is precisely what allows us to identify subtle aspects associated with the evolutionary dynamics of spatially structured populations.

The neutral effective migration rate in a mainland-island context

Genetic influx into a population often does not correspond to the real migration rate ( m ) of individuals, due to class structure within the population. The effective migration rate ( m e ) is a concept to measure gene flow in such a situation. The ratio of the effective migration rate to the real migration rate ( m e m ) is called the gene flow factor, and represents the degree of gene flow modification. Prior authors proposed different definitions of the effective migration rate. These may be categorized into two groups: the neutral effective migration rate and the selective effective migration rate. In this article, we construct a general model of a class-structured population with a mainland-island structure. Using the model, we prove that the gene flow factor of the neutral effective migration rate converges to the mean reproductive value of immigrants if the limit is taken with the real migration rate converging to zero. This limit theorem provides a novel interpretation of gene flow and can be used to derive approximation formulae of the neutral effective migration rate. We illustrate this method analyzing two examples, sex ratio distortion due to extrinsic factors and hybrid zones with underdominance.

Evolution of parental favoritism among different-aged offspring

The theories of intrafamilial conflict and parental investment have yet to examine how parents’ decisions about resource allocation are influenced by the fact that their offspring may be different in age. Two counteracting effects of offspring growth on parental allocation of resources have deterred the development of a formal model: A parent may favor older offspring due to their greater reproductive value or favor younger offspring due to their higher marginal returns from extra resources. Using an evolutionary invasion analysis in class-structured populations, I present a formal model that explores how a parent should allocate its resources among different-aged offspring from the viewpoint of the parent. The parent’s evolutionarily stable strategy is to allocate its resources such that the marginal benefit to each offspring’s survival, weighted by the survival probability to the reproductive age, is equal to the marginal cost to the parent’s residual survival. Two general situations are considered in which younger offspring obtain higher marginal returns than older offspring. In nearly all circumstances, a parent is expected to bias its resources toward older offspring. The results may account for the widespread yet puzzling phenomenon of parental bias toward older offspring in view of previous theories of intrafamilial conflict. Key words: evolutionarily stable strategy, hatching asynchrony, invasion analysis, offspring age, parental investment, reproductive value. [Behav Ecol]