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.

Metacommunities, fitness and gradual evolution.

We analyze the evolution of a multidimensional quantitative trait in a class-structured focal species interacting with other species in a wider metacommunity. The evolutionary dynamics in the focal species as well as the ecological dynamics of the whole metacommunity is described as a continuous-time process with birth, physiological development, dispersal, and death given as rates that can depend on the state of the whole metacommunity. This can accommodate complex local community and global metacommunity environmental feedbacks owing to inter- and intra-specific interactions, as well as local environmental stochastic fluctuations. For the focal species, we derive a fitness measure for a mutant allele affecting class-specific trait expression. Using classical results from geometric singular perturbation theory, we provide a detailed proof that if the effect of the mutation on phenotypic expression is small ("weak selection"), the large system of dynamical equations needed to describe selection on the mutant allele in the metacommunity can be reduced to a single ordinary differential equation on the arithmetic mean mutant allele frequency that is of constant sign. This invariance on allele frequency entails the mutant either dies out or will out-compete the ancestral resident (or wild) type. Moreover, the directional selection coefficient driving arithmetic mean allele frequency can be expressed as an inclusive fitness effect calculated from the resident metacommunity alone, and depends, as expected, on individual fitness differentials, relatedness, and reproductive values. This formalizes the Darwinian process of gradual evolution driven by random mutation and natural selection in spatially and physiologically class-structured metacommunities.

Hamilton's rule, gradual evolution, and the optimal (feedback) control of phenotypically plastic traits.

Most traits expressed by organisms, such as gene expression profiles, developmental trajectories, behavioural sequences and reaction norms are function-valued traits (colloquially "phenotypically plastic traits"), since they vary across an individual's age and in response to various internal and/or external factors (state variables). Furthermore, most organisms live in populations subject to limited genetic mixing and are thus likely to interact with their relatives. We here formalise selection on genetically determined function-valued traits of individuals interacting in a group-structured population, by deriving the marginal version of Hamilton's rule for function-valued traits. This rule simultaneously gives a condition for the invasion of an initially rare mutant function-valued trait and its ultimate fixation in the population (invasion thus implies substitution). Hamilton's rule thus underlies the gradual evolution of function-valued traits and gives rise to necessary first-order conditions for their uninvadability (evolutionary stability). We develop a novel analysis using optimal control theory and differential game theory, to simultaneously characterise and compare the first-order conditions of (i) open-loop traits - functions of time (or age) only, and (ii) closed-loop (state-feedback) traits - functions of both time and state variables. We show that closed-loop traits can be represented as the simpler open-loop traits when individuals do not interact or when they interact with clonal relatives. Our analysis delineates the role of state-dependence and interdependence between individuals for trait evolution, which has implications to both life-history theory and social evolution.

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.

Evolutionary Pathways for the Generation of New Self-Incompatibility Haplotypes in a Nonself-Recognition System.

Self-incompatibility (SI) is a genetically based recognition system that functions to prevent self-fertilization and mating among related plants. An enduring puzzle in SI is how the high diversity observed in nature arises and is maintained. Based on the underlying recognition mechanism, SI can be classified into two main groups: self-recognition (SR) and nonself-recognition (NSR). Most work has focused on diversification within SR systems despite expected differences between the two groups in the evolutionary pathways and outcomes of diversification. Here, we use a deterministic population genetic model and stochastic simulations to investigate how novel S-haplotypes evolve in a gametophytic NSR [SRNase/S Locus F-box (SLF)] SI system. For this model, the pathways for diversification involve either the maintenance or breakdown of SI and can vary in the order of mutations of the female (SRNase) and male (SLF) components. We show analytically that diversification can occur with high inbreeding depression and self-pollination, but this varies with evolutionary pathway and level of completeness (which determines the number of potential mating partners in the population), and, in general, is more likely for lower haplotype number. The conditions for diversification are broader in stochastic simulations of finite population size. However, the number of haplotypes observed under high inbreeding and moderate-to-high self-pollination is less than that commonly observed in nature. Diversification was observed through pathways that maintain SI as well as through self-compatible intermediates. Yet the lifespan of diversified haplotypes was sensitive to their level of completeness. By examining diversification in a NSR SI system, this model extends our understanding of the evolution and maintenance of haplotype diversity observed in a recognition system common in flowering plants.

Optional interactions and suspicious behaviour facilitates trustful cooperation in prisoners dilemma.

In evolutionary game theory interactions between individuals are often assumed obligatory. However, in many real-life situations, individuals can decide to opt out of an interaction depending on the information they have about the opponent. We consider a simple evolutionary game theoretic model to study such a scenario, where at each encounter between two individuals the type of the opponent (cooperator/defector) is known with some probability, and where each individual either accepts or opts out of the interaction. If the type of the opponent is unknown, a trustful individual accepts the interaction, whereas a suspicious individual opts out of the interaction. If either of the two individuals opt out both individuals remain without an interaction. We show that in the prisoners dilemma optional interactions along with suspicious behaviour facilitates the emergence of trustful cooperation.

Evolutionarily stable mating decisions for sequentially searching females and the stability of reproductive isolation by assortative mating.

We consider mating strategies for females who search for males sequentially during a season of limited length. We show that the best strategy rejects a given male type if encountered before a time-threshold but accepts him after. For frequency-independent benefits, we obtain the optimal time-thresholds explicitly for both discrete and continuous distributions of males, and allow for mistakes being made in assessing the correct male type. When the benefits are indirect (genes for the offspring) and the population is under frequency-dependent ecological selection, the benefits depend on the mating strategy of other females as well. This case is particularly relevant to speciation models that seek to explore the stability of reproductive isolation by assortative mating under frequency-dependent ecological selection. We show that the indirect benefits are to be quantified by the reproductive values of couples, and describe how the evolutionarily stable time-thresholds can be found. We conclude with an example based on the Levene model, in which we analyze the evolutionarily stable assortative mating strategies and the strength of reproductive isolation provided by them.

Chaotic dynamics of allele frequencies in condition-dependent mating systems.

I study the dynamics of allele frequencies in sexually reproducing populations where the choosy sex has a preference for condition-dependent displays of the opposite sex. The condition of an individual is assumed to be shaped by frequency-dependent selection. For sufficiently strong preferences the dynamics becomes increasingly complex, and periodic orbits and chaos are observed. Moreover, multiple attractors can exist simultaneously. The results hold also when the choosy sex is allowed to maintain a moderate level of assortative mating. Complex dynamics, a well studied phenomenon in a purely ecological setting, has been rarely observed in ecologically motivated population genetic models.

On invasion boundaries and the unprotected coexistence of two strategies.

In this paper we present, in terms of invasion fitness functions, a sufficient condition for a coexistence of two strategies which are not protected from extinction when rare. In addition, we connect the result to the local characterization of singular strategies in the theory of adaptive dynamics. We conclude with some illustrative examples.

Evolutionary branching of a magic trait.

We study the adaptive dynamics of a so-called magic trait, which is under natural selection and which also serves as a cue for assortative mating. We derive general results on the monomorphic evolutionary singularities. Next, we study the long-term evolution of single-locus genetic polymorphisms under various strengths of assortativity in a version of Levene's soft-selection model, where natural selection favours different values of a continuous trait within two habitats. If adaptive dynamics leads to a polymorphism with sufficiently different alleles, then the corresponding homozygotes cease to interbreed so that sympatric speciation occurs.