First-order effect of frequency-dependent selection on fixation probability in an age-structured population with application to a public goods game

Inclusive fitness and Hamilton's rule in a stochastic environment

The evolution of cooperation in Prisoner's Dilemmas with additive random cost and benefit for cooperation cannot be accounted for by Hamilton's rule based on mean effects transferred from recipients to donors weighted by coefficients of relatedness, which defines inclusive fitness in a constant environment. Extensions that involve higher moments of stochastic effects are possible, however, and these are connected to a concept of random inclusive fitness that is frequency-dependent. This is shown in the setting of pairwise interactions in a haploid population with the same coefficient of relatedness between interacting players. In an infinite population, fixation of cooperation is stochastically stable if a mean geometric inclusive fitness of defection when rare is negative, while fixation of defection is stochastically unstable if a mean geometric inclusive fitness of cooperation when rare is positive, and these conditions are generally not equivalent. In a finite population, the probability for cooperation to ultimately fix when represented once exceeds the probability under neutrality or the corresponding probability for defection if the mean inclusive fitness of cooperation when its frequency is 1/3 or 1/2, respectively, exceeds 1. All these results rely on the simplifying assumption of a linear fitness function. It is argued that meaningful applications of random inclusive fitness in complex settings (multi-player game, diploidy, population structure) would generally require conditions of weak selection and additive gene action.

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.

Fifty years of Theoretical Population Biology

The year 2020 marks the 50th anniversary of Theoretical Population Biology. This special issue examines the past and continuing contributions of the journal. We identify some of the most important developments that have taken place in the pages of TPB, connecting them to current research and to the numerous forms of significance achieved by theory in population biology.

Diffusion approximation for an age-class-structured population under viability and fertility selection with application to fixation probability of an advantageous mutant

In this paper, we ascertain the validity of a diffusion approximation for the frequencies of different types under recurrent mutation and frequency-dependent viability and fertility selection in a haploid population with a fixed age-class structure in the limit of a large population size. The approximation is used to study, and explain in terms of selection coefficients, reproductive values and population-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant.