Évolution dans des populations structurées en classes

Firstly, we consider an age-structured asexual haploid population with constant age-specific birth and death (or survival) parameters for each type. In this simplified framework, the rate of change in mean fitness in continuous time is expected to be exactly equal to the genetic variance in fitness (or to the genetic variance in fitness divided by the mean fitness in discrete time), which can be seen as a generalized growth-rate theorem. Fisher’s (1930) Fundamental Theorem of Natural Selection (FTNS), and in particular the development of an explicit age-structured version of the theorem, is shown to hold with the Malthusian parameter used as the definition of fitness. Then, 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. The main result is that the type frequencies as N → +∞ 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. 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 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-age-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant. In the same context, we deduce the first-order effect of frequency-dependent viability and fertility selection on the probability of fixation of a mutant in a large finite haploid population with a fixed age structure by applying a direct small perturbation method to the neutral two-timescale genealogical process. This effect is expressed in terms of fixation-fitness coefficients times ancestry coefficients that are related to the effective population size. In the case of constant selection, the fixation-fitness coefficients are functions of the coefficients of viability and fertility selection weighted by reproductive values and population-structure coefficients for the different age classes. This explains the difference between the effects of viability selection and fertility selection on fixation probability in age-structured populations. With frequency-dependent selection in the form of a public goods game, the fixation-fitness coefficients depend also on the public good allocation strategy of the population and the resource allocation strategies of the individuals. In this case, the results show that weak selection may favor the fixation of cooperation if the cooperators allocate more resources to reproduction versus survival than the defectors.

Keyphrases: Age-structured populations., Growth-rate theorem., Malthusian parameter., Reproductive value., Fundamental Theorem of Natural Selection., Class-structured populations., Demographic models., Population growth., Frequency-dependent selection., Timescale separation., Replicator equation., Perron-Frobenius theory., Evolutionary game theory., Age-class-structured population, Diffusion approximation., Fixation probability., Two timescales., Population-structure coefficient., Fixation fitness coefficient., Ancestry coefficient., Genealogical process., Public goods game.