Discrete logistic branching populations and the canonical diffusion of adaptive dynamics
Résumé
The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE, called 'canonical equation of adaptive dynamics'. However, in order to include the effect of genetic drift in this description, we have to apply a limit of weak selection to a finite stochastically fluctuating discrete population subject to competition in the logistic branching fashion. We start with the study of the particular case of two competing subpopulations resident and mutant) and seek explicit first-order formulae for the probability of fixation of the mutant, also interpreted as the mutant's fitness, in the vicinity of neutrality. In particular, the first-order term is a linear combination of products of functions of the initial mutant frequency times functions of the initial total population size, called invasibility coefficients (fertility, defence, aggressiveness, isolation, survival). Then we apply a limit of rare mutations to a population subject to mutation, birth and competition where the number of coexisting types may fluctuate, while keeping the population size finite. This leads to a jump process, the so-called 'trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. Finally, we apply a limit of weak selection (small mutation steps) to this jump process, that leads to a diffusion process of evolution, called 'canonical diffusion of adaptive dynamics', in which genetic drift is combined with directional selection driven by the fitness gradient.