Discrete logistic branching populations and the canonical diffusion of adaptive dynamics
- Champagnat, Nicolas
- Lambert, Amaury
2010 Mathematics Subject Classification
- 60J80 60J25 60J70 60J75 60J85 60K35 92D10 92D15 92D25 92D40
- Logistic branching process, population dynamics, density-dependence, competition, fixation probability, genetic drift, weak selection, adaptive dynamics, invasion fitness, timescale separation, trait substitution sequence, diffusion approximation, harmonic equations, convergence of measure-valued processes
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.