Adaptive dynamics in logistic branching populations
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. Under these assumptions, the evolution of a quantitative dominant trait in an isolated population is described by a deterministic differential equation called `canonical equation of adaptive dynamics'. In this work, in order to include the effect of genetic drift in this model, we consider instead finite, randomly fluctuating populations and weak selection. We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the current state of the population. Rescaling mutation steps (weak selection) then yields a diffusion on the trait space christened `canonical diffusion of adaptive dynamics', in which genetic drift (diffusive term) is combined with directional selection (deterministic term) driven by the fitness gradient. Finally, in order to compute the coefficients of this diffusion, we seek explicit first-order formulae for the probability of fixation of a nearly neutral mutant appearing in a resident population. The first-order term is a linear combination of products of functions of the initial mutant frequency times `invasibility coefficients' associated with fertility, defence, aggressiveness and isolation, which measure the robustness (stability with respect to selective strengths) of the resident type. Some numerical results on the canonical diffusion are also given.