  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. It was in adaptive dynamics that this kind of model was first
invented and studied, under the additional assumption of large population.
Assuming also small mutation steps, adaptive dynamics' theory provides a
deterministic ODE approximating the evolutionary dynamics of the dominant trait
of the population, called `canonical equation of adaptive dynamics'. In this
work, we want to include genetic drift in this models by keeping the population
finite. Rescaling mutation steps (weak selection) yields in this case a
diffusion on the trait space that we call `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. These formulae are expressed in terms of
`invasibility coefficients' associated with fertility, defense, aggressiveness
and isolation, which measure the robustness (stability w.r.t. selective
strengths) of the resident type. Some numerical results on the canonical
diffusion are also given.
