  The biological theory of adaptive dynamics proposes a description of the
long-term evolution of a structured asexual population. It is based on the
assumptions of large population, rare mutations and small mutation steps, that
lead to a deterministic ODE describing the evolution of the dominant type,
called the ``canonical equation of adaptive dynamics.'' Here, in order to
include the effect of stochasticity (genetic drift), we consider self-regulated
randomly fluctuating populations subject to mutation, so that the number of
coexisting types may fluctuate. We apply a limit of rare mutations to these
populations, 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. Then we apply a
limit of small mutation steps (weak selection) to this jump process, that leads
to a diffusion process that we call the ``canonical diffusion of adaptive
dynamics,'' in which genetic drift is combined with directional selection
driven by the gradient of the fixation probability, also interpreted as an
invasion fitness. Finally, we study in detail the particular case of multitype
logistic branching populations and seek explicit formulae for the invasion
fitness of a mutant deviating slightly from the resident type. In particular,
second-order terms of the fixation probability are products of functions of the
initial mutant frequency, times functions of the initial total population size,
called the invasibility coefficients of the resident by increased fertility,
defence, aggressiveness, isolation or survival.
