In the course of Darwinian evolution of a population, punctualism is an
important phenomenon whereby long periods of genetic stasis alternate with
short periods of rapid evolutionary change. This paper provides a mathematical
interpretation of punctualism as a sequence of change of basin of attraction
for a diffusion model of the theory of adaptive dynamics. Such results rely on
large deviation estimates for the diffusion process. The main difficulty lies
in the fact that this diffusion process has degenerate and non-Lipschitz
diffusion part at isolated points of the space and non-continuous drift part at
the same points. Nevertheless, we are able to prove strong existence and the
strong Markov property for these diffusions, and to give conditions under which
pathwise uniqueness holds. Next, we prove a large deviation principle involving
a rate function which has not the standard form of diffusions with small noise,
due to the specific singularities of the model. Finally, this result is used to
obtain asymptotic estimates for the time needed to exit an attracting domain,
and to identify the points where this exit is more likely to occur
