  We consider an individual-based spatially structured population for Darwinian
evolution in an asexual population. The individuals move randomly on a bounded
continuous space according to a reflected brownian motion. The dynamics
involves also a birth rate, a density-dependent logistic death rate and a
probability of mutation at each birth event. We study the convergence of the
microscopic process when the population size grows to $+\infty$ and the
mutation probability decreases to $0$. We prove a convergence towards a jump
process that jumps in the infinite dimensional space of the stable spatial
distributions. The proof requires specific studies of the microscopic model.
First, we examine the large deviation principle around the deterministic large
population limit of the microscopic process. Then, we find a lower bound on the
exit time of a neighborhood of a stationary spatial distribution. Finally, we
study the extinction time of the branching diffusion processes that approximate
small size populations.
