The spatial aspect is an important issue in ecology. We use a stochastic
model that takes into account the spatial motions of individuals and we
study it in the context of adaptive dynamics. The theory of adaptive
dynamics is used to understand the long term consequences of mutations
under two main assumptions: large population and rare mutations.
Firstly, in a large population limit, our stochastic model converges to
a deterministic model which is a system of non-linear non-local partial
differential equations. This system models the dynamics of a population
of two traits. We study it to deduce the conditions that characterize
the extinction and the coexistence of the two traits in a long time. We
also deduce a formula of the invasion fitness of a mutant individual.
Secondly, using the deterministic studies and considering large
population and rare mutations asymptotics, we show the convergence of
the stochastic model to a jump process that jumps in a space of
monomorphic stable spatial distributions.