  To describe population dynamics, it is crucial to take into account jointly
evolution mechanisms and spatial motion. However, the models which include
these both aspects, are not still well-understood. Can we extend the existing
results on type structured populations, to models of populations structured by
type and space, considering diffusion and nonlocal competition between
individuals? We study a nonlocal competitive Lotka-Volterra type system,
describing a spatially structured population which can be either monomorphic or
dimorphic. Considering spatial diffusion, intrinsic death and birth rates,
together with death rates due to intraspecific and interspecific competition
between the individuals, leading to some integral terms, we analyze the long
time behavior of the solutions. We first prove existence of steady states and
next determine the long time limits, depending on the competition rates and the
principal eigenvalues of some operators, corresponding somehow to the strength
of traits. Numerical computations illustrate that the introduction of a new
mutant population can lead to the long time evolution of the spatial niche.
