  The interplay between space and evolution is an important issue in population
dynamics, that is in particular crucial in the emergence of polymorphism and
spatial patterns. Recently, biological studies suggest that invasion and
evolution are closely related. Here we model the interplay between space and
evolution starting with an individual-based approach and show the important
role of parameter scalings on clustering and invasion. We consider a stochastic
discrete model with birth, death, competition, mutation and spatial diffusion,
where all the parameters may depend both on the position and on the trait of
individuals. The spatial motion is driven by a reflected diffusion in a bounded
domain. The interaction is modelled as a trait competition between individuals
within a given spatial interaction range. First, we give an algorithmic
construction of the process. Next, we obtain large population approximations,
as weak solutions of nonlinear reaction-diffusion equations with Neumann's
boundary conditions. As the spatial interaction range is fixed, the
nonlinearity is nonlocal. Then, we make the interaction range decrease to zero
and prove the convergence to spatially localized nonlinear reaction-diffusion
equations, with Neumann's boundary conditions. Finally, simulations based on
the microscopic individual-based model are given, illustrating the strong
effects of the spatial interaction range on the emergence of spatial and
phenotypic diversity (clustering and polymorphism) and on the interplay between
invasion and evolution. The simulations focus on the qualitative differences
between local and nonlocal interactions.
