Patterns for Competing Populations with Species Specific Nonlocal Coupling

We study a reaction-diffusion population model for two competing species, u and v, which also includes nonlocal integral terms that represent competition for resources. The integral terms have the form of a convolution of a given kernel function and the solution. They manifest the fact that consumption of resources by the species at a spatial location x depends not just on the populations at point x but rather on weighted averages of the populations in an interval about x. The kernel functions that we employ are characterized by two parameters, δ, which gives a spatial scale of the nonlocality, with δ = 0 corresponding to the local case, and α, a parameter associated with the extent of the asymmetry of the kernel, where α = 0 corresponds to a symmetric kernel, i.e., to a kernel that is an even function of its argument.