Patterns for Competing Populations with Species Specific Non local 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, delta, which gives a spatial scale of the nonlocality, with delta = 0 corresponding to the local case, and alpha, a parameter associated with the extent of the asymmetry of the kernel, where alpha = 0 corresponds to a symmetric kernel, i.e., to a kernel that is an even function of its argument. We consider the parameter regime where for delta = 0 the system admits a stable coexistence equilibrium, which is destabilized for sufficiently large delta. Deeply in the instability region, the ensuing patterns involve arrays of islands, regions of nonzero population, separated by deadzones where the populations are essentially extinct. These structures are stationary if alpha = 0 and propagate if alpha not equal 0. Unlike previous work, we study the patterns when the kernel parameters are different for the two species, focusing first on different. delta with alpha = 0 and then on different alpha with the same delta. In the first case we numerically find stationary patterns consisting of (i) plateau regions for the more local species (v), where the plateau value is essentially the v-component of the unstable equilibrium point, (ii) oscillatory patterns, where oscillations appear along the plateau and along the u-island and (iii) split patterns, where instead of an island, there is an island cluster of two or more neighbors separated by deadzones or near deadzones. In the second case we consider weakly coupled systems and show that there are two kinds of patterns, (i) bound waves, when each species propagates with the same speed and (ii) unbound patterns where each species propagates with its own speed, interacting with the other species when they intersect, i.e., when the two species occupy the same or closely spaced regions. We show that whether the patterns are bound or unbound depends on the stability of the decoupled system, i.e., the system when the coupling parameters are zero. When one of the species for the decoupled system is stable, then the ensuing patterns for weak coupling is bound, while if both species are unstable for the uncoupled systems, the resulting patterns are unbound for weak coupling.