Instability of non-constant equilibrium solutions of a system of competition-diffusion equations

The system of interaction-diffusion equations describing competition between two species is investigated. By using a version of the Perron-Frobenius theorem of positive matrices generalized to function spaces, it is proved that any non-constant equilibrium solution of the system is unstable both under Neumann boundary conditions (for the rectangular parallelepiped domain) and under periodic conditions. It is conjectured that this result extends to convex domains, and that the simple interaction-diffusion model cannot explain spatially segregated distributions of two competing species in such domains.     

Diffusive instabilities in a one-dimensional temperature-dependent model system for a mite predator-prey interaction on fruit trees: dispersal motility and aggregative preytaxis effects

A weakly nonlinear analysis relevant to the formation of one-dimensional spatial patterns generated by diffusive instabilities is performed on a particular interaction-diffusion model for a temperature-dependent predator-prey mite system on fruit trees. The bifurcation from a uniform steady state is of a subcritical nature in a low temperature-low population interval while in a high temperature-high population one there exist temperatures for which it can be supercritical resulting in a family of parallel stripes. The occurrence of such population clumping, caused both by the predator's having a sufficient dispersal advantage and by its strongly stabilizing tendency toward preytactic aggregation lying in some critical range, may help explain the inhomogeneous ecological patterns exhibited by phytophagous arthropods found on uniformly distributed vegetation or on plants grown in monocultures.