Parabolic systems with nonlinear competitive interactions

A parabolic system with nonlinear local or nonlocal interactions is considered, with specific applications to the description of competition/combat situations. Global existence and comparison results are proven under suitable assumptions on the interaction terms. Comparisons with solutions of the 'equivalent' ordinary differential (space-independent) model that has been previously used in modelling combat are of special interest. The stationary state of such a system- corresponding to a stalemate- is also considered. Under certain conditions on the interaction, domain size, and boundary conditions, a uniqueness result is obtained for the stationary state. A simple example shows that these conditions are sharp and uniqueness may be lost when they are not fulfilled.