UBC Mathematics Department
Although, for single equations, diffusion can be viewed as a smoothing and trivializing process, the situation becomes drastically different when we come to systems of diffusion equations. For example, in a system of equations modeling two interactive substances, different diffusion rates could lead to nonhomogeneous distribution of such reactants. Using an activator-inhibitor system, with slowly diffusing activator and rapidly diffusing inhibitor, we will briefly introduce the current mathematical research on those highly concentrated solutions (i.e. solutions whose graphs display narrow peaks or spikes, also known as point-condensation solutions or, spike-layers). In this example, not only is gap between the diffusion rates important, the reaction terms are also essential in producing the patterns. In contrast to this example, we are going to discuss the weak competition case in the classical competition-diffusion Lotka-Voterra system which does not have any nontrivial solutions no matter what the diffusion rates are. However, in modeling two competing species in population dynamics, since individuals are not moving around just randomly, one must take into account the population pressure created by each of the competitors. Thus we are lead to "cross-diffusion" systems. It is interesting to note that spike-layers also appear in cross-diffusion systems. In addition to discussing spike-layers, we shall focus on the effect of cross-diffusion versus that of diffusion in those systems.