Competition in a patchy environment with cross-diffusion in a three-dimensional Lotka–Volterra system

In this paper, we formulate a three-dimensional competitive Lotka–Volterra system in two patches in which the per capita migration rate of each species is influenced not only by its own density but also by another’s density. That is to say, there is cross-diffusion present in the Lotka–Volterra system. We first show that there is a critical value of the bifurcation parameter at which the system undergoes a Turing bifurcation under the effect of cross-diffusion, in theory. At the same time, we also give the results of numerical studies. Our work illustrates that the cross-migration response is an important factor that should not be ignored for this kind of system.