Competition in patchy space with cross-diffusion and toxic substances

The main goal of this paper is to continue our investigations of the important system (see [S. Aly, M. Farkas, Competition in patchy environment with cross diffusion, Nonlinear Analysis: Real World Applications 5 (2004) 589–595]), by considering a Lotka–Volterra competitive system affected by toxic substances in two patches in which the per capita migration rate of each species is influenced not only by its own but also by the other one’s density, i.e. there is cross-diffusion present and it is assumed that the individuals of a particular species will initiate toxin production at a rate proportional not only to its own but also to the other one’s density. In the absence of diffusion, we study the conditions of the existence and stability properties of the equilibrium point with toxic substances. For the full general model (with both toxic substances and diffusion) we show that at a critical value of the bifurcation parameter of diffusion the system undergoes a Turing bifurcation and numerical studies show that if the bifurcation parameter of diffusion is increased through a critical value the spatially homogeneous equilibrium loses its stability and two new stable equilibria emerge, i.e., the cross-migration response is an important factor that should not be ignored when a pattern emerges.