Turing bifurcation in activator–inhibitor (depletion) models with cross-diffusion and nonlocal terms

In this paper, we consider the instability of a constant equilibrium solution in a general activator–inhibitor (depletion) model with passive diffusion, cross-diffusion, and nonlocal terms. It is shown that nonlocal terms produce linear stability or instability, and the system may generate spatial patterns under the effect of passive diffusion and cross-diffusion. Moreover, we analyze the existence of bifurcating solutions to the general model using the bifurcation theory. At last, the theoretical results are applied to the spatial water–biomass system combined with cross-diffusion and nonlocal grazing and Holling–Tanner predator–prey model with nonlocal prey competition.