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. [ABSTRACT FROM AUTHOR]