Pattern formation in a diffusive predator–prey system with cross-diffusion effects

Abstract

We investigate the effects of cross-diffusion on pattern formation of a diffusive predator–prey system. The stability and bifurcation for the reaction diffusion system are discussed, and the conditions for the Turing instability are presented. We have found that the cross-diffusion coefficient $$d_{12}$$ , which reflects the tendency of preys leaving predators, suppresses the formation of Turing pattern, while the cross-diffusion coefficient $$d_{21}$$ , which represents the trend of predators to move away from preys, plays a facilitating role in pattern formation. The obtained conclusions are illustrated and verified via numerical simulations. Finally, we prove that under some conditions, the system admits at least a nonconstant positive steady state if $$d_{21}$$ is large enough, and this also explains why spatial patterns created by cross-diffusion can and do emerge.