Spatiotemporal Patterns Induced by Cross-Diffusion in a Three-Species Food Chain Model

Abstract

This paper focuses on a three-species Lotka–Volterra food chain model with cross-diffusion under homogeneous Neumann boundary conditions. The known results indicate that no spatiotemporal patterns happen in the corresponding reaction–diffusion system. When some cross-diffusion terms are introduced in the system, the existence of nonconstant positive steady-states as well as the Hopf bifurcation is studied. Our result shows that cross-diffusion plays a crucial role in the formation of spatiotemporal patterns, that is, it can create not only stationary patterns but also spatially inhomogeneous periodic oscillatory patterns, which is a strong contrast to the case without cross-diffusion.