Cross diffusion induced spatially inhomogeneous Hopf bifurcation for a three species Lotka–Volterra food web model with cycle

Abstract

In this article, we consider a three species Lotka–Volterra food web reaction–diffusion model with cycle, in which the intensity of the rate of preying on species 1 by species 2, the rate of preying on species 2 by species 3 and the rate of preying on species 3 by species 1 are in general asymmetrical. By suitably choosing cross diffusion coefficients as the bifurcation parameter, a spatially inhomogeneous Hopf bifurcation at a positive constant steady state is proved to occur for a sequence of critical values of the bifurcation parameter. In addition, we demonstrate that these spatially inhomogeneous bifurcating periodic solutions are stable when diffusion coefficients are located in suitable ranges. Our results show that cross diffusion plays a key role in the formation of spatially inhomogeneous periodic oscillatory patterns.