Inhomogeneous Hopf bifurcation and stability in a two-competing-prey and one-predator model with cross-diffusion

Spatial and temporal pattern formations of interacting species in ecological and chemical systems have become a central object of research in recent decades. We consider a two-competing-prey and one-predator Lotka–Volterra reaction–diffusion model. We first establish the sufficient conditions for the global attractivity of positive constant equilibrium solution. Then by taking cross-diffusion coefficients as the bifurcation parameter, we show that our model undergoes an inhomogeneous Hopf bifurcation around the positive constant equilibrium solution when the bifurcation parameter is varied through critical values. Furthermore, we prove that these inhomogeneous bifurcating periodic solutions are stable when diffusion coefficients lie in suitable ranges. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms rather than diffusion terms. We show that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross-diffusion factors. Our results demonstrate that cross-diffusion plays a crucial role in the formation of spatially inhomogeneous periodic oscillatory patterns.