Nonlinear Instability For A Predator-Prey Model Incorporating A Prey Refuge

Nonlinear evolution of spatial and temporal patterns is investigated for a predator-prey model incorporating a prey refuge. Our results can be interpreted as a rigorous mathematical characterization for early pattern formation. First, the stability and insta-bility of nonnegative equilibria and the existence of bifurcation are studied in the corre-sponding ODE model. Second, the stability and instability of nonnegative equilibria are investigated in the weakly coupled reaction-diffusion model. Finally, nonlinear instability of the positive constant equilibrium point for the cross-diffusion model in a d-dimensional box in Rd is studied, where d?3. It is proved that the unstable positive equilibrium point of the cross-diffusion model is nonlinearly unstable and its small perturbation can result in giving rise to spatial and temporal patterns.