Steady states and spatiotemporal evolution of a diffusive predator–prey model

Abstract

The existence of steady states, bifurcations and the spatiotemporal patterns are presented for the diffusive predator–prey model. First, the boundedness and positivity of solutions are justified, respectively. By employing the priori estimates, Poincare inequalities and Leray–Schauder degree, nonexistence and existence of nonconstant steady states are established, respectively. To further explore the pattern dynamics, the Hopf bifurcation and Turing instability are analyzed, the weakly nonlinear analysis is employed to establish the amplitude equations. It is found that the various complex pattern solutions can be identified from amplitude equations. The numerical results are in agreement with the theoretical analysis. We also find that the predator–prey model with Beddington–Deangelis (BD)-type functional response can admit various spatiotemporal patterns, such as labyrinthine-like patterns, the wave patterns near the Hopf–Turing bifurcation threshold, and so on. Such complex spatiotemporal patterns may be useful to help us understand the interaction among species.