Steady-state bifurcation and Hopf bifurcation in a cross-diffusion prey-predator system with Ivlev functional response

In this paper, we investigate bifurcations of stationary solutions of a cross-diffusion prey-predator system with Ivlev functional response and Neumann boundary conditions. First, we analyze the local stability and existence of a Hopf bifurcation at a coexistence stationary solution. We show that the bifurcating periodic solutions are asymptotically orbitally stable and the bifurcation direction is supercritical when the ratio of the conversion of prey captured by predator to the death rate of predator is in the interval (1,2]. Next, we derive sufficient conditions for the existence of a steady-state bifurcation from a simple eigenvalue. We establish the existence of two intersecting C-1 curves of steady-state solutions near the coexistence stationary solution. To illustrate our theoretical results, we give some numerical examples. From numerical simulations we observe that the coexistence stationary solution loses its stability via the Hopf bifurcation and a periodic solution emerges after a short time. Also by taking the bifurcation parameter value in the stable region, the effect of the initial condition disappears over time and the solution returns to the coexistence stationary solution.