Bogdanov–Takens bifurcation for a diffusive predator–prey system with nonlocal effect and prey refuge.

In this paper, a diffusive predator–prey system with nonlocal effect and prey refuge subjects to the homogeneous Neumann boundary condition is studied. Firstly, the conditions for the occurrence of Turing, Hopf, Turing–Turing and Bogdanov–Takens bifurcations are established. Then, in order to meticulously describe the spatiotemporal dynamics resulting from the Bogdanov–Takens bifurcation, we derive an algorithm for calculating the third-order truncated normal form of Bogdanov–Takens bifurcation of this system by using the center manifold theory and normal form method, which can be applied to general diffusive predator–prey system with nonlocal effect. With the aid of the derived third-order truncated normal form, the complex spatiotemporal dynamics are theoretically predicted. At last, we carry out some numerical simulations to support the theory analysis, and the stable positive constant steady state and a pair of stable spatially inhomogeneous steady states are found. [ABSTRACT FROM AUTHOR]