A semilinear parabolic predator–prey system with time delay and habitat complexity effect

Abstract

Based on the research on the predator–prey model with Holling type response function, a delayed predator–prey system with diffusion term and habitat complexity effect is established, and the effects of time delay and diffusion on dynamical behavior of the system are studied. First, taking habitat complexity as the parameter, the dynamical properties of the system without time delay are studied. By eigenvalue analysis, the sufficient conditions for locally asymptotic stability of the positive equilibrium and globally asymptotic stability of the boundary equilibrium are given, the existence conditions of Hopf bifurcation induced by diffusion term are discussed. In an appropriate range, diffusion makes a family of spatially homogeneous and inhomogeneous periodic solutions bifurcate from the positive equilibrium. Second, taking production delay as the bifurcation parameter, the existence conditions of Hopf bifurcation are given, the method to determine the bifurcation direction and the stability of bifurcating periodic solutions is given by using the center manifold theory and normal form method. Finally, the biological interpretations of the results are given, and some numerical simulations are given to verify the theoretical analysis results.