Dynamics analysis of a diffusive prey-taxis system with memory and maturation delays

In this paper, a diffusive predator-prey system considering prey-taxis term with memory and maturation delays under Neumann boundary conditions is investigated. Firstly, the existence and stability of equilibria, especially the existence, uniqueness and stability of the positive equilibrium, are studied. Secondly, we prove that: (i) there is no spatially homogeneous steady state bifurcation as the eigenvalue of the negative Laplace operator is zero; (ii) as this system is only with memory delay tau 1, the the spatially nonhomogeneous Hopf bifurcation appears; (iii) when the model is only with maturation delay tau 2, the system has spatially homogeneous and nonhomogeneous periodic solutions; (iv) for the case of two delays, the system has rich dynamics, for example, stability switches, whose curves have four forms. Finally, some numerical simulations are produced to verify and support the theoretical results.