Bifurcation analysis and Turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality

In this paper, we consider a predator-prey model with herd behavior and hyperbolic mortality subject to the homogeneous Neumann boundary condition. Firstly, we prove the existence and uniqueness of positive equilibrium for this model by analytical skills. Then we analyze the stability of the positive equilibrium, Turing instability, and the existence of Hopf, steady state bifurcations. Finally, by calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Meanwhile, for the steady state bifurcation, the possibility of pitchfork bifurcation can be concluded by the normal form, which does also determine the stability of spatially inhomogeneous steady states. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out and expand our theoretical results.