Stability, Steady-State Bifurcations, and Turing Patterns in a Predator-Prey Model with Herd Behavior and Prey-taxis

In this paper, we consider a predator-prey model with herd behavior and prey-taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey-tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey-taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots-strip pattern, strip pattern, stable nonconstant steady-state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.