Bifurcation branch of stationary solutions in a general predator–prey system with prey-taxis

In this paper, a general reaction–diffusive predator–prey system with prey-taxis subject to the homogeneous Neumann boundary condition is considered. Firstly, we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique. And then, choosing the prey-tactic sensitivity coefficient as bifurcation parameter, we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey-tactic sensitivity is repulsive. Moreover, we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions. Our results show that prey-taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns.