Steady states of a diffusive predator-prey model with prey-taxis and fear effect

Abstract

AbstractIn this paper, a diffusive predator-prey system with a prey-taxis response subject to Neumann boundary conditions is considered. The stability, the Hopf bifurcation, the existence of nonconstant steady states, and the stability of the bifurcation solutions of the system are analyzed. It is proved that a high level of prey-taxis can stabilize the system, the stability of the positive equilibrium is changed when χ crosses $\chi _{0}$ χ 0 , and the Hopf bifurcation occurs for the small s. The system admits nonconstant positive solutions around $(\bar{u}, \bar{v}, \chi _{i} )$ ( u ¯ , v ¯ , χ i ) , the stability of bifurcating solutions are controlled by $\int _{\Omega} \Phi _{i}^{3} \,\mathrm{d}x$ ∫ Ω Φ i 3 d x and $\int _{\Omega} \Phi _{i}^{4} \,\mathrm{d}x$ ∫ Ω Φ i 4 d x . Finally, numerical simulation results are carried out to verify the theoretical findings.