Steady-State Bifurcation in Previte–Hoffman Model

Abstract

Prey-taxis, which describe the directed movement of the predator species, is introduced into the Previte–Hoffman model. Steady-state bifurcation is investigated in such model with the no-flux boundary conditions and the prey-taxis. Firstly, we present the stability analysis of the unique positive equilibrium, the existence of the Hopf bifurcation, and the steady-state bifurcation, respectively. Thereafter, to determine the existence and the stability of the nonconstant steady-state, which bifurcates from the steady-state bifurcation, the Crandall–Rabinowitz local bifurcation theory is employed to complete the tasks. As a result, the stability and instability of the nonconstant steady-state could be characterized. The results show that only the repulsive prey-taxis can induce the steady-state bifurcation of the Previte–Hoffman model. The bifurcations will lead to the occurrence of spatiotemporal patterns, which are demonstrated through numerical simulations.