Steady-State Bifurcation in Previte-Hoffman Model

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.

Automated summary

This study introduces prey-taxis, which describes the directed movement of predator species, into the Previte-Hoffman model and investigates steady-state bifurcation with no-flux boundary conditions and prey-taxis. The stability analysis of the positive equilibrium, the existence of Hopf bifurcation, and steady-state bifurcation are presented. The Crandall-Rabinowitz local bifurcation theory is employed to determine the existence and stability of nonconstant steady-state bifurcation. Results show that only repulsive prey-taxis can induce steady-state bifurcation in the Previte-Hoffman model, leading to the occurrence of spatiotemporal patterns demonstrated through numerical simulations.