Stability of bifurcating solution of a predator-prey model

To explore the role of the prey-taxis in an ecological model, we investigate a predator-prey model with prey -taxis in this paper. Firstly, the local stability of the positive equilibrium and the occurrence conditions of the steady state bifurcation are given. Thereafter, we investigate the existence and stability of the bifurcating solution around the threshold. Precisely, by treating the prey-taxis constant e as the bifurcation parameter, we confirm the model possesses the steady state bifurcation at e =ekS for k & ISIN; N0/{0}. Also, we set ekS () =ekS +e1 +2e2+& BULL; & BULL; & BULL; for small > 0. We show that e1 = 0and e2 determines the stability of the bifurcating solution. Finally, the stable bifurcating solution is observed by using numerical experiments. The findings of this paper are: (i) the repulsive prey-taxis will facilitate the occurrence of the steady state bifurcation. (ii) the bifurcating solution is stable if e2 < 0 and it is unstable if e2 > 0.

Automated summary

In this paper, we investigate the role of prey-taxis in an ecological model. The local stability of the positive equilibrium and the occurrence conditions of the steady state bifurcation are given. By treating the prey-taxis constant e as the bifurcation parameter, we confirm the model possesses the steady state bifurcation at e =ekS for k & ISIN; N0/{0}. Numerical experiments show the stable bifurcating solution.