Stability of bifurcating solution of a predator–prey model

Abstract

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 ξ as the bifurcation parameter, we confirm the model possesses the steady state bifurcation at ξ=ξkS for k∈N0/{0}. Also, we set ξkS(ɛ)=ξkS+ɛξ1+ɛ2ξ2+⋅⋅⋅ for small ɛ>0. We show that ξ1=0 and ξ2 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 ξ2<0 and it is unstable if ξ2>0.