Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis

Abstract

In this paper, we deal with the Gause–Kolmogorov-type predator–prey system with indirect prey-taxis, which means that directional movement of predators is stimulated by some chemicals emitted by preys. The existence of the positive equilibrium, the effect of the indirect prey-taxis on the stability and the related bifurcations are investigated. The critical values for the occurrence of the Hopf bifurcation, Turing bifurcation, Turing–Hopf bifurcation and double-Hopf bifurcation are explicitly determined. An algorithm for calculating the normal form of the double-Hopf bifurcation for the non-resonance and weak resonance is derived. Moreover, we apply the theoretical results to the system with Holling-II type functional response, the stable region and the bifurcation curves are completely determined in the plane of the indirect prey-taxis and self saturation coefficient. The dynamical classification near the double-Hopf bifurcation point is explicitly determined. In the neighborhood of the double-Hopf bifurcation, there are stable spatially homogeneous/inhomogeneous periodic solutions, stable spatially inhomogeneous quadi-periodic solutions and the pattern transitions from one spatial–temporal patterns to another one with the changes of the indirect taxis and semi saturation coefficients. The results show that spatially inhomogeneous Hopf bifurcations are induced by an indirect prey-taxis parameter $$\chi >0$$ , which is impossible for the reaction–diffusion predator–prey model with a direct prey-taxis.