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

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.

Automated summary

This paper discusses the Gause-Kolmogorov-type predator-prey system with indirect prey-taxis, investigating the existence of positive equilibrium, the effect of indirect prey-taxis on stability, and related bifurcations including Hopf, Turing, Turing-Hopf and double-Hopf bifurcations. It is found that spatially inhomogeneous Hopf bifurcations are induced by an indirect prey-taxis parameter, contrasting with models with direct prey-taxis.