Global bifurcation in a general Leslie–Gower type predator[formula omitted]prey system with indirect prey-taxis

Abstract

The influence of the indirect prey-taxis on the spatiotemporal dynamics of a class predator−prey system with predator functional response is studied. By analyzing the corresponding characteristic equation, the local stability of the positive equilibrium and the related bifurcations of the system are discussed. The critical values of the indirect prey-taxis coefficient leading to the Hopf bifurcation, steady state bifurcation, Turing-Turing bifurcation and double-Hopf bifurcation are derived. Taking the indirect prey-taxis coefficient as the bifurcation parameter, we obtain the global structure of nonconstant steady states bifurcating from the positive equilibrium by an abstract bifurcation theorem. Moreover, the stability condition of bifurcation solutions is carried out. Our results suggest that attractive indirect prey-taxis can destabilize the positive equilibrium and induced the emergence of spatially inhomogeneous periodic solution. And the higher the secretion level of chemoattractant by the prey, the more likely the system is to exhibit spatial patterns.