Double-Hopf bifurcation and Pattern Formation of a Gause-Kolmogorov-Type system with indirect prey-taxis and direct predator-taxis

This paper focuses on a general Gause-Kolmogorov-Type predator–prey model with direct predator-taxis and indirect prey-taxis. The global existence and boundedness of solutions to the system are proved. Our approach is applicable to more general situation. Some measures of success for the constant coexistence steady state are rigorously calculated, such as decreasing the indirect prey-taxis sensitivity or the release rate of stimulus, and increasing the predator-taxis sensitivity or the decay rate of stimulus. By choosing the indirect prey-taxis coefficient as the bifurcation parameter, the existence of Hopf and double-Hopf bifurcations is established. We find that both indirect prey-taxis and direct predator-taxis can promote the complexity of the spatiotemporal pattern, e.g. spatially nonhomogeneous periodic patterns with different spatial frequency, spatially nonhomogeneous quasi-periodic patterns. Finally, we apply our theoretical analyses to a Rosenzweig–MacArthur model with indirect prey-taxis and direct predator-taxis, spatially homogeneous periodic patterns.