Dynamic complexity in a modified Leslie-Gower model with taxis mechanism and digestion delay

In this paper, we investigate the complex spatiotemporal dynamics of a modified Leslie-Gower model with taxis mechanism and digestion delay. First, the boundedness of solutions and the global stability of the positive equilibrium point without digestion delay are obtained. Then, the occurrence conditions for Hopf bifurcation and Turing-Hopf bifurcation under the combined effect of the predator-taxis and digestion delay are obtained. Theoretically, there is no Hopf bifurcation or Turing instability as the taxis mechanism and digestion delay are absent. Our results find that the predator-taxis effect governs the existence of the Turing bifurcation and the emergence of nonhomogeneous patterns, while the digestion delay determines the stability and the existence of periodic solutions. Finally, numerical simulations verify the validity of the theoretical analysis, and homogeneous pattern, nonhomogeneous steady, nonhomogeneous periodic, and mixed patterns are observed. Interestingly, it is further shown that a double Hopf bifurcation emerges, which is induced by the interaction between nonhomogeneous Hopf bifurcations with different modes.