Complex spatiotemporal dynamics in a ratio-dependent predator–prey model with nonlinear taxis mechanism

In this paper, we investigate the complex dynamics in a ratio-dependent predator–prey model with nonlinear taxis mechanism. First, the boundedness of solutions and the global stability of the coexistence equilibrium point are analyzed. Then, by treating the prey–taxis sensitivity coefficient as bifurcation parameter, we obtain the threshold of Turing bifurcation. Moreover, the supercritical and subcritical Turing bifurcation can be classified at the bifurcation onset by deducing the amplitude equation. Numerical simulations are carried out to verify the validity of theoretical results. It indicates that the taxis mechanism induces the loss of the spatially homogeneous steady state. We further show the transition of spatial pattern from non-homogeneous steady state to wave oscillation as the taxis sensitivity coefficient is far from the critical value. And if the nonlinear taxis mechanism tends to be saturated, it will slow down the appearance of non-homogeneous patterns and limit the maximum population density of predator. Finally, stripe patterns near the threshold are performed in the 2D spatial domain.