Global Solution and Spatial Patterns for a Ratio-Dependent Predator-Prey Model with Predator-Taxis

This paper analyzes the dynamic behavior of a ratio-dependent predator-prey model with predator-taxis, which the prey can move in the opposite direction of predator gradient. The first purpose is to prove rigorously the global existence and boundedness of the classical solution for the model based on the heat operator semigroup theory and some priori estimates. The another purpose is to analyze the stability of positive equilibrium, which the results will be extended to the case that the derivative of prey's functional response with prey is positive, and it will be found that large predator-taxis can stabilize equilibrium even diffusion-driven instability has occurred. Finally, the numerical simulations present that the pattern formation may arise and predator-taxis is the driving factor for the evolution of spatial inhomogeneity into homogeneity.