Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis

This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619-657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.