Global bifurcation for a Holling-Tanner predator-prey model with prey-taxis

The local existence and stability of non-constant positive steady states for the Holling-Tanner predator-prey model with prey-taxis was discussed by Wang et al. (2017) through local bifurcation analysis. It is our purpose in this paper to make a detailed description for the structure of the set of the non-constant steady states. We will prove a global bifurcation theorem by treating prey-taxis as a bifurcation parameter, which gives the existence of non-constant steady states under a rather natural condition. A priori estimates for steady state solutions will play a key role in the proof. In addition, the stability results for the homogeneous equilibrium will extend to the case that the derivative of prey's functional response with prey is positive, and it is found that attractive prey-taxis can stabilize homogeneous equilibrium even diffusion-driven instability has occurred. (C) 2018 Elsevier Ltd. All rights reserved.