Global bifurcation of coexistence states for a prey-taxis system with Dirichlet conditions

Abstract

This paper is concerned with positive solutions of boundary value problems \begin{equation*} \left\{\begin{array}{ll} {\rm div}\left(d(v)\nabla u-uχ(v)\nabla v\right)+λu-u^2 +γu F(v)=0, & x \in Ω,\\[1mm] D Δv+μv-v^2-u F(v)=0, & x \in Ω,\\[1mm] u=v=0, & x \in \partial Ω. \end{array}\right. \end{equation*} This is the stationary problem associated with the predator-prey system with prey-taxis, and $u$ (resp. $v$) denotes the population density of predator (resp. prey). In particular, the presence of $\chi (v)$ represents the tendency of predators to move toward the increasing preys gradient direction. Regarding $\lambda$ as a bifurcation parameter, we make a detailed description for the global bifurcation structure of the set of positive solutions. So that ranges of parameters are found for which the system admits positive solutions.

21 pages, 10 figures