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-u\chi(v)\nabla v\right)+\lambda u-u^2 +\gamma u F(v)=0, & x \in \Omega,\[1mm] D \Delta v+\mu v-v^2-u F(v)=0, & x \in \Omega,\[1mm] u=v=0, & x \in \partial \Omega. \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.