Positive Steady-State Solutions for a Class of Prey-Predator Systems with Indirect Prey-Taxis

This article is concerned with a class of prey-predator systems with indirect prey-taxis, which means the stimulus released by the prey causes the directional movement of the predator. Under homogeneous Neumann boundary conditions, the dynamics of such systems have been studied extensively. However, the corresponding Dirichlet problem remains largely unexplored. The purpose of this paper is to discuss the existence and nonexistence of positive steady-state solutions and make a detailed description for the global bifurcation structure of the set of positive steady-state solutions. Compared with the original systems without indirect prey-taxis, our mathematical analysis shows that the introduction of indirect prey-taxis not only makes mathematical analysis more difficult, but also affects the bifurcation structure of the set of positive steady-state solutions. We hope this work will inspire further research on prey-predator systems with indirect prey-taxis.