Dynamic Analysis of an Indirect Prey-Taxis Model With Singular Sensitivity

Abstract

In this paper, we consider an indirect prey-taxis model with singular sensitivity. One of the main obstacles in the research is the possible singularity of the system. We first study the global existence of the unique classical solution of the system in a bounded convex region with smooth boundary and Neumann boundary conditions. We further investigate the global boundedness of the solutions. Then, by constructing some proper Lyapunov functionals, we show the global asymptotic stability of the steady states and give the rate of convergence of the solution. In addition, we discuss the local stability of the predator-free steady state and positive constant steady state by using the corresponding characteristic equations. And adopting the indirect prey-taxis coefficient as the bifurcation parameter, we analyze the occurrence of Hopf bifurcation and steady-state bifurcation. Our results reveal that indirect prey-taxis can destroy the stability, with higher chemotactic intensities making the system more likely to exhibit time-periodic patterns, while lower chemotactic intensities make the system more likely to display steady-state patterns. Among other things, we conduct a comparative analysis with the nonsingular indirect prey-taxis system. Finally, several numerical simulations are presented to illustrate the findings.