Dynamics of a density-dependent stage-structured predator–prey system with Beddington–DeAngelis functional response

In this paper, we study the dynamics of a stage-structured predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. First, we derive a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of hyperbolic curves. Then, we provide a sufficient condition to assure the local asymptotic stability of this positive equilibrium by constructing a Lyapunov function. Afterward, by iteratively making use of the comparison theorem, we propose a sufficient condition to assure its global attractiveness. Finally, by investigating types of the ωω-limit set instead of making use of the persistence theory, we prove that the predator coexists with the prey permanently if and only if positive equilibria exist.