Dynamics of Stage-Structured Predator–Prey Model with Beddington–DeAngelis Functional Response and Harvesting

In this paper, we investigate the stability of equilibrium in the stage-structured and density-dependent predator–prey system with Beddington–DeAngelis functional response. First, by checking the sign of the real part for eigenvalue, local stability of origin equilibrium and boundary equilibrium are studied. Second, we explore the local stability of the positive equilibrium for $\tau =0$ and $\tau \ne 0$ (time delay $\tau$ is the time taken from immaturity to maturity predator), which shows that local stability of the positive equilibrium is dependent on parameter $\tau$. Third, we qualitatively analyze global asymptotical stability of the positive equilibrium. Based on stability theory of periodic solutions, global asymptotical stability of the positive equilibrium is obtained when $\tau =0$; by constructing Lyapunov functions, we conclude that the positive equilibrium is also globally asymptotically stable when $\tau \ne 0$. Finally, examples with numerical simulations are given to illustrate the obtained results.