Leslie-Gower predator-prey model with stage-structure, Beddington-DeAngelis functional response, and anti-predator behavior

We concern with a Leslie-Gower predator-prey model with stage-structure on predator, where the predator population is splitted into juvenile and adult predator. It is assumed that only adult predator which has ability to reproduce and to attack prey with the Beddington-DeAngelis functional response. Moreover, anti-predator behavior is also considered in this model, which is represented by the possibility of juvenile predator to be attacked by prey. To study the dynamical properties of the model, we determine the equilibria of the model and their stability behaviour. Our model has four equilibria: the extinction of both prey dan predator populations point which always exists; the prey extinction point which also always exist; the predator extinction point which conditionally exists, and the interior equilibrium point which exists under certain conditions. It is shown that both prey and predator cannot simultaneously extinct because the point of all populations extinction is always unstable. The three other equilibria are conditional asymptotically stable. Our dynamical analysis has been confirmed by some numerical simulations.