Stability of a diffusive predator–prey model with modified Leslie–Gower and Holling-type II schemes and time-delay in two dimensions

Our investigation deals with the stability and permanence of diffusive predator–prey model with modified Leslie–Gower and Holling-type II schemes and time-delay in two dimensions. Firstly, we prove that the solutions of this model are globally bounded and remain permanently in the positive quadrant. From this system, we obtain three trivial equilibrium points of which, under certain conditions, one is locally stable and the others unstable. We show that the unique point of positive internal equilibrium is locally stable. Then, by constructing an appropriate Lyapunov’s functional, we establish the main result which is the global and asymptotic stability of this model. Finally, a numerical simulation is run to illustrate all these different theoretical results.