Persistence and global stability in a delayed Leslie–Gower type three species food chain

Our investigation concerns the three-dimensional delayed continuous time dynamical system which models a predator–prey food chain. This model is based on the Holling-type II and a Leslie–Gower modified functional response. This model can be considered as a first step towards a tritrophic model (of Leslie–Gower and Holling–Tanner type) with inverse trophic relation and time delay. That is when a certain species that is usually eaten can consume immature predators. It is proved that the system is uniformly persistent under some appropriate conditions. By constructing a proper Lyapunov function, we obtain a sufficient condition for global stability of the positive equilibrium.