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Persistence and global stability in a delayed Leslie-Gower type three species food chain. (English) Zbl 1127.92046
Summary: Our investigation concerns a 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 [P. H. Leslie and J. C. Gower, Biometrika 47, 219–234 (1960; Zbl 0103.12502)]. This model can be considered as a first step towards a tritrophic model (of Leslie-Gower and Holling-Tanner type) with inverse trophic relations 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.
MSC:
92D40Ecology
34D23Global stability of ODE
34D20Stability of ODE