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.