Study of a Leslie–Gower-type tritrophic population model

A three-dimensional continuous time dynamical system is considered. It is a model for a tritrophic food chain, based on a modified version of the Leslie–Gower scheme. We establish and prove theorems on boundedness of the system, existence of an attracting set, existence and local or global stability of equilibria which represent the extinction of the top or intermediate predator. Using intensive numerical qualitative analysis we show that the model could exhibit chaotic dynamics for realistic parameter and state values. Transition to chaotic behavior is established via period doubling bifurcation, and some sequences of distinctive period-halving are found.