A Three-Species Food Chain System with Two Types of Functional Responses

In recent decades, many researchers have investigated the ecological models with three and more species to understand complex dynamical behaviors of ecological systems in nature. However,when they studied the models with three species, they have just considered the functional responses between prey and mid-predator and between mid-predator and top predator as the same type. However, in the paper, in order to describe more realistic ecological world, a three-species food chain system with two types of functional response, Holling type and Beddington-DeAngelis type, is considered. It is shown that this system is dissipative. Also, the local and global stabilityof equilibrium points of the system is established. In addition, conditions for the persistence ofthe system are found according to the existence of limit cycles. Some numerical examples aregiven to substantiate our theoretical results. Moreover, we provide numerical evidence of the existenceof chaotic phenomena by illustrating bifurcation diagrams of system and by calculatingthe largest Lyapunov exponent