Chaotic dynamics in a discrete-time predator-prey food chain

In this paper, we consider a classical discrete-time food chain model describing predators-prey interaction. The Holling type I functional response is used as the uptake for both predators. The existence and local stability of fixed points of the discrete dynamical system are analyzed algebraically. Using growth rate of prey as the bifurcation parameter, it is shown that the system undergoes a flip and Hopf bifurcations around planer or interior fixed point. It has been found that the dynamical behavior of the model is very sensitive to the parameter values and the initial conditions. Numerical simulations not only illustrate the key points of analytical findings but also exhibit complex dynamical behaviors of the model, such as the phase portraits, cascade of period-doubling bifurcation and determine the effects of operating parameters of the model on its dynamics. The Lyapunov exponents are numerically computed to characterize the asymptotic stability of the system dynamic response and estimate the amount of chaos in the system.