Synchronization of two three-species food chain system with Beddington-DeAngelis functional response using active controllers based on the Lyapunov function

In this paper master-slave scheme is used to investigate the problem of synchronization of two three species models with Beddington-DeAngelis type of functional response. Using Lyapunov stability theory the analytical expression for nonlinear active controllers have been deduced for the synchronization of (I) two identical three-species web chain but with different growth rate and (II) two identical chaotic three-species food chain with different initial conditions. The bifurcation diagram of the food chain system has been used to find the intrinsic growth rate parameter for which the dynamics of three-species food chain exhibit either regular or chaotic behavior in phase space. Based on the chosen intrinsic growth parameters that characterizes the complex dynamics of the master and slave system, the active controller functions, have been obtained, which causes global asymptotic stability of synchronization errors, for the case I and II respectively. The efficacy of the controller functions, in the synchronization process, is demonstrated using the time series plot of species population and phase plots before and after the activation.