Active Control Method for Phase and Chaos Synchronization within Two Coupled Food Chains Under Refugia and Allee Effect

This work investigates synchronization within a system of two coupled food chains. We assess the scenario if and how the chaotic dynamics of a food chain can be suppressed in the presence of other food chains and order can be restored within the system of coupled food chains. For this, we consider the extended version of Hasting Powell model (eHPM), in which two bio-controlling parameters, refugia, and Allee, are already present to suppress chaos. Through imperative mathematical properties and numerical simulations of the system, we first understand the impact of the interplay of refugia and Allee effects in defining the qualitative behaviour of the system. After analyzing the system's sensitive dependence on the initial condition, we use the master-slave scheme to attain synchronization within the system of two coupled food chains defined for three different frameworks. Following the active control method, Lyapunov stability theory is used to deduce the analytical expression for nonlinear active controllers for the synchronization of three different frameworks namely - (a) two identical eHPMs but with different qualitative behaviour (master eHPM possesses regular whereas slave eHPM possesses chaotic dynamics), (b) non-identical eHPM (master) and HP model (slave)~( eHPM possesses regular whereas HP model possesses chaotic dynamics), and (c) non-identical eHPM (master) and HP model (slave) with different initial conditions (both eHPM and HP model posses chaotic dynamics). Using techniques from active control theory, it is demonstrated that all three proposed systems can be phase and chaos synchronized. The time series plots of the error dynamics are used to validate that the slave system from all three frameworks adheres to the same control functions effectively for the synchronization process. Numerical simulation from the study can provide basis for assessing the potential significance of synchronization in a network of coupled food chains where Refugia and Allee effects are present.