Stability and Bifurcation in Fractional Order Derivatives of a Tri-Trophic Food Chain with Harvesting

Abstract

This paper primarily focuses on the study of an ecological model in which fractional differential equations (FDEs) are utilized to construct the model rather than using traditional first order differential equations.  A model of tri-trophic food chain is explored using the Beddington–DeAngelis (BD) functional response with harvesting in each species. The impact of harvesting on the system’s behaviour is observed, that brings about the occurrence of both bifurcations, Transcritical and Hopf bifurcation, which depends not only on fractional order derivative parameters, but also on the harvesting parameters. The existence, uniqueness, non-negativity, and boundedness of the solutions are explored. Next, the stability of all feasible equilibrium points is determined locally. Additionally, sufficient conditions are established to make certain the global asymptotic stability of each point of equilibrium, except the trivial equilibrium point, by selecting an appropriate Lyapunov function. Finally, numerical simulations are given to illustrate the outcomes of our dynamical analysis of the model.