On qualitative analysis of an ecological dynamics with time delay

Abstract

In this paper, we study a fractional-order predator-prey system with time delay, where the dynamics are logistic with prey population commensurate to the carrying capacity. Mainly, by linearizing the system around the equilibrium point, we first analyze the stability and then prove the existence of Hopf bifurcation. Moreover, by defining the Lyapunov function for this system, the global stability of the solution is proven. The results of this study demonstrate that the stability and Hopf bifurcation of the ecological model are remarkably affected by both the time delay and the order of fractional derivatives. Finally, to support our new theoretical results, two different numerical examples are illustrated by taking two different fractional orders, q$$ q $$.