Dynamics analysis of a predator–prey fractional-order model incorporating predator cannibalism and refuge

Abstract

In this article, we consider a predator–prey interaction incorporating cannibalism, refuge, and memory effect. To involve the memory effect, we apply Caputo fractional-order derivative operator. We verify the non-negativity, existence, uniqueness, and boundedness of the model solution. We then analyze the local and global stability of the equilibrium points. We also investigate the existence of Hopf bifurcation. The model has four equilibrium points, i.e., the origin point, prey extinction point, predator extinction point, and coexistence point. The origin point is always unstable, while the other equilibrium points are conditionally locally asymptotically stable. The stability of the coexistence point depends on the order of the Caputo derivative, α. The prey extinction point, predator extinction point, and coexistence point are conditionally globally and asymptotically stable. There exists Hopf bifurcation of coexistence point with parameter α. The analytic results of stability properties and Hopf bifurcations are confirmed by numerical simulations.