Unraveling the combined actions of a Holling type III predator–prey model incorporating Allee response and memory effects

Abstract

In this article, we have studied a memory dependent prey–predator model with logistic type prey growth subject to the Allee effect and Holling type-III functional response. To include the memory effect, we have used here a fractional order system. The model contains at most four equilibrium points among which one is trivial, two are axial, and the last one is interior. Here we have studied behavior of these four equilibrium points using eigen analysis method. Our investigation shows that the trivial equilibrium point is stable for strong Allee effect, and unstable for weak Allee effect. Between the two axial equilibrium points, one is stable under certain condition but the other is always unstable. This unstable equilibrium point exists only for strong Allee case. The interior equilibrium point arises when the axial equilibrium point is unstable. Stability condition of this interior equilibrium point depends on the parametric relationship. The schematic diagram divides the a-p plane into five sub-regions in both the cases of memory dependent and memory less system. The number and nature of the equilibrium points in different sub-regions are different. Finally, we have drawn the bifurcation diagrams considering all the equilibrium points in different sub-regions and the results are concluded at the end.