Weak Allee effect in a predator–prey model involving memory with a hump

In this paper we consider a predator–prey system which has a factor that allows for a reduction in fitness due to declining population sizes, often termed an Allee effect. We study the influence of the weak Allee effect which is included in the prey equation and we determine conditions for the occurrence of Hopf bifurcation. The prey population is limited by the carrying capacity of the environment, and the predator growth rate depends on past quantities of the prey which is represented by a weight function that specifies a moment in the past when the quantity of food is the most important from the point of view of the present growth of the predator. The stability properties of the system and the biological issues of the memory and Allee effect on the coexistence of the two species are studied. Finally we present some simulations to verify the veracity of the analytical conclusions.