Influence of multiple delays mechanisms on predator–prey model with Allee effect

Although there are many factors that influence population fluctuations, the impact of delay on population dynamics is one of the most significant ones. In this paper, we explore how the Allee effect in combination with a double delay (competition and cooperation) affects predator–prey population dynamics. One can determine the requirements for the equilibrium’s asymptotic stability and the presence of Hopf bifurcations by investigating the roots of a characteristic equation. The rigorous mathematical proofs of the stability and direction of the Hopf bifurcating periodic solutions are derived with the help of the normal form theory and the center manifold theorem. Numerical simulation shows that different delay mechanisms can generate different behaviors, such as the competitive delay that produces regular and irregular oscillations, as well as the chaotic, while the cooperative delay results in the stability switch. These theoretical and numerical results are illustrated that depending on the chosen delayed mechanism, delay may not necessarily lead to instability, and also can help for understanding the biological implications corresponding to the interaction dynamics of predator and prey.