Dynamics of predator-prey system with the consequences of double Allee effect in prey population

A underlying complex dynamical behavior of double Allee effects in predator-prey system is studied in this article to understand the predator-prey relation more intensely from different aspects. We first propose a system with the Caputo sense fractional-order predator-prey system incorporating the Allee effect in prey populations to explain how the memory effect can change the different emergent states. Local stability analysis is analyzed by applying Matignon's condition for the FDE system. Further, we consider a discrete-time system to show the influence of double Allee effects in non-overlapping generations. For discrete-time system, different bifurcations like Neimark-Sacker, flip bifurcations, irregularity in periodic oscillations, are observed. Irregularity occurs through a period-doubling cascade which is a common route to chaos in a dynamical sense. Maximum Lyapunov exponent (MLE) is shown to illustrate the irregular behaviors of discrete-time systems. The Allee effect influences system stability where the strong Allee effect enhances system stability whereas the stability is lost for the weak Allee effect. The extinction risk of populations in the presence of the Allee effect is a concerning issue. We have insight into how all populations survive along with stable extinction equilibrium. Our proposed systems exhibit different alternative states. Multiple stable attractor basins are plotted to depict the different alternative states of the FDE system as well as the discrete-time system. Initial population densities play a key role in the coexistence of all the populations otherwise there is a risk of species extinction. Besides analytical results, numerical simulation is performed to valid our analytical findings of different dynamical states like bifurcation, stability, irregularity as well as multi-stability.