Stability and bifurcation analysis of a population dynamic model with Allee effect via piecewise constant argument method

This work investigates the complex dynamics of a discrete-time predator-prey system with a nonlinear Allee effect. We obtain the discrete system using the piecewise constant argument method. The piecewise argument approach produces a more dynamically consistent discrete system than other numerical techniques for discretization. We investigate the presence and stability of fixed points. Furthermore, we have demonstrated that the system undergoes Neimark-Sacker bifurcation at the positive fixed point by utilizing the Allee effect constant as the bifurcation parameter. To reduce bifurcation and chaos, we use feedback and hybrid control strategies. Our numerical simulations demonstrate the importance of the Allee effect in determining the system's behavior. The findings indicate that an adequate Allee effect might improve social connections and cooperation across populations. However, a significant Allee effect on prey can destabilize the positive fixed point, thus resulting in the extinction of predator and prey populations.