Stability and bifurcation analysis of a discrete Leslie predator-prey system via piecewise constant argument method

The objective of this study was to analyze the complex dynamics of a discrete-time predator-prey system by using the piecewise constant argument technique. The existence and stability of fixed points were examined. It was shown that the system experienced period-doubling (PD) and Neimark-Sacker (NS) bifurcations at the positive fixed point by using the center manifold and bifurcation theory. The management of the system's bifurcating and fluctuating behavior may be controlled via the use of feedback and hybrid control approaches. Both methods were effective in controlling bifurcation and chaos. Furthermore, we used numerical simulations to empirically validate our theoretical findings. The chaotic behaviors of the system were recognized through bifurcation diagrams and maximum Lyapunov exponent graphs. The stability of the positive fixed point within the optimal prey growth rate range $A_1<a<A_2$ was highlighted by our observations. When the value of $a$ falls below a certain threshold $A_1$, it becomes challenging to effectively sustain prey populations in the face of predation, thereby affecting the survival of predators. When the growth rate surpasses a specific threshold denoted as $A_2$, it initiates a phase of rapid expansion. Predators initially benefit from this phase because it supplies them with sufficient food. Subsequently, resource depletion could occur, potentially resulting in long-term consequences for populations of both the predator and prey. Therefore, a moderate amount of prey's growth rate was beneficial for both predator and prey populations.