Influence of Breeding Delays and Memory Effects on Predator-Prey Model Amidst Fear

In this study, we present a predator-prey model that incorporates a delay in the prey’s reproduction resulting from fear induced by predators. Next, we modify our model to a fractional-order system, incorporating the effects of memory. Establishing positivity and boundedness of the solutions demonstrates the well-posedness of the system. The local and global asymptotic stability of the positive equilibria are established under certain suitable parametric conditions. Additionally, we prove the existence and uniqueness of solutions for the fractional-order system while ensuring that they remain bounded. It is observed that, depending on constraints defined by the values of the model parameters, the breeding delay in the model system has both a stabilizing and destabilizing role in the system dynamics. The maximum length of delay that preserves the stability of the limit cycle is calculated. In the presence of delay, it is noticed that the fear factor in model system dynamics plays exactly the opposite role to that of the system without delay; more preciously, when the prey species delayed their breeding, fear acts as a destabilizing factor. We also consider the modified fractional order system to reveal the impact of the forgetting process on the system dynamics. Numerical simulations capture system dynamics and reveal that the delayed model system exhibits abundant dynamics, including several stability changes and chaotic behavior. Order of fractional derivative found to be involved in changing the stability property of the system near the coexistence equilibrium state.