Predator-prey population dynamics with time delay and prey refuge effects

This study examines a predator-prey model dynamics involving two prey species and one predator, characterized by two distinct time delays (or lags). To incorporate the time required for prey populations to recover from predation, a delay is introduced into the prey's survival response, reflecting the rebound period following predation pressure. A significant challenge in delayed predator-prey systems with prey refuge lies in understanding the system's behavior when the delay parameters become substantially large and deviate from their critical values. This research aims to explore the dynamics of predator-prey interactions, focusing on achieving stability in the presence of prey refuge and time delays. The stability of the equilibrium point was established through the application of the Routh-Hurwitz criterion, and by using the Lipschitz condition, we described the existence and uniqueness of the model system. Furthermore, Hopf bifurcation analysis is performed to determine the critical delay thresholds, which indicate the onset of oscillatory behavior in the system. Analytical results are complemented by numerical simulations conducted using MATLAB's ode23 solver, providing a robust validation of the theoretical findings. The simulations reveal that predator-prey systems incorporating prey refuges can exhibit complex and chaotic dynamics, particularly when prey require significant time to recover before becoming vulnerable to predation. By integrating mathematical modeling, stability analysis, and computational methods, this study provides a comprehensive understanding of the intricate dynamics of predator-prey systems under the influence of time delays and prey refuge.