A modified predator-prey model involving dual time delays: Bifurcation mechanism and controller design

This study establishes a predator-prey model with dual time delays, aiming to rigorously investigate the mechanisms through which time delays influence system stability and periodic behaviors. The well-posedness of the solutions, including existence and uniqueness, non-negativity, and boundedness, is established by fixed-point theorem, inequality technique, and proper function creation. Furthermore, a rigorous analysis of delay-induced Hopf bifurcations is conducted across five distinct delay-combination scenarios. Utilizing the Routh-Hurwitz criterion in conjunction with stability theory and bifurcation principles, we derive explicit stability conditions and precise Hopf bifurcation thresholds. This systematically elucidates how variations in delays govern stability transitions and generate periodic solutions. Crucially, we calculate the critical time-delay length that triggers stability switching. Two innovative controllers (ordinary feedback controller and dislocated feedback controller) are designed to regulate bifurcation dynamics. These effectively modulate bifurcation behavior, significantly modify the stability region, and enable precise control over the time of Hopf bifurcation onset. Numerical simulations implemented via Matlab confirm the validity and efficacy of the theoretical results, providing fundamental theoretical insights and practical strategies for predicting ecosystem dynamics and regulating ecological equilibrium.