Bifurcation and Conservation Control in a Fear-Induced Alarm-Mediated Tiger–Deer Ecosystem

In this work, we develop a mathematical model to study the dynamics of a predator–prey system in which tigers act as predators and deer serve as prey. We also include alarm communication among the deer that is triggered by the presence of the predator. The model captures the impact of predator-induced fear on prey reproduction. Two discrete time delays are included to represent realistic biological processes. The first delay indicates the time it takes for deer to respond to alarm signals within the herd, while the second delay corresponds to the tiger population’s reproductive maturation period. The dynamical properties of the system are examined through equilibrium and stability analysis. We examine the stability criteria of equilibrium points after proving their existence. To ascertain whether the predator population can effectively invade and endure in a prey-only environment, a threshold value known as the predator invasion number [Formula: see text] is calculated. The local stability of the interior coexistence equilibrium is investigated using linear stability theory. According to our examination, time delays can change the equilibrium’s stability and cause a Hopf bifurcation, which causes the interacting populations to exhibit persistent oscillatory behavior. Different configurations are taken into account in order to further investigate the role of delays. These include cases in which one delay changes while the other remains constant, as well as cases in which both delays take on the same values. Numerical simulations are presented to validate the analytical results and to illustrate the impact of fear intensity on population dynamics. Moreover, an optimal conservation control strategy is proposed to manage the environment and enhance prey survival. The findings show that the system can be stabilized and the long-term cohabitation of tiger and deer populations can be supported by adequate control measures.