Prey and Predator Inhabitants: Analysis and Direction of Hopf Bifurcation for Optimal Control of Disease Transmission in Prey–Predator Population With Numerical Simulations

ABSTRACT In this research, we focus on addressing the eco‐epidemiological delay induced prey–predator systems to better understand the dynamic properties of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a complex one. We introduce a novel delayed prey–predator interaction with a Holling type‐II function response model, where the disease spreads among prey and predator populations susceptible to infection, without considering recovery from infection. The model applies to both species, with the assumption that infected prey, which are less moveable due to the disease, are the only prey consumed by predators both susceptible and infected. In light of these observations, we extend the two‐prey, two‐predator system to include the effects of control costs on prey reproduction and the switching behavior in predation. We explore several aspects of the model, including the positivity of resolutions, presence of multiple steady states, and the stability and bifurcation at these equilibrium points, all while considering a time delay. Our findings indicate that a Hopf bifurcation can arise near the steady points when the bifurcation parameter representing the incubation period crosses certain threshold values. We further discuss the direction and stability of the Hopf bifurcation at the interior equilibrium point. In the prey–predator system, two types of controls are considered. Isolation, which separates susceptible prey and predator from the infection, and the other provides treatment to minimize deaths caused by the disease. A finite‐time optimal control problem is established to minimize the terminal infected prey, predator, and control related costs. Using Pontryagin's maximum principle, the Hamiltonian and adjoint system are derived to characterize the optimal controls. This study highlights the impact of delayed responses such as incubation periods and immunity waning on the emergent eco‐epidemiological outcomes and identifies an optimal control strategy that reduces both the density of infected prey, predator, and the associated treatment costs. Simulations are shown to validate the accuracy and practical relevance of these theoretical results.