Multiple bifurcations and optimal control in an eco-epidemiological model driven by herd behavior and Allee effects

This paper formulates a three-dimensional eco-epidemiological model to investigate the spatial and behavioral dynamics of wildlife communities. The system incorporates prey herd behavior, predator cooperative hunting, a Beddington-DeAngelis incidence rate, and mate-finding Allee effects. We establish the positivity and boundedness of solutions, derive the basic reproduction number \(R₀\) , analyze the local stability of equilibria, and analytically derive threshold conditions for transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations using Sotomayor’s theorem, center manifold reduction, and other analytical methods. Furthermore, optimal control analysis based on Pontryagin’s maximum principle demonstrates that high-intensity selective culling suppresses epidemics but disrupts energy flows, risking a trophic cascade and predator extinction. Numerical simulations verify these theoretical results and indicate that herd behavior acts as a spatial defense mechanism to curb over-predation, maintaining predator-prey coexistence. Moreover, at low population densities, mate-finding Allee effects exacerbate the extinction risks driven by herd behavior, heightening the system’s sensitivity to initial conditions. Additionally, while the normalized sensitivity analysis of \(R₀\) confirms that defensive isolation directly mitigates the risk of initial disease invasion, global sensitivity evaluations and two-parameter bifurcation heatmaps jointly reveal paradoxical ecological feedbacks: predators’ targeted consumption of infected individuals indirectly promotes healthy population recovery. Conversely, excessive defensive isolation enacted by the healthy herd paradoxically starves predators by cutting off their infected prey supply. This energy depletion causes predator extinction, which subsequently triggers a predator-release effect and a compensatory surge in infected prey. This study quantifies the interplay between spatial behavior and disease dynamics, providing a theoretical foundation for balancing wildlife disease interventions and ecosystem integrity.