The persistence and bifurcation conditions of an eco-epidemiological model involving toxin and fear with two types of disease

Abstract

In this paper, we investigate the conditions in which local bifurcations might occur in an eco-epidemiological model that considers the combined effects of human behavior and fear. The model takes into account the SI (Susceptible–Infected) infection structure in the predator population and the SIS (Susceptible–Infected–Susceptible) disease dynamics in the prey population. The effect of fear on prey behavior and human intervention methods, including harvesting, treatment, and environmental management, is given special attention. The existence and stability of equilibrium points as well as the conditions around local bifurcations are the main topics of the qualitative study. It appears that the system can display Hopf bifurcation around the positive equilibrium, which results in the formation of periodic oscillations, as well as transcritical and saddle-node bifurcations close to certain equilibrium points. The dynamics and stability of the system are significantly shaped by the interplay between the degree of fear and the parameters of human involvement. Additionally, the presence of local bifurcations is convincingly verified using Sotomayor's theorem. Transitions between disease persistence, extinction, and coexistence stages may be brought about by changes in important factors, according to the acquired analytical results. In order to verify the theoretical results, numerical simulations are carried out for certain parameter settings and initial conditions. The results demonstrate a high degree of agreement with the system's expected dynamical behavior.