Convergence of solutions for a four-species food chain model with decaying disturbances

Ecosystems are significantly impacted by both natural and anthropogenic disturbances. This study utilizes a four-species ecosystem model to examine the asymptotic behavior of the population densities of each species, particularly the uniform boundedness of the solutions and the convergence of all solutions to an interior point. Considering the effects of disturbances, a system of differential equations with time-varying coefficients is employed to describe the mathematical model. If the magnitude and persistence of disturbances are substantial, the ecosystem may be destroyed, leading to species extinction. Thus, this study assumes that the effects of disturbances gradually diminish, depending on species adaptability and environmental resilience. This assumption is modeled using absolutely integrable time-varying coefficients. If all solutions converge to an interior point, all species coexistence within the ecosystem is achieved. Consequently, this study provides sufficient conditions for the permanence of the model. Moreover, in the scenario where the time-varying coefficients are not absolutely integrable, the potential for species extinction and survival is analyzed using a three-species ecosystem model.