The fear and toxin effect on Sokol-Howell prey-predator model involving hunting cooperation with anti-predator behaviour and disease in top predator

Abstract

This study examines the complex dynamic behavior of a multi-trophic epidemiological ecology model, encompassing prey, predators, susceptible and infected apex predators. When a predator consumes, preys according to Holling type-Ⅳ functional response, while the apex predator utilizes a functional response of the Sokol-Howl type. Among other environmental challenges, the model considers the toxic effect on all species, with toxicity levels decreasing from prey to apex predator. Cooperative hunting among predators and apex predators, as well as prey-predator behavior, enhances the model's dynamics. Additionally, the fear effect addresses the psychological impact, driving prey to fear predators and predators to fear apex predators. We conduct a comprehensive qualitative analysis, including the presence and limitations of solutions, the local and global stability of equilibrium points. Numerical analysis demonstrate how toxicity, fear, and cooperative hunting synergistically influence species survival and extinction, revealing the stability of diseased, three-trophic food webs under environmental stresses.