Exploring multistability and bifurcations in a three-species Smith growth model incorporating refuge, harvesting, and time delays

Abstract

This study delves into a tritrophic ecological model encompassing three distinct species, elucidating predator–prey dynamics through the lens of Smith growth pattern. The model integrates several pivotal ecological elements, including an additive Allee effect dictating prey growth, a ratio-dependent functional response characterizing predator–prey interactions, the provision of refuge for intermediate predators, and the incorporation of a Michaelis–Menten-type harvesting mechanism of the top predators. Moreover, we incorporate gestation and harvesting delays as novel factors to scrutinize their impact on the overall dynamics of the food web system. Through an extensive analysis of the delayed and non-delayed models, our investigation rigorously explores the equilibrium points, stability attributes, and bifurcations structures. In the absence of time delay, our findings underscore the profound influence wielded by factors such as refuge availability, Allee effect, harvesting, and the availability of environmental resources in dictating the survival prospects of the involved species. Furthermore, our exploratory analysis uncovers a rich tapestry of intricate dynamics, encompassing chaotic behavior, periodic oscillations and, multistability. These revelations underscore the profound complexity inherent in the ecosystem, particularly accentuated by the temporal delays involved in gestation and harvesting processes. The nuanced interplay between these temporal delays and ecological parameters contributes to the emergence of diverse and complex dynamics, elucidating the intricate nature of the ecological systems.