Coexistence and Stability Analysis of a Harvested Tri-Trophic Ecological Model with Hybrid Holling Responses and Intra-Specific Competition

Abstract

This paper presents and analyzes a three-species ecological model incorporating Holling type I and Holling type II functional responses together with intra-specific competition. The model describes nonlinear interactions among the species and includes density-dependent growth of the prey population. The positivity and boundedness of the solutions are established to ensure the biological feasibility of the system. The existence of equilibrium points is investigated, and the local stability of the coexistence equilibrium point is analyzed using the Routh–Hurwitz criterion. It is shown that the system is locally asymptotically stable and globally stable under suitable parametric conditions. Furthermore, the effect of intra-specific competition on the system dynamics is examined, demonstrating that it prevents unbounded growth and promotes species coexistence. Numerical simulations are performed to validate the analytical results. The time-series plots and phase portraits illustrate the stability characteristics of the system and exhibit various dynamical behaviors, including coexistence and species extinction under parameter variations. The findings provide significant insights into complex ecological interactions and emphasize the important role of competition in maintaining system stability.