Journal of Advances in Mathematics and Computer Science

A tri-trophic prey–predator system describes the interactions among prey, intermediate predators, and top predators in an ecological food chain. This study incorporates interval-valued parameters to account for uncertainties in ecological processes and investigates harvesting strategies using optimal control theory to achieve sustainable resource management and ecosystem stability. This article describes an interval-valued tri trophic food chain framework that takes biological variations and environmental variability into account by combining harvesting and parametric ambiguity. The model is composed of populations of top predators, intermediate predators and prey and all parameters are expressed as bounded intervals. To enable extensive analysis, a corresponding parametric deterministic approach is established from the interval system. The fundamental properties of the framework notably boundedness of solutions and non-negativity are confirmed. The existence of different kinds of equilibria trivial, boundary, planar and interior is analyzed and their local and global stability is computed applying the Jacobian matrix, Routh-Hurwitz criteria and Lyapunov functions. The basic reproduction number is computed using the next generation matrix procedure which specifies a threshold condition for the persistence and extinction of a species. An optimal control problems is designed to decrease control costs and maintain ecological balance while limiting the densities of hazardous populations of prey. For identifying requisite optimality circumstances, Pontryagin’s
Maximum Principle is utilized. Additionally, the concept of bionomic equilibrium is studied by integrating ecological dynamics with economic harvesting implications. Numerical simulations describe how interval uncertainty influence normal behavior of a system exhibiting how modifications to parameter boundaries profoundly impact bifurcation dynamics, stability and coexistence. The outcome illustrate an improved structure for assessing complex ecological systems and developing effective management strategies when interval uncertainty is taken into the account.