MATHEMATICAL MODELING OF STABILITY AND BIFURCATION IN LOTKA–VOLTERRA SYSTEMS UNDER PARAMETRIC UNCERTAINTY

Abstract

This paper develops a mathematical framework for studying the stability and bifurcation behavior of Lotka–Volterra dynamical systems in the presence of parametric uncertainty. The growth, mortality, self-limitation, and interaction coefficients are represented by nominal values with bounded perturbations. For a modified predator–prey Lotka–Volterra system, the positive coexistence equilibrium, feasibility condition, Jacobian matrix, trace–determinant stability criteria, sensitivity coefficients, and a robust stability margin are derived. The critical surface is shown to separate the coexistence regime from the loss of a biologically feasible positive equilibrium. A numerical experiment demonstrates how admissible parameter variation shifts the equilibrium and reduces the stability margin. The analysis also clarifies that, for the selected continuous self-limited model with positive parameters, an interior Hopf bifurcation cannot occur without an additional feedback mechanism; therefore, richer bifurcation scenarios require extensions such as delay, saturation, harvesting, or discrete-time updating. The proposed approach provides a reproducible basis for uncertainty-aware analysis of ecological and related Lotka–Volterra-type models.