AN APPROACH TO THE STABILITY OF THE GAUSE-LOTKA-VOLTERRA COMPETITION ECOLOGICAL MODEL USING THE ROUTH-HURWITZ CRITERION

Abstract

In the qualitative study of systems of ordinary differential equations, we can analyze the dynamics of the solutions by observing whether small variations or modifications in the initial conditions produce small changes in the future, this intuitive idea was formalized and studied by Lyapunov and reflects the concept of stability. For both linear and nonlinear systems of differential equations, we can analyze stability using criteria to obtain Hurwitz type polynomials. In this paper, we study the stability of a prey-predator model where three species compete considering the Gause effect generated by the Gause-Lotka-Volterra system, using an analytical stability technique to obtain Hurwitz-type polynomials called Routh-Hurwitz criterion. This criterion provides the necessary and/or sufficient conditions to analyze the dynamics of the system by studying the location of the roots of the characteristic polynomial associated with the system. This article describes and presents an analytical method to analyze the stability of the Gause-Lotka-Volterra model, based on the study of Hurwitz type polynomials. This technique allows establishing the asymptotic stability of the system by analyzing the location of the eigenvalues of the matrix associated with said system. The research results allow establishing a region of asymptotic stability of the Gause-Lotka-Volterra model that allows analyzing the interaction dynamics between the species.