Mathematical model of competition for a limited resource in ecosystems: numerical and analytical study of sustainability

This paper investigates the dynamics of interaction between two species competing for a limited resource using a mathematical model that is an autonomous system of ordinary differential equations in normal form. The model is based on Gause's principle, Volterra's hypotheses, Tilman's theory of resource competition, and the Michaelis-Menten equation to describe population growth. The system of nonlinear ordinary differential equations is analyzed for stability at stationary points using the first approximation analytical method proposed by A.A. Lyapunov, which is suitable for the study of systems consisting of two or more equations, and analytically and numerically solved for various values of model parameters. The results show that species survival and coexistence depend on the level of the limiting resource, the ratio of fertility and mortality rates and intraspecific competition, and substrate concentration. Numerical simulations correspond to scenarios of extinction of one species, dominance of one species, or their coexistence depending on environmental conditions. The results obtained in this work are consistent with natural ecological relationships and emphasize the importance of considering anthropogenic factors, such as eutrophication, when predicting changes in ecological systems.

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