2024/12/17 by Diego Rojas La Luz, La Luz, Diego Rojas, Gheorghe Crăciun +3
Mathematics · Medicine · #Advanced Differential Equations and Dynamical Systems #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.2412.13367
We study the global stability of generalized Lotka-Volterra systems with generalized polynomial right-hand side, without restrictions on the number of variables or the polynomial degree, including negative and non-integer degree. We introduce polyexponential dynamical systems, which are equivalent to the generalized Lotka-Volterra systems, and we use an analogy to the theory of mass-action kinetics to define and analyze complex balanced polyexponential systems, and implicitly analyze complex balanced generalized Lotka-Volterra systems. We prove that complex balanced generalized Lotka-Volterra systems have globally attracting states, up to standard conservation relations, which become linear for the associated polyexponential systems. In particular, complex balanced generalized Lotka-Volterra systems cannot give rise to periodic solutions, chaotic dynamics, or other complex dynamical behaviors. We describe a simple sufficient condition for complex balance in terms of an associated graph structure, and we use it to analyze specific examples.