Generalized Lotka-Volterra Systems and Complex Balanced Polyexponential Systems

TL;DR

This paper establishes the global stability of generalized Lotka-Volterra systems with broad polynomial forms by introducing polyexponential systems and analyzing complex balance, ruling out complex dynamics like chaos.

Contribution

It introduces polyexponential systems as a new framework and provides conditions for complex balance, extending stability analysis to more general Lotka-Volterra models.

Findings

Complex balanced systems have globally attracting states.

Such systems cannot exhibit periodic or chaotic behavior.

A graph-based condition for complex balance is provided.

Abstract

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.