Generalized Lotka–Volterra Systems and Complex Balanced Polyexponential Systems

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 noninteger 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.