The Integration of Three-Dimensional Lotka-Volterra Systems

TL;DR

This paper develops methods to explicitly solve complex three-dimensional Lotka-Volterra systems using special functions like elliptic and beta functions, expanding the class of integrable models.

Contribution

It introduces new solution techniques for three-dimensional Lotka-Volterra systems employing special functions and nonlinear differential equations, including parametric solutions and Painleve transcendents.

Findings

Explicit solutions for several Lotka-Volterra systems are constructed.

Solutions involve incomplete beta, elliptic functions, and Painleve transcendents.

The approach extends integrability to previously intractable systems.

Abstract

The general solutions of many three-dimensional Lotka-Volterra systems, previously known to be at least partially integrable, are constructed with the aid of special functions. Examples include certain ABC and May-Leonard systems. The special functions used are incomplete beta and elliptic functions. In some cases the solution is parametric, with the independent and dependent variables expressed as functions of a `new time' variable. This auxiliary variable satisfies a nonlinear third-order differential equation of a generalized Schwarzian type, and results of Carton-LeBrun on such equations are exploited. Several difficult Lotka-Volterra systems are successfully integrated in terms of Painleve transcendents. An appendix on incomplete beta functions is included.