The integration of three-dimensional Lotka–Volterra 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 elliptic and incomplete beta 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 the equations of this type that have the Painlevé property are exploited, so as to produce solutions in closed form. For several especially difficult Lotka–Volterra systems, the solutions are expressed in terms of Painlevé transcendents. An appendix on incomplete beta functions and closed-form expressions for their inverses is included.