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 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.
Keywords
Cite
@article{arxiv.1211.5813,
title = {The Integration of Three-Dimensional Lotka-Volterra Systems},
author = {Robert S. Maier},
journal= {arXiv preprint arXiv:1211.5813},
year = {2014}
}