## 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.

Original language | English (US) |
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Article number | 20120693 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 469 |

Issue number | 2158 |

DOIs | |

State | Published - Oct 8 2013 |

## Keywords

- Generalized Schwarzian equation
- Lotka-Volterra system
- Painlevé property

## ASJC Scopus subject areas

- General Mathematics
- General Engineering
- General Physics and Astronomy