### 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) |
---|---|

Article number | 20120693 |

Journal | Proceedings of The Royal Society of London, Series A: Mathematical and Physical Sciences |

Volume | 469 |

Issue number | 2158 |

DOIs | |

State | Published - Oct 8 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

**The integration of three-dimensional Lotka-Volterra systems.** / Maier, Robert S.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The integration of three-dimensional Lotka-Volterra systems

AU - Maier, Robert S

PY - 2013/10/8

Y1 - 2013/10/8

N2 - 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.

AB - 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.

KW - Generalized Schwarzian equation

KW - Lotka-Volterra system

KW - Painlevé property

UR - http://www.scopus.com/inward/record.url?scp=84884157157&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884157157&partnerID=8YFLogxK

U2 - 10.1098/rspa.2012.0693

DO - 10.1098/rspa.2012.0693

M3 - Article

AN - SCOPUS:84884157157

VL - 469

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0962-8444

IS - 2158

M1 - 20120693

ER -