The integration of three-dimensional Lotka-Volterra systems

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10 Scopus citations


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 languageEnglish (US)
Article number20120693
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2158
StatePublished - Oct 8 2013


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

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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