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.

References in corpus (6)

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• The Integration of Three-Dimensional Lotka-Volterra Systems
• Lagrangian structures, integrability and chaos for 3D dynamical equations
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Cited by in corpus (4)

• Hypergeometric First Integrals of the Duffing and van der Pol Oscillators
• The Integration of Three-Dimensional Lotka-Volterra Systems
• Integration of differential equations by ${\mathcal{C}}^{\infty }$-structures
• Preservation of the invariants of Lotka-Volterra equations by iterated deferred correction methods