On a Hamiltonian version of a 3D Lotka-Volterra system

Tudoran, Răzvan M.; Gîrban, Anania

doi:10.48550/arxiv.1106.1377

Cited by 1 publication

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“…They are essentially the first integrals introduced as I 1 , I 2 , I 3 in (1.9) above (where x is to be read as y). Starting from (3.32), one can confirm that the first integrals found by Llibre & Valls (2011) and Tudoran & Gîrban (2012) are also time-independent.…”

Section: Two-dimensional Integrationmentioning

confidence: 67%

“…The system with α = β = −1, which is mutualistic rather than competitive, will be shown to have the Painlevé property. May-Leonard systems have been analysed from the Painlevé point of view (Bountis & Segur 1982;Sachdev & Ramanan 1997) and multiple first integrals have been discovered (Llibre & Valls 2011;Tudoran & Gîrban 2012). But the results below are much more explicit.…”

Section: Two-dimensional Integrationmentioning

confidence: 99%

“…If the growth rates are equal, x is expressed in terms of the Weierstrassian elliptic function ℘(τ ); and if a single rate (λ) is nonzero, the Painlevé transcendent w V (e λτ ). This case, with equal growth rates, was studied geometrically by Tudoran & Gîrban (2012), but the elliptic general solution is new. The appearance of a Painlevé function is also novel.…”

Section: Unequal Growth Rates and Painlevé Transcendentsmentioning

confidence: 99%

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The integration of three-dimensional Lotka–Volterra systems

Maier

2013

Proc. R. Soc. A.

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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.

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Section: Two-dimensional Integrationmentioning

confidence: 67%