On a Quadratic Poisson Algebra and Integrable Lotka – Volterra Systems with Solutions in Terms of Lambert’s $W$ Function
Author(s):
van der Kamp P. H., McLaren D. I., Quispel G. R. W.
We study a class of integrable inhomogeneous Lotka – Volterra systems whose
quadratic terms are defined by an antisymmetric matrix and whose linear terms consist
of three blocks. We provide the Poisson algebra of their Darboux polynomials and prove
a contraction theorem. We then use these results to classify the systems according to the
number of functionally independent (and, for some, commuting) integrals. We also establish
separability/solvability by quadratures, given the solutions to the 2- and 3-dimensional systems,
which we provide in terms of the Lambert $W$ function.
Keywords:
Poisson algebra, integrability, Lotka – Volterra system, Lambert $W$ function, Darboux polynomial
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