On the Integrability of Lotka–Volterra Equations: An Update

Abstract

In 2004, Christopher and Rousseau considered various results around the integrability of the origin for the Lotka–Volterra equations $$\displaystyle{\dot{x} = x(1 + ax + by),\qquad \dot{y} = y(-\lambda + cx + dy),}$$ for rational values of λ. In particular, for \(\lambda = p/q\) with p + q ≤ 12, they showed that all the integrability conditions were given by either the Darboux method or a monodromy argument.