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.

In this paper we consider the integrability of the critical points which do not lie at the origin. For those on one of the axes, we classify all integrable critical points with ratio of eigenvalues $-p'/q'$ with p′ + q′ ≤ 17; and for those not on the axes, we consider all critical points with ratio of eigenvalues $-p''/q''$ with p″ + q″ ≤ 10. We also extend the classification of integrable critical points at the origin for p + q ≤ 20.

In all these cases, we are able to show that the monodromy method is sufficient to prove integrability except when $\lambda ab + (1-\lambda )ad - cd = 0$, for which the system has an invariant line. However, to do this, we need to extend the monodromy method to include the monodromy about some of the invariant algebraic curves of the system as well as the axes.

To Christiane—with thanks for being able to sharein your mathematical interests in a small way.