A POWER SERIES IN SMALL ENERGY FOR THE PERIOD OF THE LOTKA-VOLTERRA SYSTEM

Abstract

The classical Lotka-Volterra system of two first-order nonlinear differential equations was investigated by Lotka [25] on chemical reactions, Lotka [26] on rhythmical reactions in physiology, Lotka [27] on parasitology, Volterra [46] on fishing activity in the upper Adriatic Sea, and Kozyreff [23] and Erneux and Kozyreff [10] on laser dynamics in the vicinity of the Hopf bifurcation. A functional relation between two dependent variables has been known to describe its periodic behavior in the phase plane. We first solve this equation explicitly for one dependent variable in terms of the other, and then obtain two integral representations of the period having a singularity of the square root type at each endpoint of the integration. Our notations are based on Lambert's W functions, which are two inverse functions of $x \to x\exp(x)$ restricted to $(-\infty, -1]$ and $[-1, \infty)$, respectively. A power series of the period is constructed in small energy for arbitrary number of terms by virtue of expansions of Lambert's W functions near the branch point $x = -\exp(-1)$. Our result settles the discrepancy of various approximate results in the literature, and are further compared with numerical results of computing the period by applying the Gauss-Tschebyscheff integration rule of the first kind.