ANALYTIC SOLUTIONS TO A CLASS OF TWO-DIMENSIONAL LOTKA-VOLTERRA DYNAMICAL SYSTEMS

Abstract. A novel coordinate transformation is used to reduce a simple generalization of the Lotka-Volterra dynamical system to a single second-order autonomous ordinary differential equation. Formal analytic solutions to this differential equation are presented which are shown to reduce to the recently obtained solution to the Lotka-Volterra system [C. M. Evans and G. L. Findley, J. Math. Chem. 25 (1999) 105–110]. An initial analysis of the analytic solution to this latter system results in the specification of a new family of Lotka-Volterra related differential equations. 1. Introduction. The Lotka-Volterra (LV) system consists of the following pair of first-order autonomous ordinary differential equations: where x1(t) and x2(t) are real functions of time, = dxi/dt, and a, b, c are positive real constants. This system was originally introduced by Lotka [7] in 1920 as a model of undamped oscillations in autocatalytic chemical reactions, and wa