Analytic Solutions to the Lotka-volterra Model for Sustained Chemical Oscillations

The Lotka-Volterra (LV) model of oscillating chemical reactions, characterized by the rate equations has been an active area of research since it was originally posed in the 1920s. In this Review, we present a simple transformation which reduces the two-dimensional LV system to a one-dimensional system modeled by a second-order nonlinear autonomous ordinary differential equation. The formal analytic solutions to the LV problem are then derived for the first time. An introductory analysis of these solutions is given which leads to the development of an LV-related family of dynamical systems. Analytic solutions to this family of differential equations are presented, for certain cases, in terms of known functions that exhibit oscillatory behavior. The Hamiltonian nature of both the LV model and the related family of differential equations is developed, and future extensions of this treatment are discussed.