Chapter 18 - Oscillatory reactions

Abstract

Non-linear systems, described by sets of differential equations involving second- or higher-order reactions, are very difficult to solve but can give rise to very interesting phenomena. Such systems are first illustrated by the Lotka–Volterra predator–prey model, which can lead to oscillatory reactions. Next, this model is adapted to describe the competition between two species (e.g., rabbits and sheep) and show that most often when two species compete for the same resources, one drives the other to extinction. A parallel is drawn with the extinction of Neanderthals when the Home sapiens sapiens arrived in Europe. The sensitivity of non-linear systems to the initial conditions can lead to chaos. This is illustrated with a discrete-time analogue of the logistics equation and with the Lorenz equations describing convection in the atmosphere. The occurrence of oscillations in chemical systems is illustrated with the Belousov–Zhabotinsky reaction. Models describing oscillatory reactions include the Brusselator, the Oregonator and the Coimbrator. This latter model shows that chemical oscillators can occur in a closed system at constant pressure and temperature provided that photochemical reactions can occur. This is illustrated by Earth as a closed system, where photosynthesis may provide the driving force for oscillations that have an impact in climate.