Complexity in biological systems and hamiltonian dynamics

Abstract

It is well known that in biology and the life sciences in general there are numerous instances of complex behaviour arising from apparently deterministic processes. One is familiar with the ‘fractal’ nature of the shapes of leaves and snowflakes. Such shapes may be described as the outcome of simple deterministic evolutionary dynamics. In another area the stochastic or chaotic response of stimulated cardiac nerve cells is well known and can be modelled by nonlinear deterministic systems of ordinary differential or difference equations. In most examples describing complex behaviour in biological systems the underlying models are either ordinary differential or difference equations leading to an analysis of temporal behaviour and its dependence on certain parameter values such as growth rates, generation times or reaction constants. In this paper we explore both the temporal and spatial nature of complex phenomena in biological systems. The examples involve partial dif­ferential or partial difference equations and are drawn from models of population genetics, mollusc shell patterning and excitable systems. In each situation we demonstrate aspects of complexity and show that non-integrable hamiltonian dynamical systems play a crucial role. This brings into the realms of biology such concepts as structural stability and Kolmogorov─Arnold─Moser (KAM) theory, which lie at the heart of current developments in the theory of dynamical systems.