OVERVIEW

Because populations of molecules, cells, or organisms are rarely distributed evenly over a featureless environment, their motions, migrations, and redistributions are of some interest. At the level of the individual, movement might result from special mechanochemical processes, from macroscopic contractions of muscles, or from amoeboid streaming. On the population level, these different mechanisms may have less bearing on net migration than other aspects such as (1) variations in the environment, (2) population densities and degree of overcrowding, and (3) motion of the fluid or air in which the organisms live.

On the collective level it is often appropriate to make a continuum assumption, that is, to depict discrete cells or organisms by continuous density distributions. This leads to partial differential equation models that are quite often analogous to classical models for molecular diffusion, convection, or attraction.

Historically, biological models involving partial differential equations (PDEs) date back to the work of K. Pearson and J. Blakeman in the early 1900s. In the 1930s others, including R.A. Fisher, applied PDEs to the spatial spread of genes and of diseases. The 1950s witnessed several important developments including the work of A.M. Turing on pattern formation (see Chapter 11) and the analysis of Skellam (1951), who was among the first to formally apply the diffusion equation in modeling the random dispersal of a population in nature.