Reaction Diffusion, Chemotaxis and Non-local Mechanisms

Abstract

In an assemblage of particles, for example cells, bacteria, chemicals, animals and so on, each particle usually moves around in a random way. The particles spread out as a result of this irregular individual motion of each particle. When this microscopic irregular movement results in some macroscopic or gross regular motion of the group we think of it as a diffusion process. Of course there may be interaction between particles for example, or the environment may give some bias in which case the gross movement is not simple diffusion. To get the macroscopic behaviour from a knowledge of the individual microscopic behaviour is much too hard so we derive a continuum model equation for the global behaviour in terms of a particle density or concentration. It is instructive to start with a random process which we shall study probabilistically in an elementary way, and then derive a deterministic model.