Reaction Diffusion, Chemotaxis, and Nonlocal 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 particle's motion. When this microscopic irregular movement results in some macroscopic or gross regular motion of the group we can 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 look at probabilistically in an elementary way, and then derive a deterministic model. For simplicity we consider initially only one-dimensional motion and the simplest random walk process. The generalisation to higher dimensions is then intuitively clear from the one-dimensional equation. Suppose a particle moves randomly backward and forward along a line in fixed steps x that are taken in a fixed time t. If the motion is unbiased then it is equally probable that the particle takes a step to the right or left. After time N t the particle can be anywhere from −N x to N x if we take the starting point of the particle as the origin. The spatial distribution is clearly not going to be uniform if we release a group of particles about x = 0 since the probability of a particle reaching x = N x after N steps is very small compared with that for x nearer x = 0. We want the probability p(m, n) that a particle reaches a point m space steps to the right (that is, to x = m x) after n time-steps (that is, after a time n t). Let us suppose that to reach m x it has moved a steps to the right and b to the left. Then m = a − b, a + b = n ⇒ a = n + m 2 , b = n − a. The number of possible paths that a particle can reach this point x = m x is