Many population models ignore the details of the dispersal phase. Too little is usually known about dispersal to model it in anything other than the simplest way (but see the work of Pacala on neighbourhood models for an exception to this, e.g. Pacala & Silander 1990). One simple approximation which leads to interesting results is to assume that the individuals diffuse out from a source population. This may describe the expansion of an invading species across suitable habitat or the movement of individuals between local populations across uncolonizable habitat. Diffusion is a random and continuous process with each particle or individual going on a random walk from its source position. Although the concept is straightforward, diffusion models are complex because they require a method of summarizing all the random movements at each point in time. Applications of spatial diffusion models in ecology include the work of Morris (1993) on pollen dispersal and insect movement and marine ecologists studying the movement of algae in water bodies (see below). Segel (1984) gives an introduction to diffusion models of bacteria movement.
Diffusion can be described by partial differential equations, or PDEs. These are needed because movement and/or abundance of individuals is dependent on two variables: spatial position and time. Maynard Smith (1968) provides an accessible introduction to partial differentiation applied to biological problems. He describes the diffusion of a substance along a tube. The change in concentration (x) with time (t) is related to the change in concentration with distance (s) according to the following PDE:
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