Random Walks and Diffusion in Ecology

Last week I had my very first paper accepted! It’s some work that came out of my masters, which I did in the mathematical biology group at Melbourne University. To mark the occasion, I’m going to write a few posts about random walks and how they relate to biology and ecology.

One of the most common models for animal movement in ecology is diffusion. It has been used by people like Ronald Fisher and James Murray, and it continues to be used extensively today. However, the reason why diffusion is a good model of animal movement is not obvious. Particularly when you consider that the diffusion equation also describes heat flow.

The diffusion equation is derived assuming that movement is random. The concept of a random walk is quite simple. Imagine a piece on an (infinitely large) chessboard. If you move the piece to a random adjacent square every second, then the piece performing a random walk. Getting from this simple idea to the diffusion equation takes a little sweat, which I won’t put you through. However, the basic idea is that if you make the squares on your chessboard really small and start moving your piece really often, then, out pops the diffusion equation.

Diffusion Equation

You can read all about the derivation here. Visually, the process of shrinking the lattice looks like this:

Random walk with large steps

Random walk with large steps

Random walk with small steps

Random walk with small steps

Random walk with tiny steps

Random walk with tiny steps

The diffusion equation gives average behavior: it will give you the probability that a particle on a random walk will be at a specific site at a given time. It will give you the expected behaviour of a large group, but will not be able to tell you where specific individual is.

Of course you can’t use the diffusion equation to model anything. You need the animals to be moving fairly randomly, without strong interactions. For example, you could not model individuals in a school of fish. This actually correspond to negative diffusivity, which is a different kettle of fish.

The diffusion equation actually works quite well for some systems which violate the assumptions. We assumed that the step sizes were infinitely small, but the equation works perfectly well when the step sizes are relatively large. There are all kinds of variations on the basic random walk. You can always derive a nonlinear diffusion equation for a random walk, but the standard diffusion will often suffice. For example, the diffusion equation will do just fine if you are modelling animals which are on a persistent random walk.

The diffusion equation is an exceptionally useful equation. However, it is important to understand the assumptions that were made in the derivation when you use it. Even if you’re going to ignore the assumptions and use it anyway.

This entry was posted in Biology, Diffusion, Ecology, Mathematics, Random Walks. Bookmark the permalink.

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