**Published on** September 17th, 2013 |
*by Daphane Ng*

# On the Correlated Random Walk with Exclusion

By Aaron Chong, The University of Melbourne

*This student took part in the 2012/13 AMSI Vacation Research Scholarship program. For more information on this year’s program please click here.*

**INTRODUCTION**

A random walk is a process where an agent moves in a randomly decided direction on each step. A correlated random walk is a random walk in which the probability of the agent moving in a particular direction is based on the direction it moved previously. This model has applications in many fields, particularly biology; for example, moving cells are likely to have a persistence of direction where they are more likely to move in the direction they are facing.

In my research I combined this with simple exclusion, in which agents abort moves into occupied locations, so as to examine the behaviour of many interacting agents moving in this way. My project involved analysing this system and deriving a partial differential equation that could describe the behaviour of the population as a whole – an equation that relates the rate of change of concentration over time to the distribution of concentration over the lattice.

**THE MODEL**

Let pf be the probability that the agent moves in the same direction it last moved if it has the opportunity, and pb be the probability that it moves in the opposite direction. To simplify I assumed that pf + pb = 1.

I began by deriving conservation-of-mass statements to describe the evolution of the discrete system, then introduced a time step τ and lattice spacing Δ to convert it into a continuous system, and some manipulation allowed me to derive two linked partial differential equations. I discovered that by scaling the variables I could derive a PDE that approximated the evolution of the system:

The PDE corresponds to a non-linear diffusion equation – in other words, the system behaves like a system of particles diffusing at a rate dependent on concentration.

A comparison between actual simulations of the system (red) and the PDE approximation (blue) at times 0, 50, 100 and 200 for pf = 0.6 is given in the chart. It can be seen that the model provides a reasonably good match, with the fit becoming closer as the time increases.

**THE VACATION SCHOLARSHIP EXPERIENCE**

The vacation scholarship program was an enjoyable experience that gave me a taste of independent academic research. I met with challenges along the way, but thanks to the guidance of my supervisors I was able to overcome them, and the feeling when I find a solution to a problem is exhilarating. I now have a better understanding of how research works and have learned new problem solving techniques.

As part of the AMSI Vacation Scholarship Program I had the opportunity to attend the CSIRO Big Day In after my research, at which I presented my research results to AMSI and CSIRO vacation students from around Australia. This experience allowed me to meet a range of like-minded students, learn about a variety of projects on the cutting-edge of scientific research, and gain an insight into the wider mathematics and scientific community.

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