By Axel Almet
Picture a crowd of people standing at a concert, cars moving on a highway and a group of cancerous cells migrating through a blood vessel during metastasis. Now shrink down the people and the cars so that they’re the same size as the cells. Then remove all complicated aspects so that we are left knowing only two things: how these objects move and how they interact with each other. All of these scenarios, when simplified, can be modeled by what we call agent-based models (ABMs). They are extremely useful for modeling complex phenomena including but not limited to the above scenarios, and can be effectively simulated on computers.
What’s more, we can make these ABMs even easier to simulate, by simplifying how they move, and the environment in which they move. Suppose that we have a set of points evenly spaced apart on a line (let’s stick to one dimension for convenience). Now suppose we have agents on a number of these points; for each agent, we tell it to move to the left or right adjacent site and the site it moves to is chosen at random. This is the simplest version of what’s known as a random walk. Finally, we impose the condition that a maximum of one agent can occupy each site. We call these ABMs exclusion processes.
The specific nature of exclusion processes means that when we use them, we only have to worry about one aspect: how we let agents interact. The simplest interaction rule is to let agents move as they please until they have to move to a site that is already occupied. When that happens, they abort their move and stay where they are. Appropriately, this is called simple exclusion and it has been shown to provide great insight in studying cell motility and traffic flow. A key characteristic of simple exclusion is that the higher the number of the agents we have, the higher the possibility that agents will abort their moves as they walk around.
However, what if we decided that we didn’t want agents to abort their moves, but we still wanted to maintain the ‘exclusion condition’? After all, these models are used in real world applications and the real world isn’t so polite. One way to do this is to introduce shoving, which is exactly as it sounds: agents are now able to push other agents off the sites that they move onto. Think of people in a mosh pit.
While the exclusion processes are discrete processes, they have a continuous analogue in the form of partial differential equations, equations modeling change in time and space. Simple exclusion can be accurately modeled by what most people in science would call the heat equation or simple linear diffusion. For shoving? We still end up with a diffusion equation but it’s far stranger and harder to solve. The bright side is that it still accurately describes the discrete process.
Axel Almet was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.