Random Walks and Ant Trails byMelissa Afi-Tennille Brown and Kelesitse Phiri

A random walk is a sequence of steps with the next step being completely independent of the previous. A random walk is a deviation from an expected outcome. Random Walks are also known as a Drunkard Walk or a Sailor Walk. It can be one-dimensional or multi-dimensional. These dimensions may be unequally or equally weighted. A random walk can be related to ant trails because each step that the ant takes is independent of the previous step. If the ant moves forward 2 steps then he retreats 2 steps, the retreat that the ant has made is not dependent of the previous 2 forward steps. Random Walks address many areas in the physical sciences. In Biology, random walks are useful when analysing sex-linked genes and in Chemistry, random walks can be used to model diffusion. We are using Random Walks to model ant trails; we have considered to models: the ant population and a random walk to food and the percentage of time in an hour and a random walk to food. We would use Random Walks to model ant trails for many reasons. One such reason is to contine a particular species of plant, if an ant walk can be modeled such that the ant crosses the plant in its path to pollinate the flowers to continue the species. Another reason is to limit certain deadly ants, such as the Argentine ant which annihilates other ant populations. Through random walks we can model the paths of the predator ant and the prey ant to see if they would ever meet and at which point to avoid the extinction of the prey ant species.

Ants, Diff Eq'ns, Linear Algebra and Random Walks

We created a model that showed how ants use its senses to communicate with one another to find food. The model looked not only at the ant as it looked for food but also, how the ants communicated with one another using Linear Algebra, and how the food finding process could be modeled using Differential Equations.In the Differential Model, we can adjust the concentration equation to show the stretch and shrink of the graph which illustrates how far the ant must go away from the colony to get food. In the Linear Algebra, we can change the message, the encoding and decoding matrices. These adjustments will affect how the ants transmit the message and the size of the message. In the random walks, we can adjust the parameters for the initial percentage of time per state,as well as the initial size of the colony per state. All these adjustments are related to how the individual ant spends its time among the states and how the colony as a whole travels and gets food. The Linear Algebra gives an idea of how ants communicate and the Differential Equations tell us how the numerous paths that the ant can take to get to its food source.

Modeling the Ant message using Linear Algebra

This is the first step of our model that encodes a message passed between 2 ants. If you have ever observed ants, you would have noticed that they appear to touch heads as they move along the trail. This notebook shows how the first ant encodes the message and the second ant decodes the message. The ants encode the message that reveals the location of the food so that ants from other colonies cannot locate the food.

Using Differential Equations to Model the Ant Path

The second notebook demonstrates an alternative method to model how an ant would find food using Differential Equations. Unlike the population ant random walk that is described later, the differential equation model only looks at one ant and the possibility of more than one food source. This model shows how the ant would move toward the food not in the form of a schematic diagram but in the form of a graph.

Random Walks and the Ant Trail

Now we will show how random walks can be used to model ant trails. These models are particuarly helpful because much information can be gained about ant behavior and the future of ant populations.The first model shows the path of a single ant in an hour while the second model shows the population of an ant colony at the various stages of the walk.

Interesting Remarks

We will describe the interesting facts.
• Interesting Remarks on the Project

This is a summary of what we have learnt from our project.
• Conclusion

• Bryn Mawr College
• Send an email to Melissa Brown

• 
• Webbit 3!
• Webbit 4!
• Monte Carlo Notebook
• Webbit 5!Chaos

---

This page is maintained by Melissa A. Brown,m2brown@brynmawr.edu. Last modified 04-07-03.