# "Order in Chaos: Mathematical Models of Social Animal Interactions" by Beamlak Lefebo

Regardless of one’s field of expertise, it is fair to assume that a strong quantitative background will prove invaluable. Mathematics is inescapable in many natural and social sciences, and with good reason: mathematical models have provided an excellent means through which to assess and predict phenomena in physics, biology, sociology, and countless other disciplines. Those well-versed in the language of mathematics can create functional yet elegant models useful in their studies. In many cases, scientists find ways to make sense of seemingly chaotic and random occurrences in nature, finding order in the tumultuous and chaotic.

In particular, the application of mathematical models to social interactions has yielded excellent outcomes. Economists and sociologists take full advantage of these methods in their studies, but an often-overlooked achievement in the use of computational techniques comes from the biological sciences.

One can’t help but wonder how insects such as ants and termites cooperate so effectively to guarantee the longevity of the nest. Luckily, many biologists with considerable knowledge of applied mathematics have wondered the same.

Among insects, ants display some of the most sophisticated social networks. Studies in ant behavior have yielded incredible discoveries regarding the mechanisms behind their capacity for cooperation in gathering food and caring for their offspring.

As recently as 2015, scientists have been making connections between ants’ movement patterns and established mathematical principles. For example, a study of the collective routes of a species of Argentine ant led to the discovery that the paths the ants follow when gathering food fit statistical distributions of probability. The team of Spanish and American researchers studied the movements of *Linepithema humile*as they explore an empty space.

Scientists used a Petri dish to simulate an open environment for the Argentine ants to explore and tracked the path each ant took through the space. By tracking the individual movement of the ants and aggregating the data to study the overall trends, the team of researchers was able to create models to describe the collective movement of the nest.

The data suggested that the ants’ movement followed Gaussian and Pareto distributions, two probability functions often used by statisticians. Specifically, these distributions accurately predicted the amount the ants turned in either direction with each step as well as the direction in which the ants will travel. The distributions of the Argentine ants’ foraging paths led scientists to the conclusion that two mechanisms are responsible for their behavior: persistence and reinforcement.

As the name may suggest, persistence refers to the ants’ tendencies to follow a linear path and not change direction until they encounter an obstacle. As the insects move through their environment, they mark the paths they have already taken with a pheromone trail, resulting in the “reinforcement” occurring in these areas.

In addition to shedding light on some of the mechanisms responsible for ant behavior, these findings could lead to exciting developments in technological fields. A possible application of the results of the research is in the programming of micro-robots designed to clean toxic areas. The scientists involved in the study of Argentine ants believe the mathematical models associated with colony behavior could be the key to coordinating the movement of the robots.

Another intriguing connection between mathematics and animal behavior can be found in the family tree of bees. Bee hives operate in the same way as ant nests in the sense that they are both collaborative. A large collective of small insects is able to survive and fend off predators through cooperation and task specialization. The assignment of these tasks, however random they may appear, ensures the longevity of the Queen Bee, and therefore the rest of the hive. A close study of the tasks assigned reveals that there is a mathematical pattern hidden in this process that is also found throughout nature: the Fibonacci sequence.

1, 1, 2, 3, 5, 8, and 13 are the first seven integers in this famous and all-pervading infinite sequence. At first glance, the Fibonacci numbers appear unremarkable, and their pattern is simple enough to grasp: begin the recursive sequence with two 1s and attain each successive value by adding the previous two. Only when its applications became known did the Fibonacci sequence gain notoriety. It can be found throughout nature, whether it be in the number of petals in a flower, the spiral patterns of distant galaxies, and, of course, the family tree of bees.

This pattern begins with the task to which a bee is assigned. First, the Queen Bee -- the matriarch at the top of the hive’s hierarchy -- lays her eggs. If an egg is fertilized by a male bee, it hatches into a female worker bee. If not, the egg is destined to hatch as a male “drone”. If one constructs a family tree for a male bee, the first two levels would each consist of one bee: the single parent of the male, that is, the Queen Bee that laid the egg, as well as the drone itself. Since the queen comes from a fertilized egg, she has two parents and the male therefore has two grandparents. The queen’s father has one parent since it was unfertilized, and the mother has two parents. This means the male has three great-grandparents. One great-grandparent is a male with one predecessor, while the other two are female and each have two parents. This brings the original bee’s count of great-great-grandparents to 5. The pattern continues such that the number of great-great-great-grandparents is 8.

At this stage in the reconstruction of the family tree, the sequence of the number of subjects, parents, grandparents, great-grandparents, great-great-grandparents and so on is 1, 1, 2, 3, 5.... This astonishing result is but one example of the Fibonacci sequence occurring in nature.

The uncanny correlation between mathematical models and animal behavior answers as many questions as it raises. While statistical distributions and concepts such as the Fibonacci sequence do provide a useful means through which to study natural phenomena, how are these models so accurate? How do these abstractions from pure mathematics make recurring appearances in the realm of biology? Is mathematics truly the key to unlocking the secrets of the natural world? Regardless of the answers to these questions, mathematical models of animal behavior have yielded fascinating results and promise exciting applications for years to come.

### References

[1] Spanish Foundation for Science and Technology (FECYT). “Ants’ movements hide mathematical patterns.” *phys.org*, 12 May 2015, https://phys.org/news/2015-05-ants-movements-mathematical-patterns.html.

[2] Sandle, Tim. “Ant movements correspond to mysterious math.” *Digital Journal*, 19 May 2015, http://www.digitaljournal.com/science/movement-of-ants-hides-statistical-patterns/article/433598.

[3] “Fibonacci and bees.” *Wild Maths, *n.d. https://wild.maths.org/fibonacci-and-bees.

[4] Horgan, Denis. “Drone parent numbers, Fibonacci sequence (Golden Ratio).” *David A. Cushman, *9 July 2004, http://www.dave-cushman.net/bee/fibonacci.html.

[5] Reich, Dan. “The Fibonacci sequence, spirals, and the golden mean.” *Department of Mathematics, Temple University, *n.d.,https://math.temple.edu/~reich/Fib/fibo.html.