Home » Why Animals Form Patterns in Nature: Exploring the Mathematical Underpinnings of Collective Behavior

Why Animals Form Patterns in Nature: Exploring the Mathematical Underpinnings of Collective Behavior

Nature is full of intricate and fascinating patterns, many of which are formed by the collective behavior of animals. From schools of fish that move in unison to flocks of birds that soar through the sky in perfect formation, these patterns have captivated scientists and nature enthusiasts alike for centuries. But why do animals form these patterns, and what mathematical principles underlie their behavior? In this blog post, we’ll explore the science behind collective animal behavior and the patterns it creates.

Collective Behavior in Animals: An Overview

Collective animal behavior refers to the actions of groups of animals that interact with each other in a coordinated manner. This can take many forms, from the synchronized movement of a school of fish to the rhythmic flapping of a flock of birds. The study of collective behavior is a relatively new field, but it has already yielded some fascinating insights into the natural world.

One of the most striking examples of collective animal behavior is the synchronized movement of schools of fish. This behavior, known as shoaling, is seen in many different species of fish, including sardines, herring, and anchovies. When these fish school, they move together in a coordinated manner, often spiraling or forming other intricate patterns.

Flocking birds rise in formation through the sky

The formation of flocks of birds is another example of collective animal behavior. Birds that flock together often fly in a V-shaped formation, with each bird taking advantage of the upwash of air created by the bird in front of it. This reduces the amount of energy each bird needs to expend in order to fly, allowing the flock to cover greater distances with less effort.

The Mathematical Underpinnings of Collective Animal Behavior

While the behavior of animals in groups may seem random and chaotic, there are actually mathematical principles that underlie their actions. One of the most important of these is the concept of self-organization.

Self-organization refers to the way that complex patterns can emerge from the actions of individual agents that follow simple rules. In the case of collective animal behavior, the agents are the animals themselves, and the rules they follow are often based on their interactions with their neighbors.

For example, in the case of shoaling fish, each individual fish follows a set of simple rules, such as “stay close to your neighbors” and “move in the same direction as the group.” When all of the fish follow these rules, a complex pattern emerges, as the group moves together in a coordinated manner.

Schools of sardines forming large swirling patterns

Similarly, in the case of flocking birds, each bird follows a set of rules that allows it to stay in formation with the rest of the flock. These rules include “stay close to the bird in front of you” and “match the speed and direction of the flock.” When all of the birds follow these rules, a V-shaped formation emerges, with each bird taking advantage of the upwash of air created by the bird in front of it.

Another important mathematical concept in collective animal behavior is synchronization. Synchronization refers to the way that the behavior of individual agents can become coordinated over time, leading to patterns that are more complex than the actions of any individual.

For example, in the case of fireflies, each individual firefly flashes its light at a slightly different rate. However, over time, the fireflies become synchronized, with all of them flashing their lights in unison. This creates a striking pattern that is more complex than the behavior of any individual firefly.


The patterns created by collective animal behavior are a testament to the power of self-organization and synchronization. While the behavior of individual animals may seem simple, when many animals come together and follow simple rules, complex patterns can emerge.

As scientists continue to study collective animal behavior, they are uncovering new insights into