They discover a “chaos butterfly effect” in swarms and herds

Researchers at the Carlos III University of Madrid (UC3M) and the Complutense University of Madrid (UCM) discovered a phase shift between chaotic states which may appear in herds of animals and, in particular, in swarms of insects. This advance could help to better understand their behavior or be applied to the study of cell or tumor movement.

A phase shift Occurs when conditions in a system change dramatically, for example, when water changes from a liquid to a solid state upon freezing. In this research, recently published in the journal Physical Review Ethis group of mathematicians found such a phenomenon in swarms.

“The swarm insects are in a limited volume, even if they are in a park or in an open space. To explain this, we assume that there is a harmonic potential, a kind of restoring force that confines them (like that of a spring that tries to return to its resting position when we stretch or contract it)”, explains one of the authors of the study, Luis L. Bonilladirector of the Gregorio Millán Barbany Institute at UC3M.

This confinement of insects responds to a constant proportionality between force and displacement. The researchers verified that for low confinement values, the movement of insects in the swarm is chaotic (their movements change a lot if the initial conditions are changed).

In this context, the phase change occurs when the swarm is divided into several that, however, are closely related to each other, as there are insects that pass from one to the other. On the critical line between phases of this change, the distance between two insects in the swarm that are influenced by each other is proportional to the size of the swarm, even though the number of insects in the swarm grows indefinitely. This is called “rolling chaos” and has not been discovered until now, researchers say.

“As insect numbers grow, the critical line moves towards zero confinement. What happens is that the maximum distance between two insects that still feel each other’s influence is proportional to the size of the swarm. It doesn’t matter how many bugs we put in it. And this represents an absolute novelty that we have discovered”, points out Bonilla.

Chaotic behavior without scale

Specifically, what these mathematicians predict through numerical simulations is that certain swarms of insects (specifically a class of small flies) have a chaotic behavior without scale, which translates into certain power laws with exponents similar to those that have been measured in nature. They also found a simplified mean field theory that supports the phase shift of scaleless chaos.

“It would be good to look for and find this phase shift between chaotic phases that we predict, either in observations in the middle of nature or in controlled laboratory studies”, says another of the research authors, UCM mathematician, Rafael Gonzalez Albaladejo.

The formation of herds is one of the manifestations of the so-called “active matter”, composed of something like self-propelled individuals that make up a whole, explain the researchers. It could be a swarm of insects, a flock of sheep, a flock of birds, a school of fish, but also moving bacteria, melanocytes (the cells that distribute pigment in the skin) or artificial systems like irregular grains or seeds. .

“The herd formation mechanisms play a role in some of these systems, so the results we got can be related to biology, to the study of cells and, in addition, to the study of tumors and other diseases”, adds González.

move accordingly

How do so many animals move together in unison? These researchers explain that each individual feels only its neighbors and moves accordingly, even if it does not have a perspective of the movement of the entire herd. And depending on whether they use vision, hearing or the vibrations of the fluid in which they are immersed, the concept of neighbor can change a lot. Some sheep moving together see and sense those around them, while some birds in a flock see their nearest neighbors even if they are quite far apart.

“O de move accordingly, it could mean that they move in the same direction as their neighbors (generally) or they can adopt different strategies depending on the situation. For example, if a crowd is trying to get out of a crowded area with more than one doorway, there are times when not following neighbors is advantageous.”

It took these mathematicians about two years to carry out this research work. At first, they planned to explain some experiments that study the conventional phase change between an infinity of insects that fill a space with constant density and that are ordered when a critical value of the control parameter passes (for example, when the noise decreases). But then they decided to add a harmonic potential to confine the swarm and explore what happens when the force of attraction between individuals decreases.

“We found many periodic, quasi-periodic and finally chaotic states for a fixed number of insects that were increasing. What is surprising is the transition between chaotic states that we did not know or supposed to exist and we manage to find the correct arguments and tests to support their existence”, indicates another of the study’s authors, Ana Carpiofrom the UCM Department of Mathematical Analysis and Applied Mathematics, which specifies that there is still much to be done based on this work.

“From the search for experimental confirmation of our predictions and better adaptation of the model to experimental observations, to carrying out theoretical and mathematical investigations that go beyond our numerical simulations”, he concludes.