A study published in the Journal of Statistical Mechanics: Theory and Experiment (JSTAT), inspired by the behavior of bacterial colonies observed in previous experiments, uses mathematical models to describe the expansion of competing populations. The aim is to understand which factors determine the success of one population over another.
According to the study, fitness — understood here as a reproductive advantage — is not the only factor involved, nor is it necessarily the decisive one: expansion speed and position along the growth front also matter. The shapes of the expanding front predicted by the model closely resemble those observed in earlier experiments of microbial colonies.
The study also makes a new prediction: that spatial structure leaves a measurable fingerprint on how fitness is distributed across a population, which new experiments could look for.
Combining equations from biology and physics
“The work was inspired by experiments on expanding bacterial colonies, in which populations that were initially mixed eventually separated into distinct sectors,” explains Sergio Eraso, an MIT PhD student and co-author of the study with MIT physicist Mehran Kardar.
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What particularly caught the researchers’ attention was that the boundaries between these sectors did not simply wander at random: they moved faster than ordinary diffusion would predict, following the mathematical scaling predicted by the KPZ equation, introduced in 1986 by Mehran Kardar, Giorgio Parisi and Yi-Cheng Zhang, to describe how rough growth fronts evolve over time. Kardar received the 2025 Boltzmann Medal for his contributions to nonequilibrium statistical physics, including his work on KPZ.
One way to picture this is to imagine setting fire to one edge of a sheet of paper. As the flame advances, the burning edge does not remain straight but develops an irregular shape. Something similar happens in bacterial colonies, where cells do not all reproduce at the same time. This makes the growth front uneven and affects the movement of the internal boundaries.
“In this work, the central model combines the KPZ equation with the Fisher equation, a classic model in evolutionary biology that describes how a population with a selective advantage spreads through space,” Eraso continues.
Competitive advantage and expansion speed
In the model, Eraso and Kardar distinguish between competitive advantage, or fitness, and expansion speed. The two properties may be related, but they do not necessarily coincide: a population may prevail in local competition while advancing more slowly.
Depending on the balance between these factors, the model produces three different front shapes: a rounded bulge, a composite bulge with sloping sides, or a V-shaped dent. In the last case, the competitively stronger population continues to invade its rival from the sides, but falls behind at the center because it expands more slowly.
“It may seem surprising, but what we find is that greater expansion speed does not necessarily lead to greater success in colonizing new territory,” Eraso explains.
In the right place at the right time
The study also shows that a less fit population, which would disappear on a flat front, may persist if it occupies a favorable position, such as a peak or protrusion. In the model, these initial positions can create niches that remain dominated for long periods by populations with lower fitness.
“You could say that the ancestors happened to be in the right place to ensure the success of their descendants,” Eraso says. “Had the same population started from a less favorable position, its adaptive disadvantage might have led to its extinction.”
Similarities with experiments
Eraso and Kardar’s model is theoretical and simplified, but it produces bulges and dents that closely resemble those observed in earlier experiments on microbial colonies.
“This does not mean that we have proved that the underlying mechanism is exactly the same, but it shows that only a few ingredients — differences in competitive ability and expansion speed — are enough to reproduce these shapes qualitatively,” Eraso explains.
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The next step will be to test experimentally under which conditions the same mechanisms operate in real populations. “We tried to identify the minimum number of ingredients needed to reproduce these shapes,” Eraso concludes. “Experiments can now test how far the same mechanisms are at work in real populations.”
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