Universidad
Politécnica de Madrid

15.04.2026

What happens when two species coexist in the same environment and are highly competitive with each other? How can mathematics help both species survive? Are there factors that more decisively influence their chances of survival? When we imagine hundreds of species of plants, bacteria, or animals competing for resources, we would normally think that a very complex dynamic would arise, with populations that could even become extinct as a result of this competition. Or that the behavior of different species would become more unpredictable. However, an international study, in which a researcher from the UPM (Universidad Politécnica de Madrid) participated and which was recently published in the journal Nature Communications, demonstrates that we can expect the opposite in highly diverse communities.

The article studies the role of self-regulation (that is, competition within the species itself) in the coexistence of species in large ecological communities, using the Lotka-Volterra mathematical model to describe species dynamics, following the spirit of previous work by mathematical biologist Roberto M. May. The Australian researcher studied the effect of self-regulation on the stability of complex systems, showing that there is a threshold of self-regulation above which system stability is guaranteed. The work of the UPM researchers complements this by taking into account other conditions, necessary for coexistence, not previously considered.

“Robert May’s work demonstrated in the 1970s that, in large and complex ecological systems, stability is lost when interactions between species are too strong or numerous. But his analysis had a significant limitation: May studied the mathematical stability of the equilibrium, but he didn’t verify whether that equilibrium was biologically possible, that is, whether all species had positive abundances,” explains UPM researcher Jose Ángel Capitán, co-author of this study.

How the system changes if self-regulation increases

“Our study examines the combined effect of increasing self-regulation on the stability and feasibility (which means that whether all species have positive abundances) of the Lotka-Volterra dynamics equilibrium points. Therefore, analyzes how the system changes when self-regulation (the “internal brake” of each species) increased, trying to determine not only whether the resulting equilibrium is stable, but also whether it is feasible,” he adds.

Specifically, the study answers several questions. The first question is whether there is a critical level of self-regulation that guarantees an equilibrium in which all species survive with positive population numbers. Furthermore, it calculates the probability of this equilibrium existing in systems with random interactions, which can be treated mathematically.

The study also analyzes the relationship between the feasibility threshold (the minimum level of self-regulation that allows all species to have positive population balances) and the stability threshold (the point at which the system's equilibrium becomes stable). “Before this threshold, even if an equilibrium exists, it is fragile: if one population undergoes a small change, the populations can change drastically. By studying this relationship, the study investigates what dynamic behavior (stable equilibrium, extinctions, oscillations, or chaos) can be expected in complex communities under these hypotheses,” explains UPM researcher José Ángel Capitán.

“The study demonstrates that there is a critical level of self-regulation above which a biologically feasible equilibrium is guaranteed (where all populations are strictly positive), and that, in large systems with random competitive interactions, the threshold that ensures dynamic stability is reached before the feasibility threshold. Or, in other words, it shows that the system becomes stable before all species can coexist. This means that if an equilibrium is ever reached where all species are present, that equilibrium will almost certainly be stable,” says the UPM researcher.

Furthermore, the study demonstrates that even if initially not all species can coexist and extinctions occur, the dynamics gradually eliminate species from the community, maintaining the same order among the thresholds (first the stability threshold, and then the feasibility threshold), ultimately leading to a subset of species that coexist robustly in equilibrium. In other words, the community is not destabilized by the extinction of some species but rather adjusts until a set of species remains that can coexist stably.

Consequently, the study predicts that large, competitive communities tend to be stable and that coexistence through chaotic cycles or oscillations is highly improbable. “Large ecological communities tend to stabilize themselves, eliminating species that don't fit in and ending up in a robust equilibrium,” concludes the UPM researcher.