Mathematical formulas to anticipate catastrophic events
Mathematical formulas to anticipate catastrophic events
The bifurcations are a mathematical phenomenon that allows describing qualitative changes in the dynamics of a system when a control parameter changes. For example, in a growth model of a population of bacteria we can have survival or extinction. One of these two states can occur when changing the mortality rate, which acts as a control parameter.
The bifurcations are found in a large number of physical phenomena, chemical reactions, lasers, laboratory experiments with cells, climate models, in mathematical models of ecosystems, and so on. However, in mathematical models, the bifurcations explain the dynamics of the system in the stationary regime, that is, considering its evolution during an infinite time. In natural situations, observable time is always limited.
Now, an interdisciplinary group of scientists from the Autonomous University of Barcelona (UAB), the Center for Mathematical Research (CRM) and the Barcelona Graduate School of Mathematics (BGSMath) (Catalonia, Spain) has found some general formulas that allow describing the bifurcations in a more realistic way, that is, not at infinite time but for finite times, achievable in practice.
"The mathematical formulas identified are universal and will allow us to make very specific predictions about the phenomena we are observing and if they are approaching a bifurcation," explains biologist Josep Sardanyés, one of the three authors of the article.
The biologist Josep Sardanyès, the physicist Álvaro Corral and the mathematician Lluís Alsedà, authors of the study. (Photo: BGSMath)
"For phenomena such as the extinction of a species or climate change," he adds, "we can only observe evolution in a limited time. Thanks to our method, these data are enough for short times to establish whether a given system approaching a change will have a 'soft' bifurcation, that is gradual, or a 'catastrophic' bifurcation, that is, it will reach a point that will generate a abrupt and irreversible phase change ".
In other words, the laws described in this paper will allow to give 'warning signals' through the analysis of finite temporal series, as is the case of those obtained for ecological systems, before an irreversible catastrophic event (a extinction, or an extreme chemical reaction, or the melting of the polar layers, etc.) takes place.
These formulas have universality, that is, although the equation that describes a phenomenon is complicated, if a certain bifurcation underlies it, its description at finite time will be unique and also simple.
The phenomenon of bifurcation also presents 'self-similarity', in such a way that the description at a given time is a 'scaled' replica of what happens to another time. This property is analogous to what occurs in thermodynamic phase transitions, specifically near the so-called critical point.
In the study, published in the journal Scientific Reports, the physicist Álvaro Corral, researcher at the Center for Mathematical Research, the Barcelona Graduate School of Mathematics, the Mathematics Department of the UAB and the Complexity Science Hub of Vienna participated; the biologist Josep Sardanyés, from the Center for Mathematical Research and the Graduate School of Mathematics; and the mathematician Lluís Alsedà, from the Mathematics Department of the UAB and the Barcelona Graduate School of Mathematics. According to the authors, this research, funded by the "La Caixa" Foundation, could also be applied to other dynamic systems. (Source: UAB-CRM-BGSMath)
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