Mathematical law could anticipate the possibility of mass extinctions in ecosystems

09/10/2018

Bifurcations are a mathematical phenomenon which allow describing qualitative changes within the dynamics of a system when one of its control parameters changes. For example, in bacteria population growth there can be survival or extinction; one of these two states can appear after a change in the mortality rate, which acts as a control parameter.

Bifurcations can be found in a large amount of physical phenomena: chemical reactions, lasers, laboratory experiments with cells, climate models, mathematical models of ecosystems, etc. However in mathematical models bifurcations explain the dynamics of a system under a steady state condition, i.e., considering its evolution in an infinite time frame. In natural situations, observable time is always limited.

An interdisciplinary group of scientists formed by a mathematician, a physicist and a biologist from the Universitat Autònoma de Barcelona (UAB), the Mathematics Research Centre (CRM) and the Barcelona Graduate School of Mathematics (BGSMath), funded by the “la Caixa” Foundation, has discovered general formulas which can be used to describe realistic bifurcations, not in an infinite time frame but in a finite one, obtainable in practice.

“The mathematical formulas identified are universal and allow us to make very specific predictions on the phenomena we observe and whether there will be a bifurcation in the near future”, explains Josep Sardanyés, one of the three authors of the study. “For phenomena such as the extinction of a species or climate change, we can only observe evolution within a limited time frame. Thanks to our method, all we need are these short-term data to establish whether a system is nearing a change which will produce a 'soft' or gradual bifurcation, or a 'catastrophic' one, in which a point is reached when an abrupt and irreversible change occurs”.

In other words, the laws described by researchers in this study can aid in transmitting warning signals through the analysis of short time series, as is the case of those obtained by ecological systems, before an irreversible catastrophic event (extinction, extreme chemical reaction, melting of the polar caps, etc.) takes place.

These are universal formulas, i.e., although the equation describing a phenomenon is complicated, if there is an underlying bifurcation, its description in a finite time frame will be unique and simple.

The bifurcation phenomenon also shows a "repetition" of sorts, in which the description made at a given time is a “scaled” replica of what happens at another time. This property is analogous to what is seen in thermodynamic transitions, and specifically when approaching what is known as the critical point.

The study, published in the journal Scientific Reports of the Nature group, included the participation of Álvaro Corral from the Mathematics Research Centre, the Barcelona Graduate School of Mathematics, the UAB Department of Mathematics and the Complexity Science Hub Vienna; Josep Sardanyés from the Mathematics Research Centre and the Graduate School of Mathematics; and Lluís Alsedà from the UAB Department of Mathematics and the Barcelona Graduate School of Mathematics. The research has received funding from the “la Caixa” Foundation.

Original article:
Álvaro Corral, Josep Sardanyés and Lluís Alsedà "Finite-time scaling in local bifurcations" Scientific Reports (2018) 8: 11783 DOI: 10.1038/s41598-018-30136-y