The mathematics of mass extinction

With the exception of the asteroid-induced Cretaceous–Paleogene event 66 million years ago, mass extinctions tend not to have clearly delineated start-points. Investigating how and why they happen, therefore, is a challenging business.

Now mathematicians and biologists have joined forces in an attempt to identify the factors and forces that combine to tip broadly stable ecosystems into chaos.

In a paper in the journal Science, the team, led by mathematicians Sergei Petrovskii and Andrew Morozov from the University of Leicester in the UK, review records of sudden ecological transitions that occurred in ancient history and more recent periods, looking for common factors.

Broadly speaking, mass extinctions occur when these comparatively rapid transitions occur. The changes arising – and cascading – from deviations from stable initial conditions produce significant alterations to biodiversity and the consequent collapse of food webs, resulting in widespread species loss.

What prompts these changes, however, remains a mystery. Most mass extinctions lack an easily identifiable catalyst.

Combining ecological theory, empirical data and mathematical modelling, Perovskii, Morozov and colleagues identify common factors in what they term the “transient dynamics” that shift a system between one state and another.

The result, they write, is “that hitherto idiosyncratic and individual patters can be classified into coherent framework, with important lessons and directions for future study”.

One of the critical forces involved, they say, are “ghost attractors” – a term that does not, despite first impressions, imply supernatural involvement.

The idea takes it start from the standard mathematical concept of an “attractor” – understood to be a set of numerical values toward which any dynamical system tends to evolve. The attractor, thus, represents the stable end-point of a system, found after a very long, even infinite, span of time.

It is a robust phenomenon – well able to withstand small perturbations and return to its original state.

A ghost attractor, in contrast, behaves in the same manner, but only for a shorter, finite period of time. At its conclusion, the system will experience a rapid transition into a different state, which may express very different properties – resulting in catastrophic biodiversity loss.

The researchers also identified another phenomenon, which they dubbed a “crawl-by”. They defined this as an eventually catastrophic change to the dynamism of a system, but one which happens over a protracted period.

“Our research shows that a healthy ecosystem will not necessarily remain healthy, even in the absence of any significant environmental change,” says Petrovskii.

“Therefore, better monitoring of the state of an ecosystem is required to mitigate potential disasters.

“We also can predict an approaching catastrophe in the sense that our study advises where to look for its signs and what is the relevant time scale: the environmental change (whether it is natural or man-made) that will finally lead to big changes might have happened a very long time ago.”

Mass extinctions in the present period, he adds, “may be a debt that we have to pay for the actions or mistakes – for example unsustainable use of natural resources – made many generations ago.”

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