Mary Lou Zeeman's MAA talk is featured in MPE2013 blog

Mathematics of Tipping Points

A lake that used to be clear, with a rich vegetation and a diverse aquatic life, suddenly becomes turbid, with much less vegetation and only bottom dwelling fish remaining. It turns out that the change comes from increased nutrient loading, but when the runoff leading to the nutrient inflow is reduced, the lake doesn’t become clear again – it remains murky.

A dry land area with patchy vegetation becomes completely barren after an especially dry season, but when normal rain patterns return, it remains a desert.

An entire planet that used to have varied climate zones, ranging from tropical areas to icecaps near the poles, freezes over completely, perhaps due to variations in the solar energy output, with all oceans frozen except near some thermal vents and all continents covered by thick ice sheets. When the solar output increases again, the planet remains in its frozen state.

These are examples of transitions of ecological systems past “tipping points” – the subject of a fascinating talk given by Mary Lou Zeeman on March 28 of this year in the Carriage House lecture hall of the Mathematical Association of America (MAA) in Washington, DC. Mary Lou, one of six children of the well known British mathematician Sir Christopher Zeeman, is a professor of mathematics at Bowdoin College and works on dynamical systems, with applications in ecology and biology. The Carriage House auditorium was full when she gave her talk. The audience included students, residents of the Washington area who are interested in science, and local mathematicians – just the ecological mix that the MAA lecture series tries to achieve.

There is a commonality to all these scenarios that can be described with mathematical methods from bifurcation theory. Mary Lou used the “Snowball Earth” scenario of the third example to illustrate this. According to geological evidence, this “mother of all tipping points” actually occurred on Earth not just once, but several times about 600 million years ago. Each complete glaciation lasted many millions of years and ended only when carbon dioxide in the atmosphere accumulated due to volcanic emissions to levels which were much higher than today, leading to a monstrous greenhouse effect and a rapid transition from “snowball” to “hothouse” Earth. Mary Lou presented a fairly simple energy balance model that is capable of explaining the fact that both a moderate and a frozen climate state are possible and stable on the same planet, with the same solar output. These different climate states are possible since a planet with a moderate climate tends to have a low albedo (most of the sunlight is absorbed by oceans and continents and keeps the planet warm) while a frozen planet has a high albedo (sunlight is reflected back by ice packs and snowfields, keeping the planet cold). The model is flexible enough to explain also the transitions between “snowball” and “hothouse” states. Intriguingly, the so-called Cambrian explosion, during which many of today’s animal phyla first appeared, happened not long after these snowball episodes.

Relatively simple mathematical models offer common explanations of such multiple stable states. Transitions between such states tend to be rapid and surprising, which is a scary thought. Mathematical insights can also lead to better detection mechanisms for such transitions and even suggest experiments to assess the resilience of an ecological system against random perturbations. For example, near such a transition point, such a system will return to its stable state more slowly after a perturbation, and its response to such a perturbation will also show more variance. Mary Lou specifically pointed to the work of Marten Scheffer and his co-authors on early warning signs for such critical transitions (Nature 2009, Science 2012).

The mathematical sciences therefore can contribute to decision support for managers and policymakers. The speaker suggested that when ecological systems are observed and managed for sustainability, such a goal should include resilience. In mathematical terms, this means one should not just identify stable equilibrium states but also understand the “size” of their basin of attraction and their sensitivity to changes of external parameters.

And here’s another term that I remember from this talk: Mathematical scientists should show “interdisciplinary courage” and instill this in their students. This includes not just a willingness to learn the language and problems of another discipline. In the privacy of their office, mathematicians are already used to dead ends and unsuccessful attempts before coming up with good ideas. As members of interdisciplinary reserach teams, they also need to risk having “bad ideas in public”. That’s a resilience that all of us should acquire.

Hans Engler
Department of Mathematics and Statistics
Georgetown University
Washington, DC 20057

Originally posted at


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