Tipping points in mathematics, ecology and geography

Today in front of the abyss, tomorrow one step ahead?

12. April 2023 by Daniel Schenz
All complex systems can reach critical threshold values. If they are exceeded, this usually has grave and irreversible consequences. Understanding these tipping points is essential to saving the relationship between the planet and humans. We have interviewed the mathematicians Sara Merino-Aceituno and Henk Bruin, the ecologist Christina Kaiser and the geographer Harald Sterly.
If certain tipping points are reached in complex systems, there is often no way back. At the University of Vienna academics from different disciplines are conducting research to understand and identify these tipping points. © Scott Goodwill (via Unsplash)

Caterpillars feeding on leaves from a tree and getting eaten by birds – these are the basic ingredients of a famous complex system, which was modelled by US-American researchers in the 1970s. Through this model, the concept of tipping points found its way into ecosystem biology. Be it a food chain, an ecosystem or the planet: All complex systems, to which also humans themselves belong, have commonalities.

First, they consist of clearly separable components that are characterised by interrelationships (trees, caterpillars, birds, etc.). Then, complex system have so-called states of equilibrium. These are states which are constant for longer periods, in which 'forces' in the system are balancing each other and minor changes are corrected: If caterpillars are reproducing too much, birds are detecting them more easily and the 'surplus' is eaten. If the caterpillars are too few, searching for them does not pay for the birds and the caterpillars can recover.

A brownish caterpillar is crawling on a coniferous tree's twig.
The eastern spruce budworm (Choristoneura fumiferana) feeds on the needles of spruce and fir trees. Roughly every 50 years, the eastern spruce budworm populations rapidly increase, resulting in wide-scale tree mortality in the Great Lakes region in North America. This phenomenon is the object of a famous model created by researchers at the University of British Columbia in the 1970s. Through this model, the concept of tipping points found its way into ecosystem biology. © Jerald E. Dewey, USDA Forest Service (via Wikipedia) CC BY 3.0

And third, these systems have tipping points: If a part of a system exceeds a certain threshold, the entire system is apparently changing massively and rapidly. If the trees on which the caterpillars are living are becoming very tall, the caterpillars remain hidden from their predators and reproduce massively. In the end, the number of budworms grows so high that the birds cannot keep up eating them and the caterpillars devour the entire forest. The system has tipped over.

As reaching tipping points in complex systems is often accompanied by diseases, catastrophes and destruction, for example through flooding, drought, species extinction and their social consequences, such as hunger, epidemics and war, we often try to avoid such a scenario. Also social transformations, such as the transport transition or energy transition, are characterised by tipping points. Regardless of whether we regard certain tipping points as positive or negative, we have to understand and be able to predict them if we would like to influence complex systems.

Looking at the world through the eyes of a mathematician

This is where mathematics comes into play. Because: "If you know the equation, you can understand a system's behaviour. And then it does not matter at all whether you are dealing with caterpillars, the climate or movements of groups," explains Sara Merino-Aceituno. The mathematician is conducting research at the University of Vienna addressing emergence, i.e. the question of how the interaction of elements within a system are generating larger structures, for example, how the behaviour of individual birds results in the movement of a flock.

"The wonderful thing about mathematics is that it is so versatile so that you can use the same tools for, at first glance, completely different systems," the scientist describes her special 'mathematical perspective' at the world.

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© Amalio Fernández-Pacheco
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Sara Merino-Aceituno is convinced that academic research is indispensable, but not sufficient for a healthy planet: "Many people do not pay heed to academic findings. On the other hand, there are people who evade responsibility by relying on technological developments not yet developed. But we need both science-based politics and the active participation of society. Science cannot invent a pill allowing humans to live an unhealthy life while remaining healthy."

Merino-Aceituno is applying her research in the area of kinetic theory to the investigation of new phenomena in biology, medicine and social sciences. She obtained her PhD degree at the University of Cambridge. Since 2018, she is working at the Faculty of Mathematics at the University of Vienna.

All complex systems have one thing in common, the so-called states of equilibrium, which depend on certain parameters, Merino-Aceituno explains: "Parameters are values that are determined by the environment and that usually do not change. If they do change, they do so much slower than the system variables. They are the adjusting screws that control the development of the system".

A blurred crowd of humans moving in an underground passage. On the right side, people are moving into the picture, on the left side out of the picture.
Pedestrian flow as an example of an equilibrium in emergent systems: As soon as the number of people exceeds a certain threshold, self-organised lanes for the different directions develop. Sara Merino-Aceituno explains, "This is a good example of the importance of parameters for the behaviour of an emergent system. If parameters, such as the density of people, are changing their value, the behaviour of the system can be completely different, even if the equations are exactly the same." © Martin Adams (via Unsplash)

The way back is much longer than the way there

And these parameters are playing a central role for another property, which is important for understanding complex systems: the so-called hysteresis. This is the inability of a system to return to its original configuration after a change caused by a small change of the parameters, i.e. reaching the tipping point, by a minor change back. On the contrary, one would have to take the parameters back many steps to bring the system to its initial state. Merino-Aceituno adds, "These reversals are thus possible from a mathematical and theoretical point, but in practice they usually are not".

Her colleague from mathematics, Henk Bruin, who is also working as a mathematician at the University of Vienna and specialises in researching dynamical systems,o explains why this is not possible based on the example of the Antarctic ice: "A warming of the atmosphere by one or two degrees Celsius can result in losing the West-Antarctic Ice Sheet. However, to get back the perpetual ice, one would have to cool down the Earth by a significantly higher temperature difference."

Melting, already transparent blocks of ice in the sea glistening by the sun. In the background, snow-capped mountains.
If the Earth warms by 1-2°C, the West-Antarctic Ice Sheet will melt. However, cooling it by 1-2°C will not bring it back. © Jeremy Ricketts (via Unsplash)

We have already seen an example for hysteresis in the beginning illustrated by the caterpillars: Only when the forest is bare and a massive number of caterpillars are subsequently die off, the system can recover, with small trees on which birds can quite easily find surplus caterpillars.

Simple preconditions for complex behaviour

Henk Bruin is writing a very simple equation on a sheet of paper (see figure) and explains, "This is the normal form for systems with such tipping points. It is very simple and only has one variable and two parameters. Mathematical models in climate research, ecology and also in economy, however, have numerous variables and parameters. In these complex systems, it is almost certain that you will find a combination of parameters at which a tipping point is reached.

Hand-written notes on a pad: "Normal Form for the Cusp Bifurcation. du/dt = r + ku + u^3. Stationary points du/dt = r + ku + u^3 = 0 (1). Saddle node bifurcations when the derivative of the RHS k + 3u^2 = 0. (2)"
The normal form of a system with tipping points, i.e. cusp bifurcation to use the proper mathematical term. It contains only one variable (u, e.g. the number of caterpillars) and two parameters (r and k, e.g. the size of the trees and the predation by birds). Henk Bruin explains, "Mathematical models in climate research, ecology and also in economy, however, have numerous variables and parameters. In these complex systems, it is almost certain that you will find a combination of parameters at which a tipping point is reached. © Hand writing: Henk Bruin. Image: Kelly Sikkema (via Unsplash)
© Barbara Mair
© Barbara Mair
Henk Bruin demands a global approach to our way of handling greenhouse gas emissions and waste of any kind. For their avoidance and processing, we should also consider countries with a weaker economy and financial situation. Henk Bruin says, "The fact yielding my hope a little is that we have done something about CFCs and the ozone layer depletion. Hopefully, new biochemical technologies also result in better opportunities for processing plastic waste."

Henk Bruin obtained his doctoral degree at the Delft University of Technology. Since 2012, he has been working at the Faculty of Mathematics at the University of Vienna. He is specialising in dynamical systems, ergodic theory, complex dynamics and continuum theory.

But what if you do not know a system's equation? The true stroke of genius in good models is often not the interpretation of an equation but identifying which factors play a role at all. Often, we do not yet know these factors, neither in highly complex systems, such as the global climate, nor in still very complex subsystems, such as the ecosystem of the soil. How can we then identify tipping points, ideally before we reach them?

Microbial communities are determining the health of our planet

An expert who can answer these questions is Christina Kaiser. She is one of the Directors of the new Cluster of Excellence "Microbiomes Drive Planetary Health" at the University of Vienna and is investigating microbial communities as dynamical systems.

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"We cannot know the ideal equation for ecosystems. For this, our measured data are too incomplete. And often we do not even know which factors play which role," says Christina Kaiser about the challenges in the field of ecology, especially in describing global nutrient cycles. This lack of clarity is an especially serious problem where processes at the smallest scale in the soil – so small that the maze-like spatial structure of the soil is becoming the process-determining factor – are linked closely with the truly planetary systems, such as the large permafrost areas, the forests and oceans.

Tipster for tipping points

There are very clear indications for when a system is approaching one or several tipping points: Close to a tipping point, the 'forces' that keep a system in a state of equilibrium are weaker. Therefore, it is becoming harder and harder for a system which is subjected to more or less random fluctuations to regain a state of equilibrium. Christina Kaiser is referring to this as 'critical slowing down' of systems.

At the same time, these fluctuations also make it possibole to jump to other states of equilibrium. Systems close to a tipping point therefore often switch back and forth between two states of equilibrium – a behaviour for which ecosystem researchers are using the term 'flickering'. "This phenomenon is even more interesting when we talk about spatial patterns," says the ecologist, "There, close to tipping points very regular, maze-like patterns form."

A raster (6x4) of simulation images. The left images and images at the bottom are showing the noise of many small points. The others are showing spiral and curved lines of densely organised points on an otherwise uniform background.
Graphical output of a computer model simulating microorganisms which are degrading organic substrate together (for example, in the soil): Christina Kaiser explains, "Along two independent parameter gradients (shown on the x and y axis), there are points at which the system suddenly transitions from an homogeneous distribution of microorganisms to a state in which the microorganisms organise themselves in patterns. In the simulation, this also has massive consequences for the internal processes and nutrient cycles in the system." © Christina Kaiser

And we can actually observe all this in real ecosystems, according to Christina Kaiser. For example, in historical data: 34 million years ago, the Earth abruptly changed from a so-called greenhouse climate to the current icehouse climate, which is characterised by the glaciation of Antarctica. And just before this leap, so-called proxy data, i.e. preserved traces and remains based on which we draw conclusions about the past temperatures, show that temperature fluctuations at the time increasingly became more 'sluggish'.

© Pamela Nölleke-Przybylski
© Pamela Nölleke-Przybylski
Christina Kaiser is researching the mechanisms underlying the large cycles on Earth driven by microbial communities. She is convinced that we have to use our understanding of these mechanisms to keep the ecosystems, if still possible, in their current states of balance to allow for a good life for all (still remaining) living beings.

Christina Kaiser completed her doctorate at the University of Vienna. Following positions at the University of Western Australia and at the International Institute for Applied System Analysis, she has been Group Leader since 2014, and associated professor since 2022 at the Centre for Microbiology and Environmental Systems Science. She is a member of the Environment and Climate Research Hub.

Sujet Environment and Climate Hub an illustration of a hand holding a tree surrounded by birds

Environment and Climate Research Hub (ECH)

Christina Kaiser is a member of the Environment and Climate Research Hub (ECH), the new multidisciplinary research network within the University of Vienna. It is dedicated to connecting researchers addressing environment, climate, and sustainability from different academic viewpoints. More about the objectives of the network.

Currently, the ECH has 65 members from different faculties and departments of the University of Vienna. All of them carry out research in the field of environment and climate.

In complex systems, the details determine the state of the whole

"Understanding these dynamics better is essential to take countermeasures," says Christina Kaiser about her motivation to understand soils as complex systems, where often tiny details emerge as game changers for the entire whole. "Because for a healthy planet, it is important to retain systemic balances so that all species can live well on it."

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In the end, crossing ecological tipping points results in catastrophic changes that make good life on Earth completely impossible. "To prevent such a scenario, it is high time to counteract resolutely and quickly, which can only be achieved by means of massive social and political change," the ecologist emphasises.

Habitability as an interactive property

Massive social and political change are imminent in any case, also in the worst-case scenario of our driving ecosystems beyond their tipping points. At the Department of Geography and Regional Research, Harald Sterly is addressing the question of how environmental changes are influencing the habitability of regions and how humans react to them.

In the project HABITABLE, Harald Sterly is asking very basic questions, such as: What does "habitable" actually mean? "A dry savannah is indeed inhabitable for rice paddy farmers, but the nomadic pastoralists are well adapted to these conditions and can survive there rather well," starts the human geographer on the question of the definition: Conversely, in eastern Germany, for example, there are many areas in which agriculture is undoubtedly possible but which are increasingly depopulated. "Habitability is a property defined by the interaction between humans and a place," says Sterly.

© Harald Sterly
© Harald Sterly
Harald Sterly addresses regional and social aspects of the nexus between climate and environmental changes and different forms of mobility and migration. For a healthy planet that enables healthy life, his wish is: Global solidarity in addressing the climate crisis; a reorientation away from the focus on financial wealth to a life in interpersonal relationships; and abandoning the idea of constant growth.

In his doctoral thesis, Sterly wrote about mobile communication and translocal social relationships of migrants between the countryside and cities in Bangladesh. He then coordinated a research project about megaurban dynamics and informality in Bangladesh and China as well as the interrelations between migration, translocal connections and social resilience in Thailand. Now he is working as a human geographer at the Department of Geography and Regional Research of the University of Vienna. The behaviour is clear, its interpretation difficult.

Despite all ambiguity surrounding their definition, we can clearly observe tipping points in many societies. According to Harald Sterly, these can be observed easily, if behaviours, such as emigration, birth rates, life styles, etc. suddenly exhibit profound changes. "However, whether these changes are good or bad, is also a question of prerogative of interpretation," adds Sterly and warns us of a settlement bias in this context. Usually, we are basically considering migration and mobility as something extraordinary in contrast to settling down. Sterly says, "We are inclined to perceive emigration always as something negative. However, in many high-risk areas, people would often indeed like to move away, but the conditions do not allow for that." Therefore, also immobility can be a problem, especially in view of the deteriorating environmental conditions.

On the other hand, there is also emigration which is not motivated by the emigrants themselves but which is initiated by politics. For example, the Philippine government, after the typhoon Haiyan in 2013, has assessed some areas as 'insecure' and prohibited the evacuated population to return to these areas.

The systemic and the political perspective

"Social systems partially follow other principles than, for example, ecological ones," explains Sterly, "For these human systems, power, symbols and discourse play a significant role." The researcher from the University of Vienna calls attention to the fact that a systemic perspective in fact makes sense in geography but it additionally needs a political and ecological perspective, which foregrounds conflicts and power dynamics.

An image of a wide landscape. Hills used for agriculture. The soil is brown and bare. At the bottom, terraces, as used for growing rice, are still visible but here no rice is growing.
Harald Sterly illustrates the fact that social and ecological tipping points can be entangled by means of cornfields in north-western Thailand: "Here, we see the effects of educational migration, among others, on the landscape and on agriculture: As young people are moving to cities for their education and university studies, more and more people are forced to change from traditional and sustainable land use practices to the cultivation of corn. But the change back is then hardly possible due to economic dependencies on traders of fertilisers and seeds." © Harald Sterly

At least in democratic systems, which are shaped by 'competing mainstream narratives', this anyhow leads to new tipping points, referred to as transitions, especially where previously probably dismissed opinions and niche interests suddenly gain ground among a majority and shap the political discourse. The examples given by Sterly are the availability of vegan products in the supermarket, riding a bicycle in the city or the wide-spread installation of photovoltaic systems. Eventually, a point is reached, where it is suddenly interesting from an economic point of view or is just accepted by society to build up the necessary infrastructures.

For Christina Kaiser, these social tipping points are a reason for hope. "We are currently running the risk of failing all our efforts, first of all, the 1.5-degree target, because there are so many opposing forces despite very important movements which created much pressure, a lot of attention and also much real sympathy. If we reach such a 'positive' social tipping point and then maybe it also becomes an economic issue, it could be that changes go quickly and take hold."

The next step is certain

Tipping points are all around us and there is no alternative to crossing some of them. At the University of Vienna, academics from different faculties are conducting research to understand what is needed to still change current developments. Sara Merino-Aceituno, Henk Bruin, Christina Kaiser and Harald Sterly agree that this is still not enough. In addition to the academic findings, we also need a political consensus over the severity of the situation and a broad part of society participating in tackling the crisis. Here, especially, there are still major frictional losses.

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It is now a question of reaching social tipping points. And according to the four experts from the University of Vienna, the questions we have to face include the following: Can we muster the same political will, with which we were able to overcome the COVID-19 pandemic, quickly enough to avoid the most dangerous developments of environmental change and to cushion the worst consequences? And in doing so, will we also manage to consider people thousands of kilometers away who are struggling with the consequences of our actions here? Or do we remain in our current political equilibrium until first our environment and then our social environment create new facts? (ds)