Tipping points

The concept of tipping points is powerful, but sometimes a bit muddled. Things that get described as tipping points often sound to me like mere dramatic events or nonlinear effects, simple thermodynamic irreversibilities, or exponential signals emerging unexpectedly from noise. These may play a role in tipping points, and lead to surprises, but I don’t think they capture the essence of the idea. You can see examples (good and bad) if you sift through the images describing tipping points on google.

I think of tipping points as a feedback phenomenon: positive feedback that amplifies a disturbance, such that change takes off, even if the disturbance is removed. The key outcome is a system that is stable or resistant to disturbances up to a point, beyond which surprising things may happen.

A simple example is sitting in a chair. The system has two stable equilibria: sitting upright, and lying flat on your back (tipped over). There’s also an unstable equilibrium – the precarious moment when you’re balanced on the back legs of the chair, and the force of gravity is neutral. As long as you lean just a little bit, gravity is a restoring force – it will pull you back to the desirable upright equilibrium if you pick up your feet. Lean a bit further, past the unstable tipping point, and gravity begins to pull you over backwards. Gravity gains leverage the further you lean – a positive feedback. Waving your arms and legs won’t help much; you’re going to be flat on your back.

A more generalized explanation is given  in catastrophe theory. The interesting twist is that a seemingly-stable system may acquire tipping points unexpectedly as its parameters drift into regimes that create new stable and unstable points, leading to surprises. Even without structural change to the system, its behavior mode can change unexpectedly as the state of the system moves from locally-stable territory to locally-unstable territory, which occurs due to shifting loop dominance from nonlinearities. (Think of the financial crisis and some kinds of aircraft accidents, for example.)

Anyone know some nice, simple tipping point models? I think I’ll have to mine my archives for some concrete examples…

11 thoughts on “Tipping points”

    1. Bill – Yes – I think I’ve even seen him talk about it somewhere. I’d forgotten about it though. Their definition: The term “tipping point” commonly refers to a critical threshold at which a tiny perturbation can qualitatively alter the state or development of a system. Here we introduce the term “tipping element” to describe large-scale components of the Earth system that may pass a tipping point.

      There’s a collection of eco-tipping points at http://www.ecotippingpoints.com/, though I think there’s not much formalization of the concept. From their site:

      Origin of the “Tipping Point” Phrase

      The “tip point” phrase was coined more than fifty years ago to indicate a threshold for dramatic change in neighborhood demographics (Grodzins 1957). A “tip point” was the percentage of non-white residents in a previously white neighborhood that would precipitate a “white flight,” switching the neighborhood to total occupation by non-whites. Wolf (1963) used the phrase “tipping point” to describe the same phenomenon, and Schelling (1978) applied “tipping point” to other social phenomena as well.

      The “tipping point” phrase was later popularized by Malcolm Gladwell’s bestselling book The Tipping Point: How Little Things Can Make a Big Difference (Gladwell 2000). It used “tipping point” to represent the point in time when a new idea “takes off,” spreading rapidly through a society. Though Grozdins, Wolf, Schelling, and Gladwell did not use systems jargon such as “feedback loops,” their use of “tipping point” reflected the amplifying effects of feedback loops and the power of feedback loops to engender change.

      We use “tipping point” to mean a “lever” that can tip an eco-social system from one set of mutually reinforcing processes, called a “system domain,” to a different domain. The “tip” sets the system on a completely new course of change. EcoTipping Points are levers that set an eco-social system on a positive course of change. They are catalytic, turning the system from decline to a course of restoration and sustainability (Marten 2005, 2007, 2008; Marten et al. 2005).

      In both cases, the term seems to encompass both dynamic (positive feedback to multiple equilibria) and static (nonlinear threshold) versions of the story.

  1. One example could be a simple ice sheet model (weertman 1964?). If you perturb surface the mass balance by shifting the equilibrium line a little upwards, then the ice sheet will respond to the decreased mass balance by shrinking. As the ice sheet gets thinner, then it will also be exposed to a smaller area of accumulation. This is the height-mass balance feedback. Eventually if you push the equilibrium line too far, then you will put the whole ice sheet in the ablation zone. This can trigger a run away decay and the ice sheet won’t grow back if you reset the equilibrium line.

    In reality there are some stabilizing mechanisms. E.g. there are mountains that can act as seeds for regrowth and the desert effect decreases as the ice sheet gets smaller.

    1. Cool. It looks like there’s a neat recent look at that in Nature Geoscience, with a nice phase plot showing stable and unstable points. I’ll have to check it out.

  2. As I recall from Gladwell’s book, there were examples that had a lot to do with buzz and perceptions of coolness about certain products — weren’t Hush Puppy shoes mention? Talk about unexpected. Recently stumbled across a field called Belief Propagation, which has largely relied on Bayesian networks. Lots of work on repeatable, converging solutions in a linear domain, but some acknowledgement of “loopy” trees and instability. May be worth a look.

  3. Also see paper by Elena Bennett, Graeme Cumming and I – . It was part of a special feature on resilience indicators

    Bennett, E.M., Cumming, G.S. & Peterson, G.D. (2005) A Systems Model Approach to Determining Resilience Surrogates for Case Studies. Ecosystems, 8, 945-957.

    Most models of resilience have a tipping points part to them. A classic model is:

    Ludwig, D., Jones, D.D. & Holling, C.S. (1978) Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest. The Journal of Animal Ecology, 47, 315-332.

    A very simple model of a forest/fire tipping point is in an appendix in:

    Peterson, G. D. 2002. Estimating resilience across landscapes. Conservation Ecology 6(1): 17. [online] URL: http://www.consecol.org/vol6/iss1/art17/

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