From 2010 through the first two-thirds of 2015, at least 211 scientific articles with the term “tipping point” and 109 with “regime shift” in the title were published (according to the Web of Science database, as of 23 November 2015). These span a broad range of science, technology, and engineering, but the geosciences are well represented. In recent years the concept of tipping points in the global environment related to climate change, regime shifts, ecosystem collapse and other phenomena has garnered a great deal of both scientific and public attention. “Tipping point” is often used in public (and sometimes scientific) discourse to refer to impending doom, or at least major environmental changes with uncertain and potentially negative impacts. However, tipping points are not necessarily associated with negative impacts on humans. Nor are they inevitably associated with direct or indirect human agency, as Earth history is marked by numerous tipping points and regime shifts.

Tipping points are a type of threshold phenomenon. In systems theory (and Earth and environmental sciences) a threshold is a boundary separating different behaviors or states (qualitatively different conditions) of a system. Tipping points are thresholds (but not all thresholds are tipping points) that result in rapid or abrupt state changes relative to the time scale under consideration. Regime shifts are threshold-driven state changes that may be gradual or abrupt. Regime shifts are a subset of thresholds that are generally understood to apply at a broad landscape or ecosystem scale.

Understanding scientific concepts is heavily dependent on the metaphors we use to visualize, analyze, and communicate them. Recognizing that tipping point itself is a metaphor, I got to wondering about what other metaphors might be useful in exploring these abrupt shifts in Earth systems.

Balance

The tipping point notion is at least implicitly based on a metaphor analogous to a balance or scale. In fluvial geomorphology this analogy is often used with respect to aggradation or degradation with a diagram or conceptual model like the one below:

www.fgmorph.com/fg_2_9.php

My interpretation of this, by the way, is quite different from the traditional one. Some assume (on the basis of precious little evidence) that fluvial systems seek to equalize sediment supply and transport capacity and keep the scale balanced. My view is that the scale is usually tipped to one side or the other, but increases or decreases in sediment supply or transport capacity can cause it to tip in the other direction.

Seesaw

Since the idea of a balance as a weighing device is indeed to achieve balance, I prefer the seesaw metaphor for the situation described above. This is a pretty good metaphor for a fairly common situation in Earth systems, where an unstable equilibrium separates two alternative stable states.

The Cliff

This analogy is firmly related to a doom-based view of tipping points. The TP occurs when the system is pushed to the edge, and a precipitous decline.

http://livinggreenmag.com/2012/06/13/climate-change/scientists-warn-that-earth-is-close-to-climate-tipping-point/

Dominoes

In this case a line of dominoes is the analog of a complex, interlinked environmental system with many components. Tipping one domino (a local tipping point) will cause some or all of the others to fall as well (a global or regional TP).

http://thedominoeffectmovie.com

Any of these metaphors, and no doubt others, have their advantages in communicating certain ideas and analyzing certain problems. The dominoes analogy seems to me to have particular promise.

Suppose each domino is designated x_i, with i = 1, 2, . . . , n total dominoes. The dominoes are all assumed to be adjacent to at least one other domino. We further assume that a domino tipping either left or right will knock over exactly one other domino (except for the first and last, x₁ and x_n, which could tip in one direction without disturbing other dominoes).

This characterizes (or caricatures) a situation where a local tipping point anywhere is the chain of dominoes has a unique effect. The two end dominoes are all-or-nothing: If they tip one way, nothing else happens. The other way, and every other domino goes down. For all the others, the number of dominoes that fall also depend on which way an individual tips. Using a simple left-to-right description the line of dominoes (i.e., x₁ is the first, left-most and x_n the last, right-most domino), the number of fallen dominoes is n – i + 1 if it tips right, and dominoes x_i through x_n fall. A left tip, and dominoes x₁ through x_i  go over, with the total number tipped equal to i.

Any domino is highly sensitive to any left-tips for of any dominoes to its right, and insensitive to any left-tips to its left, and vice-versa. We can also look at probabilities. If every piece has an equal probability of being disturbed, and falling left or right is equally probable, then p(x_i, R) = p(x_i, L) = 1/2n—that is, the probability of any given domino tipping left or right is 1/2n.

The probability of a tipover at domino j, given the falling or domino i, is as follows:

p(x_j:x_i) = 0 if j > i and i tips left.

p(x_j:x_i) = 0 if j < i and i tips right.

p(x_j:x_i) = 1 if j < i and i tips left.

p(x_j:x_i) = 1 if j < i and i tips right.

This seems to me to convey the idea that tipping points, like pretty much everything else in the geosciences are difficult to generalize about without geographical and historical context!