The stability in systems can be a local phenomenon when parameter changes within a certain range do not cause severe changes in the system's behavior. Most modeled systems with just one stable attractor, as for example the logistic-growth model, tend to return to this attractor when perturbed in their dynamics. Systems with more than one stable attractor however, are prone to so called critical transitions. They "jump" from one stable state to the other.
An illustrative example for such jumps gives the behavior of lakes in relation to their nutrient load, turbidity and vegetation coverage (Scheffer et al. 1993). Good empirical evidence exists that lakes tend to have two different stable states, one with clear water and a relatively rich ecology of submerged water plants and one with no or few plants and high turbidity of the water. A simplified explanation for this stresses that turbidity, caused by high nutrient loads consumed by phytoplankton, impedes the diffusion of light in the lake which plants need to grow. The higher turbidity hence, the lower the plant coverage. On the other hand, water plants absorb nutrients and give home to zooplankton which feeds on phytoplankton. Water plants thus tend to reduce the turbidity of the water. If the lake is not overly structured, with different levels of water depths for example, it will thus be driven by two contrasting positive feedback cycles: one in which turbidity enhances turbidity, and another one in which vegetation enhances vegetation.
Scheffer, M., S. H. Hosper, M. L. Meijer, B. Moss, E. Jeppesen (1993). Alternative Equilibria in Shallow Lakes. Trends in Ecology & Evolution 8, no. 8: 275-79.
Scheffer, Marten (2009). Critical Transitions in Nature and Society. Princeton Studies in Complexity. Princeton UP.
In shallow lakes hence, with similar water depth everywhere, vegetation for the most part will disappear at a certain critical turbidity. Assuming less feedback influence in equilibrium turbidity and plotting how this is affected by the vegetation cover could produce an image like the one below to the left. On the other hand, the equilibrium vegetation coverage could follow a more sigmoidal form, like the image below to the right, saying that the vegetation is high and stable at low turbidity and low and stable at high turbidity. In between it is in a transient state.
Combining the information of these two plots, like in the image to the right, shows the equilibria (solid dots for stable, open dots for unstable equilibria) resulting from the dynamic interaction between turbidity and vegetation. The arrows indicate in which direction the dynamics will develop when temporarily perturbed. The two solid dots hence indicate that the ecosystem of such lakes has two alternative stable states and a good chance for rather abrupt transitions between these states. The existence of such critical transitions is well confirmed by empirical limnology.
The plot also indicates that the existence of two alternative stable states depends on the steepness of the drop in the vegetation curve. One parameter determining this steepness is the depth of the lake. In deeper lakes, plants have less impact on clarity than in shallow lakes. Gradual slopes in deeper lakes imply a less abruptly declining plant cover as light gets gradually limited. As a result, multiple intersections of the equilibrium curves are less likely as lakes are deeper. The left plot below shows a rather gradual declining plant cover as it is typical for deeper lakes, now in relation to the nutrient load of the lake. The right plot shows a (stylized) more-than-abrupt decline of plant cover, assumed typical for shallow lakes.
Considering the depth of the lake a third dimension in these plots, demonstrates how the stability plane seems to fold when following the depth of the lake from deep to shallow (from back to front in the image below). Systems that show this kind of gradual development towards bi-stability are called subjected to a fold- or cusp catastrophe (see below).
The image to the right demonstrates a peculiar characteristic of systems with two or more alternative attractors. Following the red line from the left upper corner down towards the point F1 indicates vegetation coverage that slowly declines with increasing nutrient load. When nutrient load reaches the point F1 vegetation coverage suddenly "jumps" to a much lower level, indicated by the lower right part of the red line. Vegetation suddenly deteriorated, a catastrophe occurred. At this point, if one would want to recover vegetation by reducing the nutrient load one would follow the lower part of the red line backwards until the point F2 is reached. At this point the system jumps again back to the higher level of vegetation. However, between the points F1 and F2 lies a range of nutrient loads at which the system, the lake in this case, can have two different states, depending on the direction from which these states are approached. This phenomenon is called hysteresis, and is quite common in systems with alternative attractors.
More formally the above described performance can be demonstrated with a curve of the form \(f(x) = ax + x^4 - x^2\) which is interpreted as a surface holding a ball (left figure below). The ball marks a stable state as it is laying relatively stable on a small platform, a local minimum which is formed by the curve with the parameter \(a = -0.55\).
When \(a\) is successively reduced, the curve stretches and deepens its minimum on the right side. The ball however keeps its position on the platform. When \(a = -0.69\) its position has hardly changed. At first glance, it seems just as stable as with \(a = -0.55\). However, when altering the parameter only about one hundredths more to \(a = -0.7\) (right figure above) the ball exceeds a tipping point and rolls to the new, substantially lower minimum on the other, the positive side of the x-axis. A tiny change, which didn't differ from the other minimal changes before, now has a large and unforeseen effect - a critical transition occurred.
If \(a\) now is gradually altered backwards, one needs to increase it up to \(a = 0.7\) to carry the ball back to the left side, to the negative side of the x-axis. Between \(a = -0.69\) and \(a = +0.69\) the ball shows no reaction with respect to its position to the left or right of zero. If this position would be the primary observational distinction (i.e. the question whether x is smaller than or greater than zero), the position of the ball seems to suggest conflicting results: at the same values for e.g. \(a = -0.69\) the ball once is to the left of the zero point and once its to the right, depending on whether it is approached from the left or the right. This hysteresis indicates that the output of a system is dependent not only on its current input, but also on its history of past inputs.
Another, more abstract example can be seen if the a-parameter in the Lorenz-attractor if it is reduced from its initial value 10 to 5.0. The "butterfly" changes its form, but remains cognizable. Once it is reduced just a bit further however, to 4.9 for instance, the butterfly looses its distinct double-loop form. Systems hence can show unpredictable, surprising behavior, although they are stable over wide arrays of their parameter range. Such local stability is often referred to as "metastable" or as "punctuated equilibrium", since it does not indicate whether a system is close to collapse. To many experts for example, the social situation in Eastern Europe prior to the year 1990 were metastable in this sense. The upheavals and the socialist breakdown came as a complete surprise. Hardly anyone then suspected that the socialist system would crumble so soon.
A more theoretical, though not less famous example of a metastable equilibrium provides the sandpile experiment as described by Per Bak and Kan Sneppen in 1993. In this experiment, sand trickles evenly onto the center of a round plate and over time forms a pile of sand. Sand grains fall on top of each other, slip off to the side and so form a heap according to a Gaussian bell. Over time, the slope of this heap gets steeper. The steeper the slope, the more grains of sand slip down the slope, forming avalanches the average size of which increases with the steepness of slopes.
Bak, Per / Sneppen, Kan (1993): Punctuated Equilibrium and Criticality in a Simple Model of Evolution; in: Physical Review Letters 71/1993, S. 4083-4087.
The catastrophic aspect of this is the difficulty to predict the actual occurrence of avalanches. The number of sand grains added before an avalanche occurs can vary considerably. From a certain size onwards, the system changes no longer gradually and therefore (semi) predictable, but in the form of leaps from one level of order to the next. Mathematically this is described in the so-called Catastrophe theory, a theory of discontinuities and singularities, which analyses degenerate critical points in a potential function - points where not just the first derivative, but one or more higher derivatives of the potential function are also zero.
Coming back to the above-mentioned sandpile. If more sand is trickled onto it, with time the sand will fill the entire plate and the first avalanches will exceed its edges and fall down to the floor. If finally on average the same amount of sand grains as falling down to the floor is added, the sandpile will no longer grow. The system has reached a critical state. Mathematically, this is characterized by a power law relationship between the number of sand grains displaced by each newly added grain and the frequency of avalanches of different sizes.
This critical state acts as an attractor. If the steepness of the slopes - perhaps accidentally - should once outgrow this condition, to form a so-called "supercritical state", the subsequent avalanches will be much larger than in the critical state. More sand grains will fall down than are added. The "supercritical" heap therefore shrinks in its size until it again reaches the critical size.
On the other hand, if the sandpile - also by chance - should decimate below this critical size, this again has an effect on the average size of the avalanches. They become smaller relative to the added amount of sand, which in its turn causes the sand pile to grow. The system thus approaches its critical condition from the "bottom" as well as from the "top". It organizes its critical state by itself. One therefore speaks of self-organized criticality.
Bak, Per (1996): How Nature Works: The Science of Self-Organized Criticality. New York. Copernicus.
A particular example for self-organized criticality is known under the expression efficient market hypothesis. However, it could be more correct in this case to speak of self-organizing uncertainty. Market success is said to be hard to predict. Only in rare cases, for example if one happens to be in the position to offer a much demanded product as a sole supplier, one could be sure to have success on the market. However, just the very fact of success in this situation is a guarantee for not staying a sole supplier for long. More and more competitors will offer similar products, up to a supply level at which every new provider faces high uncertainty whether he will be successful. On the other hand, if the uncertainty gets too high, some suppliers will retreat from the market, which might raise the probability for success again. In this sense, markets are said to be efficient in organizing their own uncertainty.