Many complex systems are homeostatic (aka autopoietic aka self-maintaining). Such systems are, by their nature, resistant to change because they contain mechanisms to return system dynamics to a particular pattern. However it is not the case that such systems only have one possible self-maintaining pattern. If a homeostatic system receives a large enough perturbation then the system may settle to a different pattern. We would expect that as a homeostatic system experiences a series of shocks it would spend a much greater amount of time in each homeostatic pattern compared to the transition time between patterns. That is precisely the phenomena called punctuated equilibria.

Punctuated equilibria have been identified in a number of systems: animal biology, ecosystems, evolutionary processes, and gene interaction to name a few. It is recognizable as fits and starts in the system's characteristics. It was, as far as I know, first discussed by Stephen Jay Gould who noticed that the fossil record shows long periods wherein the set of species found is roughly the same interspersed with short periods of rapid evolutionary change. This is why finding intermediate species is so hard; the small probability of any animal becoming fossilized combined with the short duration of the intermediate evolutionary stages. This was a major break from the gradualism proposed by Darwin, although Darwin's mechanism can easily be made to fit the punctuated pattern. In a fantastic and brilliant paper called "Topology of the Possible" Walter Fontana demonstrates how the presence of fitness neutral genomic changes (which are known to exist) naturally produces punctuated phenotypic changes from gradual genomic changes. However, analogous mechanism may not be at the heart of punctuated equilibria in other systems.

An interesting aspect of the so-called punctuated equilibria phenomenon is that they are not equilibria at all. The coherent pattern of behavior they describe is resistant to small, medium, and some large perturbations, but it can also collapse or shift to a new pattern. In my methodology using dynamics represented by a Markov model to measure robustness, the definition I developed for the robustness of a set cumulatively adds the probability mass of states in that set as it flows from all initial states. Sets that are strongly connected components, called cores, may maintain their mass for long periods of time as it transitions within the set. It may also regain mass after losing it and draw mass in from outside. If each core represents a different characteristic (e.g. phenotype, utility value, or any qualitative feature) and there are multiple cores connected to each other through relatively short paths then such a system will exhibit punctuated equilibria-looking dynamics.

The sorts of feedback mechanisms capable of maintaining a homeostatic system are, it seems, exactly those sorts of mechanisms that will produce punctuated equilibria. Does that pan out in practice? One widely recognized homeostatic system is the human body. By the reasoning presented in this paper, then, it should experience punctuated equilibria in its development. Does it? Well, Im not really sure. I can say that for muscle gain and fat loss there does seem to be tendency of the body to maintain its usual levels unless one really pushes for a prolonged period of time. And juvenile growth seems to be in spurts for many individuals. But the human body is a rather noisy system considering all the things we put it through. Ecosystems also seem homeostatic in the way I describe, but I would need to look at some real data to determine whether there are natural extended periods of time for each of varying population levels (rather than a continuous increasing and decreasing as the differential equation models produce).

Perhaps the most promising area for uncovering these patterns of behavior is the social sciences. The census and the plethora of polls (political, market, etc.) collect a great deal of diversified data on many individuals across time and in many different regions. These data may reveal punctuated equilibria dynamics in myriad social systems. As Dimitri Pappas once said, The status quo pretty much stays the same most of the time. Indeed it does, but when it shifts my hypothesis is that is it shifts big and fast. Political revolts and economic depression are two versions of such punctuated coherent patterns of behavior. Understanding the duration of plateaus and the steepness of transitions may improve our prediction of these events and help design social institutions more capable of adjusting to them.

In the near future I will be applying the methodology I developed to several real systems and then hopefully I will have more things to say about how homeostatic systems and punctuated equilibria are related.