Of Pond Scum and Wall Street Bankers

I woke up at 3:00 in the morning a couple of weeks ago and couldn’t get back to sleep. I was thinking about biodiversity and beyond. This has happened to me a number of times since I read an article earlier in the millennium entitled “The Forest and the Trees: Romancing the J-Curve” by A.K.Dewdney. in The Mathematical Intelligencer (Volume 23, Number 3, 2001, pp 27-34) . The unfortunate consequence of reading that article is that the more I think about it, the more I am drawn into it and the problem expands into deeper and more far reaching questions about the complexity of the world. After some time I reach an impasse…some limit to my ability to comprehend it, some limit to the amount of time I can spend on it…some limit and I move on without resolving anything. It is at the same time both intriguing and frustrating. Now here it was again. This time I got up and wrote down a couple of simple experiments I could do to help me look at it again. I hope to describe the experiments and results on my website at http://amateurgeophysics.wordpress.com in the future.

If you think about an ecosystem, say a pond or a woodlot, you might think about all of the species that occur there and how many individuals represent each of those species. You would realize that there were more individuals in some species than in others. If you have some basic statistics in your background you might visualize something like a normal distribution, a bell curve, of species abundances with a mean and standard deviation…many species having an average abundance with some having more and others having less. At this point you would be wrong.

What you would really see is a distribution with many species having low abundances and a small number of species with great abundances. The rest would fall somewhere in between. This histogram is what Dewdney calls the J-Curve. While Dewdney spends a lot of time pursuing the best mathematical function to fit to biodiversity data, two aspects of this “J-Curve” capture my attention and lead me elsewhere. First is the simple ecosystem model that Dewdney describes which when run on the computer yields a “J-Curve” species-abundance histogram. His model is one of mutual predation among species with predator and prey being selected at random. “Well,” you might say, “that is not a real system!” to which I respond that no model is a real system. Models are created to help you think about a real system and that is exactly what Dewdney’s model does. It will form the basis for one of my experiments.

The second aspect of this “J-Curve” is that it falls into the class of power law  distributions having the same general shape that shows up in many diverse applications. From what I read, actually proving a fit to some power law is difficult but for me that is secondary to the ubiquitous occurrence of the general trend of the histogram. Power law distributions describe situations in which small events are common while large ones are quite rare. A few examples of possible power law distributions:

• In ecosystems, a few species have high abundances while many have only a few members.
• There are many villages with small populations but only a few very large cities.
• A few words show up many times in a large document while many words show up rarely.
• Small earthquakes are quite common whereas the largest ones are quite rare.
• Financial wealth is disproportionately held by a small number of people relative to the great masses of poor.

Aside from the observation that certain species found in pond scum and Wall Street investment bankers may occupy the same part of their respective distributions, what underlying features and drivers can cause such a diverse range of naturally occurring phenomena to all show this kind of distribution? And what are the implications that these distributions apparently result from stochastic processes? Do we live in a mostly random world over which we have little control? While there are many deterministic processes in the physical world, I think this might be more true than most people want to admit. I think that Dewdney’s simple model will suggest that, contrary to what the Occupy Wall Street movement would like to believe, even if the nation’s wealth was evenly distributed among the people, after some period of time the wealth would naturally return to its current uneven distribution. Some species are going extinct? Yes, that would be expected looking at the J-Curve species-abundance histogram. I suspect that we cannot change either of these examples any more than we can change the distribution of earthquake magnitudes.

This is the kind of thing that, every so often, keeps me awake at night!

Advertisement