Some thoughts on ecology, evolution and economics

A Tale of Tails

By far, the most recurring theme in this blog is an exploration of the lopsided nature of evolutionary success – what I have called Haldane’s Rule.  Is this a function of the process of speciation or is it something more mundane – something that we would expect in any branching process?  I have come to the realization that the expectation that species groups should be approximately equal is all wrong.  It is unlikely to happen in a rarely occurring branching process that accumulates over time.

Let’s begin with a coin toss.

coin tossFor any succession of coin tosses, the expected outcome is that half of them will be “heads”. So, for 10 tosses you get a probability curve like this with a maximum value at 5 “heads”.

equal odds

one tenthIf the coin is weighted or you are dealing with some other random outcome with less than equal odds, the peak migrates to one end or the other of the distribution and trails behind it a long tail.  Whenever the expected outcome is equal to one or fewer tosses then all you get is the tail.

For the evolutionary lineages we have been looking at, for example the perching birds. the expected number of branching events is one or less.  This does not rule out the long tail of possible outcomes with many – even hundreds – of branching events.  What you will not find as these lopsided odds get compounded over the eons is an evenly divided group of different kinds of organisms.  You must expect some groups to be much larger than others, even if the tendancy to have many branching events is not handed down through the lineages.  This is entirely because speciation is a rare event and need not be explained by any bias in how species are formed.

As an illustration, I simulated two evolutionary lineages, each of them a taxonomic Class with 5 Orders.  One lineage branched into Families, Genera and Species with a binomial distribution without tails and one lineage branched with the same average frequency but with a long tail.  The results looked like this:

Distribution Orders/Class Families/Order Species/Family Total # of species
Tail 5 11.8 77 4551
No tail 5 5.8 74 2167

The tailed distribution of a rarely occurring event produces the lopsided results that we have been witnessing in the birds, flowering plants and insects. The no tail distribution never produces a lack of branching events – an outcome that is common in all lineages. With a sufficiently long tail, the idea of an incredibly species rich group like the Arthropods – and eventually the beetles is not so very strange.  Indeed, perhaps J.B.S.Haldane should have anticipated God’s fondness for beetles.

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