Fossils Histograms

Skewed Histograms

Paleontologists have done many computer simulations in order to gain a feel for the range of outcomes of random walks in biodiversity, with both equal and unequal rates of speciation and extinction. Some simulated groups expand to the point of swamping the computer's memory; other groups go extinct rather soon. Extinction of a group is most common when the group starts out small, just as a casino gambler is most likely to go broke quickly if he starts with a small stake, close to the absorbing boundary.

In evolution, a group such as a genus of family must, by definition, start with a singles species. For the fledgling group to survive, the founding species must speciate before it goes extinct. Because new evolutionary groups start small, they usually don't last long. This, in turn, yields an important facet of the history of life: most groups of species have life spans shorter than the average of all groups. Figure 3-2 shows a histogram of life spans of fossil genera. It has a skewed shape, with many short durations and only a few long ones.

The skewed (asymmetrical) shape of variation is typical of important biological properties germane to the extinction question. These include
- number of species in a genus
- life spans of species
- number of individuals in a species
- geographic ranges of species

In each category, the small "thing" is most abundant. Let me give another example. There are about 4,000 living species of mammals, grouped into about 1,000 genera. About half of these genera have only a single species, and about 15 percent have only two species. The numbers drop off smoothly (see Figure 3-3), so there are only a few genera with more than 25 species. The most speciose living mammal genus (a small insectivore) has about 160 species. The overall average is 4 species per genus (4,000/1,000), but because of the asymmetry, fully three-quarters of the genera have 1, 2, or 3 species and are thus below average.

Let me summarize a few of the foregoing points. For reasons that develop from both theory and observation -- some depending on the Gambler's Ruin problem -- we can make the following generalizations:

1. Most species and genera are short-lived (compared with the averages).
2. Most species have few individuals.
3. Most genera have few species.
4. Most species occupy small geographic areas.

Skewed variation is extremely common in nature. Strangely, however, most of us have been trained to believe that variation in natural phenomena is bell shaped, having just as many items above as below the average -- whether we are talking about heights or weights of people, wealth  or baseall averages. Nothing could be further from the truth.

Classic examples of skewed variations include incubation times of infectious diseases and life expectancies of cancer patients. In both, the majority of cases fall below the average, because the average is constructed by summing many short time intervals and a few long ones. It would be far better to use the median time -- that time exceeded by half the individuals.

Of course, bell-shaped curves (called normal or Gaussian distributions by mathematicians) do sometimes occur in nature. It is just that other shapes are more common. Statisticians wrestle with this problem because many of the best statistical tests are designed for the bell-shaped curve. Often they avoid the problem by transforming the raw data -- that is, distorting the scale of measurement so that they can treat the results as if they had a bell-shaped distribution. One such transformation that sometimes works is to convert all measurements to their logarithms (or even square roots). If the transformed numbers have a bell-shaped distribution, the analyst can proceed with tests that assume this shape.


Other Models

The Gambler's Ruin problem has led us to generalizations about species that are relevant to the extinction problem. However, many of the patters, especially the skewed distributions, can also be approximated by processes having nothing to do with gambling or biology.

Suppose you take a stick that is 100 inches long and break it at 25 random points -- not favoring the middle or any other part. When you are done, you will have 26 short sticks. Now, measure and count the short sticks and construct a histogram. The shape of variation in stick length will took very much like those I have shown for species and genera: a hump or spike to the left and a long tail extending to the right. Figure 3-4 shows the results of a computer simulation.

Variation in population size of cities in the United States shows a pattern like this, as do many other things we can measure or count. The so-called broken stick model is one of several that have been applied to these patterns, and many attempts have been made to find out which model makes most sense or fits the observations best. For the purposes of this book, the important thing is that many of the distributions are skewed. They are not even close to the symmetrical, bell-shaped curve that we have all heard about.

One lesson about extinction to be learned from this is that some plants and animals are much more likely, a priori, to go extinct than others. The majority of species living today have small populations and live in restricted geographic areas. There are the ones we rarely see. The abundant and widespread species are commonly seen but are surprisingly few in number. For this reason, it is possible to write useful field guides to mammals and insects in volumes of manageable size. It stands to reason that when things get environmentally tough, either biological or physically, the many rare species are the most vulnerable to extinction. So, when we say that a given extinction event eliminated 40 or 80 percent of biodiversity, we should also say which 40 or 80 percent. The significance of the event will depend heavily on whether the victims were abundant, cosmopolitan species or local endemics.