How many extinct species?

The number of described living species still increases dramatically in groups like bacteria, mites and insects. In groups like vertebrates or spermatophytes, the number of described living species shows signs of saturation. How about the number of extinct species? The conditions allowing fossilization to occur are rare. Most layers containing fossils are inaccessible. The number of discovered accessible layers gradually increases with research intensity. Here we will try to use preliminary knowledge about the number of living species for estimating the number of extinct ones. Suppose that a species gives rise to a new one with a constant probability rate, say , and that it becomes extinct with another constant probability rate, say $\mu$. The latter implies that the existence time of a species follows an exponential distribution, with a mean existence time of . If denotes the number of living species at time $t$, the cumulative number of species and the simultaneous probability that the number of living species equals and that the cumulative number of species equals , then can be expressed as a function of :

When we transport the term from the left to the right, divide both sides by and let , we arrive at

This simple model defines the probability evolution of both numbers of species, if we specify the initial condition, e.g. . So we start with one single species for sure. The marginal probability evolution of (, obtained by dropping the index in (12.2),) is known as the linear birth and death process. We can extract useful information from (12.2), without actually solving the complete probability distribution as a function of time. The expected number of living species, , is found from (12.2) by multiplying both sides with and summing over all and . It is for this purpose convenient to define for negative values of and as well, but put them equal to zero. This allows us to sum over We then arrive at

(12.3)

For , we can solve :

(12.4)

In a similar way, we can multiply both sides of (12.2) by , sum over all and , arriving at

(12.5)

For , we can solve :

As an aside, we can evaluate the variance of as follows. First we multiply both sides of (12.2) by and sum over all and , arriving at

We then subtract the derivative of , which is :

(12.8)

For and noting, in general that has as solution, we find

(12.9)

This means that the variation coefficient of is given by

(12.10)

So, as for branching processes, the variation coefficient approaches a constant value. This is not mere coincidence, because the linear birth and death process can be conceived as a branching process by letting the generation time shrink to an infinitesimally small time increment , while the offspring distribution for 0,1 or 2 young has been tied to this increment via the probabilities , and . In a similar way, we obtain

(12.11)

which gives for :

(12.12)

and

(12.13)

which gives for :

(12.14)

from which follows that ) for large $t$, and that the correlation coefficient between and approaches 1 for large $t$. The expected number of extinct species is given by . Using (12.6), we find that . Since $\mu$ equals (mean species existence time) and equals (ln expected no of living species)/(evolution time), we have that

For vertebrates, which fossilize relatively well, due to their hard bones, the evolution time took some 600 Ma to arrive at some 42000 living species. The mean existence time of vertebrate species is estimated to be some 2-3 Ma, based on fossil records, which means that for each living species, we expect some 600/(2.5 ln 42000) 23 extinct ones. The values for and $\mu$ might differ between phyla. If we roughly estimate 30 million living eukaryotic species to exist, which took some 2000 Ma to evolve, and if we adopt the vertebrate mean species existence time, we expect some 46 extinct species for each living one. This model is irrealistically simple. This is obvious from the fact that, for this model, the extinction probability of a richly diversified group is extremely small. Yet such groups, like trilobites, ammonites and dinosaurs, became extinct, nonetheless. At present, the possible occurrence of periodic mass extinctions is in hot debate. Such complicating phenomena would only increase the number of extinct species to be expected. The above calculations could therefore be considered as a lower bound on the number of extinct species. During the last 300 year, 150 vertebrate species have been recorded to became extinct, mainly due to direct or indirect action of man. The expected extinction rate equals , which amounts to some 5.6 species in the last 300 year. The proper answer to the question posed in the title is therefore: too many! Further reading on vertebrate evolution: [#!Carr88!#]; On 'natural' and man-induced extinctions: [#!Stan87!#,#!Lewi86!#]. Example 6 for extinction and variation coefficients in branching processes.