Mathematical modelling of the evolution of lineages goes back at least to Yule1 who developed the eponymous Yule process (homogeneous pure birth process) in which speciations occur independently and at random. Yule's model did not include extinctions per se, because he believed that they resulted only from cataclysmic events. This issue was discussed at greater length by Raup2, who distinguished between background and episodic extinctions. Raup started from a homomogeneous birth-and-death process model (in which background extinctions occur, like speciations, independently and at random) for which he presented mathematical results, and described more complex models of extinction including episodic extinctions and a mixture of episodic and background extinctions. However he gave no mathematical results for these models. Stoyan3 considered a time in-homogeneous birth-and death process, in which speciation and background extinction rates varied with time, based on the idea that younger paraclades have higher speciation rates, while older ones have higher background extinction rates.

There has been considerable discussion (e.g. Raup2; Patzkowsky4; Przeworski and Wall5) about the suitability of the null birth-and-death process model (with constant birth and death rates) as a macroevolutionary model of species diversification. In order to truly assess the validity of such a model it is necessary to have a full understanding of its properties which can then be compared with the fossil record. Specifically analysis is needed to generate hypotheses, which can be tested against available data. To date such an analysis is incomplete, relying on the partial analytic results of Raup2 and the simulation results of Patzkowsky4 and Przeworski and Wall5.

Analytic results are clearly superior to simulation ones. In particular with analytic results for the size distribution of a clade one can fit the model via a multinomial likelihood, using observed size distributions, and thence test the adequacy of the underlying birth-and-death model using a statistical goodness-of-fit test. In addition analytic results are preferable to simulation ones, in that it is much easier to interpret a parametric formula than a collection of simulation results; and one does not have to distinguish between sampling variation due to a finite number of runs (noise) and signal.

It is the purpose of this paper to conduct a more thorough analysis of the birth-and-death model than that previosly carried out by Raup2. In particular we obtain results for size distributions of taxa and probabilities of monotypic taxa. In this paper we confine attention to obtaining analytic results and defer actual fitting and testing of the fit, using observed fossil data, to a future paper.

We develop the mathematical model presented by Raup2 (and used in simulations by the above authors) to include the possibility of episodic, cataclysmic extinctions in which complete lineages are destroyed. We consider a hiearchy of models, which can include both cataclysmic and background extinctions of species and examine the resulting size distributions of extinct genera. We start (following section), as did Yule, by considering cataclysmic extinction only. Furthermore like Patzkowsky4 and Przeworski and Wall 5, we assume that at any time an existing species can split, yielding a new species so radically different from existing ones that it becomes the founding member of a new genus. Thus we assume that the probability of a new genus being formed in an infinitesimal interval (t, t + dt) is proportional to the total number of species in existence at time t. We derive results for the size distribution of extinct genera.

In the third and fourth sections we do the same assuming only background extinctions (but no cataclysmic extinction); and both cataclysmic and background extinctions (although the results here are limited). The fifth section is devoted to the distribution of the number of genera derived from the pioneering species and in the final section the probability of a monotypic genus is compared with that of a monogeneric family.

Yule1 considered the evolution of a genus begining with one species at time t = 0, which thenceforth evolves as a homogeneous pure birth process (Yule process) with speciation rate (birth parameter) λ. He then showed that N_t, the number of species alive at time t, follows a geometric distribution with probability mass function (pmf)

p_n(t; 1) = Pr{N_t = n|N₀ = 1} = e^-λt(1 - e^-λt)^{n - 1}     (1)

for n = 1,2,.... If instead there are initially n₀ species then from standard results (e.g. Bailey, 1964) the distribution of N_t is negative binomial with pmf

p n ( t ; n 0 ) = Pr ⁡ { N t = n | N 0 = n 0 } = ( n − 1 n 0 − 1 ) e − λ n 0 t ( 1 − e − λ t ) n − n 0       ( 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6DE5@

for n = n₀, n₀ + 1,....

We now consider evolution of genera, and of species within genera, over an epoch between cataclysmic events. Let the time origin be the time of the previous cataclysm, and suppose only a single genus (containing n₀ species) survived that cataclysm. Let τ be the time of the succeeding cataclysm. Yule assumed that new genera were formed from old in a process analogous to that of speciation, thereby establishing that the time in existence of any genus would follow a truncated exponential distribution, with parameter equal to the rate at which new genera are formed from old. But it is more realistic to assume that a new genus is formed when a speciation within an existing genus is of such a radical form as to qualify the new species as belonging to a completely new genus. Thus the probabilty of a new genus being formed in an infinitesimal interval (t, t + dt) should be proportional to the existing number of species in all existing genera in the family (and not to the existing number of genera in the family). We let

K_t denote the number of genera at time t, evolved from the pioneeering n₀ species;

L_t denote the number of species at time t in all genera, evolved from the pioneeering n₀ species; and

N_t denote the number of species in the pioneering genus at time t.

We assume that speciations (within a genus) occur at the rate λ and new genera are formed from existing species at the rate γ. Then to order o(dt) the following state transitions (of K_t, L_t, N_t) can occur in (t, t + dt):

(k, l - 1, n - 1) → (k, l, n) with probability λ(n - 1)dt

(k, l - 1, n) → (k, l, n) with probability λ(l - 1 - n)dt

(k - 1, l - 1, n) → (k, l, n) with probability γ(l - 1)dt

(k, l, n) → (k, l, n) with probability 1 - (λ + γ)ldt.

Letting p_{k, l, n}(t) = P(K_t = k, L_t = l, N_t = n), the following differential-difference equations can be established from the above:

d d t p k , l , n ( t ) = λ ( n − 1 ) p k , l − 1 , n − 1 ( t ) + λ ( l − 1 − n ) p k , l − 1 , n ( t ) + γ ( l − 1 ) p k − 1 , l − 1 , n ( t ) − ( λ + γ ) l p k , l , n ( t ) .       ( 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeGadaaabaWaaSaaaeaacqWGKbazaeaacqWGKbazcqWG0baDaaGaemiCaa3aaSbaaSqaaiabdUgaRjabcYcaSiabdYgaSjabcYcaSiabd6gaUbqabaGccqGGOaakcqWG0baDcqGGPaqkaeaacqGH9aqpaeaaiiGacqWF7oaBcqGGOaakcqWGUbGBcqGHsislcqaIXaqmcqGGPaqkcqWGWbaCdaWgaaWcbaGaem4AaSMaeiilaWIaemiBaWMaeyOeI0IaeGymaeJaeiilaWIaemOBa4MaeyOeI0IaeGymaedabeaakiabcIcaOiabdsha0jabcMcaPiabgUcaRiab=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@954A@

Using the generating function

Φ ( x , y , z ; t ) = ∑ k = 1 ∞ ∑ l = 1 ∞ ∑ n = 1 ∞ p k , l , n ( t ) x k y l z n ,       ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHMoGrcqGGOaakcqWG4baEcqGGSaalcqWG5bqEcqGGSaalcqWG6bGEcqGG7aWocqWG0baDcqGGPaqkcqGH9aqpdaaeWbqaamaaqahabaWaaabCaeaacqWGWbaCdaWgaaWcbaGaem4AaSMaeiilaWIaemiBaWMaeiilaWIaemOBa4gabeaaaeaacqWGUbGBcqGH9aqpcqaIXaqmaeaacqGHEisPa0GaeyyeIuoakiabcIcaOiabdsha0jabcMcaPiabdIha4naaCaaaleqabaGaem4AaSgaaOGaemyEaK3aaWbaaSqabeaacqWGSbaBaaGccqWG6bGEdaahaaWcbeqaaiabd6gaUbaaaeaacqWGSbaBcqGH9aqpcqaIXaqmaeaacqGHEisPa0GaeyyeIuoaaSqaaiabdUgaRjabg2da9iabigdaXaqaaiabg6HiLcqdcqGHris5aOGaeiilaWIaaCzcaiaaxMaadaqadaqaaiabisda0aGaayjkaiaawMcaaaaa@670D@

multiplying (3) by x^ky^lzⁿ and summing yields the following partial differential equation

Φ_t = y(λy + γxy - (λ + γ)) Φ_y + λyz(z - 1) Φ_z,     (5)

which can be solved by the method of characteristics (e.g. Bailey,6) with initial condition ϕ(x, y, z; 0) = xyn0zn0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEcqWG5bqEdaahaaWcbeqaaiabd6gaUnaaBaaameaacqaIWaamaeqaaaaakiabdQha6naaCaaaleqabaGaemOBa42aaSbaaWqaaiabicdaWaqabaaaaaaa@3681@. From the solution the generating functions of K_t, L_t and N_t can be derived. They are

Φ K ( x , t ) = E ( x K t ) = x { p ( t ) 1 − x [ 1 − p ( t ) ] } n 0 ,       ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHMoGrdaWgaaWcbaGaem4saSeabeaakiabcIcaOiabdIha4jabcYcaSiabdsha0jabcMcaPiabg2da9iabdweafjabcIcaOiabdIha4naaCaaaleqabaGaem4saS0aaSbaaWqaaiabdsha0bqabaaaaOGaeiykaKIaeyypa0JaemiEaG3aaiWabeaadaWcaaqaaiabdchaWjabcIcaOiabdsha0jabcMcaPaqaaiabigdaXiabgkHiTiabdIha4jabcUfaBjabigdaXiabgkHiTiabdchaWjabcIcaOiabdsha0jabcMcaPiabc2faDbaaaiaawUhacaGL9baadaahaaWcbeqaaiabd6gaUnaaBaaameaacqaIWaamaeqaaaaakiabcYcaSiaaxMaacaWLjaWaaeWaaeaacqaI2aGnaiaawIcacaGLPaaaaaa@5A19@

Φ L ( y , t ) = E ( y L t ) = { y e − ( λ + γ ) t 1 − y [ 1 − e − ( λ + γ ) t ] } n 0 ,       ( 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@64B5@

Φ N ( z , t ) = E ( z N t ) = { z e − λ t 1 − z ( 1 − e − λ t ) } n 0 ,       ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5B8D@

where

p ( t ) = ( λ + γ ) e − ( λ + γ ) t γ + λ e − ( λ + γ ) t .       ( 9 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiabcIcaOGGaciab=T7aSjabgUcaRiab=n7aNjabcMcaPiabdwgaLnaaCaaaleqabaGaeyOeI0IaeiikaGIae83UdWMaey4kaSIae83SdCMaeiykaKIaemiDaqhaaaGcbaGae83SdCMaey4kaSIae83UdWMaemyzau2aaWbaaSqabeaacqGHsislcqGGOaakcqWF7oaBcqGHRaWkcqWFZoWzcqGGPaqkcqWG0baDaaaaaOGaeiOla4IaaCzcaiaaxMaadaqadaqaaiabiMda5aGaayjkaiaawMcaaaaa@54BD@

From this it is clear that both the total number of species, L_t, and the number of species in the pioneering genus, N_t, have negative binomial distributions (with parameters n₀ and e^{-(λ+ γ)t} and n₀ and e^-λt respectively); while the number of genera K_t has a distribution related to the negative binomial – precisely K_t + n₀ - 1 has a negative binomial distribution with parameters n₀ and p(t). The expected number of genera at time t is

E ( K t ) = 1 + n 0 γ λ + γ [ e ( λ + γ ) t − 1 ] .       ( 10 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieaacqWFfbqrcqGGOaakcqWGlbWsdaWgaaWcbaGaemiDaqhabeaakiabcMcaPiabg2da9iabigdaXiabgUcaRmaalaaabaGaemOBa42aaSbaaSqaaiabicdaWaqabaacciGccqGFZoWzaeaacqGF7oaBcqGHRaWkcqGFZoWzaaWaamWaaeaacqWGLbqzdaahaaWcbeqaaiabcIcaOiab+T7aSjabgUcaRiab+n7aNjabcMcaPiabdsha0baakiabgkHiTiabigdaXaGaay5waiaaw2faaiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIWaamaiaawIcacaGLPaaaaaa@4FC7@

It can be shown (see Appendix) that the times of formation of derived genera constitute an order statistic process. This means that they can be considered as the order statisics of a collection of independent, identically distributed (iid) random variables. From this it is shown that at any fixed time τ, the times t₁, t₂,...,t_k that the derived genera have been in existence are iid random variables with probability density function (pdf)

f k ( t ) = ( λ + γ ) e − ( λ + γ ) t 1 − e − ( λ + γ ) τ , 0 < t < τ .       ( 11 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemOzay2aaSbaaSqaaiabdUgaRbqabaGccqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiabcIcaOGGaciab=T7aSjabgUcaRiab=n7aNjabcMcaPiabdwgaLnaaCaaaleqabaGaeyOeI0IaeiikaGIae83UdWMaey4kaSIae83SdCMaeiykaKIaemiDaqhaaaGcbaGaeGymaeJaeyOeI0Iaemyzau2aaWbaaSqabeaacqGHsislcqGGOaakcqWF7oaBcqGHRaWkcqWFZoWzcqGGPaqkcqWFepaDaaaaaOGaeiilaWcabaGaeGimaaJaeyipaWJaemiDaqNaeyipaWJae8hXdqhaaiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIXaqmaiaawIcacaGLPaaaaaa@5C2B@

By summing (3) over k and l one can show that N_t is a pure birth process with birthrate λ; and by summing over k and n that L_t is a pure birth process with birthrate λ + γ. From the fact that a pure birth process is an order statistic process it can be shown (see Appendix) that at time τ the times since establishment of all non-pioneering species in the pioneering genus are independently distributed random variables, with a truncated exponential distribution with pdf

f N ( t ) = λ   e − λ t 1 − e − λ τ , 0 < t < τ ;       ( 12 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemOzay2aaSbaaSqaaiabd6eaobqabaGccqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaGGaciab=T7aSjabbccaGiabdwgaLnaaCaaaleqabaGaeyOeI0Iae83UdWMaemiDaqhaaaGcbaGaeGymaeJaeyOeI0Iaemyzau2aaWbaaSqabeaacqGHsislcqWF7oaBcqWFepaDaaaaaOGaeiilaWcabaGaeGimaaJaeyipaWJaemiDaqNaeyipaWJae8hXdqNaei4oaSdaaiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIYaGmaiaawIcacaGLPaaaaaa@5032@

and that the times since establishment of all non-pioneering species in the pioneering family are independently distributed random variables, with a truncated exponential distribution with pdf

f L ( t ) = ( λ + γ ) e − ( λ + γ ) t 1 − e − ( λ + γ ) τ , 0 < t < τ .       ( 13 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemOzay2aaSbaaSqaaiabdYeambqabaGccqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiabcIcaOGGaciab=T7aSjabgUcaRiab=n7aNjabcMcaPiabdwgaLnaaCaaaleqabaGaeyOeI0IaeiikaGIae83UdWMaey4kaSIae83SdCMaeiykaKIaemiDaqhaaaGcbaGaeGymaeJaeyOeI0Iaemyzau2aaWbaaSqabeaacqGHsislcqGGOaakcqWF7oaBcqGHRaWkcqWFZoWzcqGGPaqkcqWFepaDaaaaaOGaeiilaWcabaGaeGimaaJaeyipaWJaemiDaqNaeyipaWJae8hXdqhaaiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIZaWmaiaawIcacaGLPaaaaaa@5BF1@

Note the fact that f_L(t) ≡ f_K(t) i.e. the marginal distribution of the time since establishment of a derived genus in the family is the same as that of a derived species in the family.

Consider now the case when τ is the time of the first cataclysm since the appearance of the pioneering genus. The size distribution of all derived (non-pioneering) genera at the time of the cataclysm can be obtained by integrating the geometric pmf p_n(t; 1) in (1) with respect to the truncated exponential distribution f_K(t) between 0 and τ. This yields the pmf

q n d e r i v = ∫ 0 τ p n ( t ; 1 ) f K ( t ) d t = 1 + γ / λ 1 + e − ( λ + γ ) τ [ B ( 2 + γ / λ , n ) − B e − λ τ ( 2 + γ / λ , n ) ] ,       ( 14 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaGaemyCae3aa0baaSqaaiabd6gaUbqaaGqaaiab=rgaKjab=vgaLjab=jhaYjab=LgaPjab=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@856F@

where

B ( a , b ) = Γ ( a ) Γ ( b ) Γ ( a + b ) , B x ( a , b ) = ∫ 0 x z a − 1 ( 1 − z ) b − 1 d z MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@61D9@

are the beta function and incomplete beta functions, respectively. Alternatively the term in square brackets can be expressed in terms of the cumulative distribution function (cdf) F(x; a, b) of the beta distribution with parameters a and b leading to

q n d e r i v = ( 1 + γ / λ ) B ( 2 + γ / λ , n ) 1 − e − ( λ + γ ) τ [ 1 − F ( e − λ τ ; 2 + γ / λ , n ) ] .       ( 15 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@71C6@

This can be readily computed using standard statistical software.

The distribution of the size of the pioneering genus at time τ has pmf qnpion MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaqhaaWcbaGaemOBa4gabaacbaGae8hCaaNae8xAaKMae83Ba8Mae8NBa4gaaaaa@3532@ = p_n(τ; n₀) where p_n is negative binomial pmf given by (2). The distribution of the size of all existing genera at time τ is simply a mixture of qnpion MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaqhaaWcbaGaemOBa4gabaacbaGae8hCaaNae8xAaKMae83Ba8Mae8NBa4gaaaaa@3532@ and qnderiv MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaqhaaWcbaGaemOBa4gabaacbaGae8hzaqMae8xzauMae8NCaiNae8xAaKMae8NDayhaaaaa@367F@. Precisely

q n = π K ( τ ) q n p i o n + [ 1 − π K ( τ ) ] q n d e r i v ,       ( 16 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaWgaaWcbaGaemOBa4gabeaakiabg2da9GGaciab=b8aWnaaBaaaleaacqWGlbWsaeqaaOGaeiikaGIae8hXdqNaeiykaKIaemyCae3aa0baaSqaaiabd6gaUbqaaGqaaiab+bhaWjab+LgaPjab+9gaVjab+5gaUbaakiabgUcaRiabcUfaBjabigdaXiabgkHiTiab=b8aWnaaBaaaleaacqWGlbWsaeqaaOGaeiikaGIae8hXdqNaeiykaKIaeiyxa0LaemyCae3aa0baaSqaaiabd6gaUbqaaiab+rgaKjab+vgaLjab+jhaYjab+LgaPjab+zha2baakiabcYcaSiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI2aGnaiaawIcacaGLPaaaaaa@5AF3@

where π_K(τ) is the probability that a genus in existence at time τ is the pioneering genus, i.e.

π K ( τ ) = E ( 1 K τ ) = ∫ 0 1 Φ K ( s , τ ) s d s ,       ( 17 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFapaCdaWgaaWcbaGaem4saSeabeaakiabcIcaOiab=r8a0jabcMcaPiabg2da9Gqaaiab+veafnaabmaabaWaaSaaaeaacqaIXaqmaeaacqWGlbWsdaWgaaWcbaGae8hXdqhabeaaaaaakiaawIcacaGLPaaacqGH9aqpdaWdXaqaamaalaaabaGaeuOPdy0aaSbaaSqaaiabdUealbqabaGccqGGOaakcqWGZbWCcqGGSaalcqWFepaDcqGGPaqkaeaacqWGZbWCaaaaleaacqaIWaamaeaacqaIXaqma0Gaey4kIipakiabdsgaKjabdohaZjabcYcaSiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI3aWnaiaawIcacaGLPaaaaaa@5271@

which can be evaluated as

π K ( τ ) = ( λ + γ ) e − ( λ + γ ) τ γ [ 1 − e − ( λ + γ ) τ ] log ⁡ ( γ + λ e − ( λ + γ ) τ ( λ + γ ) e − ( λ + γ ) τ ) .       ( 18 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFapaCdaWgaaWcbaGaem4saSeabeaakiabcIcaOiab=r8a0jabcMcaPiabg2da9maalaaabaGaeiikaGIae83UdWMaey4kaSIae83SdCMaeiykaKIaemyzau2aaWbaaSqabeaacqGHsislcqGGOaakcqWF7oaBcqGHRaWkcqWFZoWzcqGGPaqkcqWFepaDaaaakeaacqWFZoWzcqGGBbWwcqaIXaqmcqGHsislcqWGLbqzdaahaaWcbeqaaiabgkHiTiabcIcaOiab=T7aSjabgUcaRiab=n7aNjabcMcaPiab=r8a0baakiabc2faDbaacyGGSbaBcqGGVbWBcqGGNbWzdaqadaqaamaalaaabaGae83SdCMaey4kaSIae83UdWMaemyzau2aaWbaaSqabeaacqGHsislcqGGOaakcqWF7oaBcqGHRaWkcqWFZoWzcqGGPaqkcqWFepaDaaaakeaacqGGOaakcqWF7oaBcqGHRaWkcqWFZoWzcqGGPaqkcqWGLbqzdaahaaWcbeqaaiabgkHiTiabcIcaOiab=T7aSjabgUcaRiab=n7aNjabcMcaPiab=r8a0baaaaaakiaawIcacaGLPaaacqGGUaGlcaWLjaGaaCzcamaabmaabaGaeGymaeJaeGioaGdacaGLOaGaayzkaaaaaa@7E0F@

Note that as τ → ∞, π_K(τ) → 0 and

q n → ( γ / λ + 1 ) Γ ( γ / λ + 2 ) Γ ( n ) Γ ( γ / λ + n + 2 ) .       ( 19 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaWgaaWcbaGaemOBa4gabeaakiabgkziUoaalaaabaGaeiikaGccciGae83SdCMaei4la8Iae83UdWMaey4kaSIaeGymaeJaeiykaKIaeu4KdCKaeiikaGIae83SdCMaei4la8Iae83UdWMaey4kaSIaeGOmaiJaeiykaKIaeu4KdCKaeiikaGIaemOBa4MaeiykaKcabaGaeu4KdCKaeiikaGIae83SdCMaei4la8Iae83UdWMaey4kaSIaemOBa4Maey4kaSIaeGOmaiJaeiykaKcaaiabc6caUiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI5aqoaiaawIcacaGLPaaaaaa@5827@

This distribution was obtained by Yule1 and is now known as the Yule distribution; for this distribution q_n behaves asymptotically like a power-law, i.e.,

q_n ~ (γ/λ + 1)Γ(γ/λ + 2) × n^{-(2 + γ/λ)}

as n → ∞, yielding the asymptotic straight line when q_n is plotted against n on logarithmic axes. We note in passing that setting γ = 0 in (19) does not yield the size distribution (as τ → ∞) of a single genus, since when γ = 0, π_K ≡ 1. In this case N_τ → ∞ with probability one.

Figure 1 shows the size distribution of pioneering and derived genera, along with the mixed distribution of all genera, calculated from the above formulae, for different values of n₀ and τ. They show how the results of Yule 1 need to be modified to take into account the effects of: (a) the evolution of new genera ; (b) pioneering genera of size (n₀) greater than one; and (c) the time, τ, until cataclysmic extinction. Large values of τ (right-hand panels), resulting in straight-line plots on the log-log scale, correspond most closely to the situation considered initially by Yule. In this case approximate power-law (fractal) distributions occur. The deviations from such a power-law distribution are greatest when cataclysmic extinction occurs earlier (smaller τ) and when the number of species in the pioneering genus (n₀) differs greatly from one (lower panels). The distribution of derived genera (dotted lines) is unaffected by the initial size (n₀) of the pioneering genus. However the overall size distribution is affected (especially at values immediately above n₀) because of the fact that the pioneering genus size has support on {n₀, n₀ + 1,...} while that of derived genera is on {1, 2,...}. This effect becomes less important when a long time elapses before the cataclysmic extinction event (because when τ is large, π_K(τ) is small–derived genera will in probability outnumber the pioneering one).

In this section we consider the size distribution of a fossil genus, starting with a single species (the case of a genus beginning with n₀ species is considered later in this section), subject to speciations at rate λ and background (individual) extinctions occurring independently and at random, at rate μ.

Thus N_t, the number of species alive t time units after the origin of the genus, follows a homogeneous birth and death process. Let M_t denote the total number of species in the genus that have existed by time t (i.e. M_t = 1 + number of speciations). The size of an extinct genus is a random variable M_T, where T itself is a random variable, denoting the time of extinction. Since no speciations can occur in a genus once it is extinct, we have that for t ≥ T, M_t ≡ M_T. However T may not be finite (N_t > 0 for all t). Thus finding the distribution of the size of an extinct genus will involve conditioning on T < ∞ (or N_∞ = 0). Clearly it is given by the distribution of M_∞ conditional on N_∞ = 0.

Now let

p_{m, n}(t) = Pr(M_t = m, N_t = n).     (20)

It was shown by Kendall7 that p_{m, n} satisfies the differential-difference equations

d d t p m , n ( t ) = − ( λ + μ ) n p m , n ( t ) + λ ( n − 1 ) p m − 1 , n − 1 ( t ) + μ ( n + 1 ) p m , n + 1 ( t )       ( 21 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKbqaaiabdsgaKjabdsha0baacqWGWbaCdaWgaaWcbaGaemyBa0MaeiilaWIaemOBa4gabeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iabgkHiTiabcIcaOGGaciab=T7aSjabgUcaRiab=X7aTjabcMcaPiabd6gaUjabdchaWnaaBaaaleaacqWGTbqBcqGGSaalcqWGUbGBaeqaaOGaeiikaGIaemiDaqNaeiykaKIaey4kaSIae83UdWMaeiikaGIaemOBa4MaeyOeI0IaeGymaeJaeiykaKIaemiCaa3aaSbaaSqaaiabd2gaTjabgkHiTiabigdaXiabcYcaSiabd6gaUjabgkHiTiabigdaXaqabaGccqGGOaakcqWG0baDcqGGPaqkcqGHRaWkcqWF8oqBcqGGOaakcqWGUbGBcqGHRaWkcqaIXaqmcqGGPaqkcqWGWbaCdaWgaaWcbaGaemyBa0MaeiilaWIaemOBa4Maey4kaSIaeGymaedabeaakiabcIcaOiabdsha0jabcMcaPiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaIXaqmaiaawIcacaGLPaaaaaa@750B@

with initial condition

p_{m, n}(0) = 1 if m = n = 1; p_{m, n}(0) = 0 otherwise.

Let

Ψ ( s , z ; t ) = ∑ m = 1 ∞ ∑ n = 0 ∞ p m , n ( t ) s m z n       ( 22 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHOoqwcqGGOaakcqWGZbWCcqGGSaalcqWG6bGEcqGG7aWocqWG0baDcqGGPaqkcqGH9aqpdaaeWbqaamaaqahabaGaemiCaa3aaSbaaSqaaiabd2gaTjabcYcaSiabd6gaUbqabaGccqGGOaakcqWG0baDcqGGPaqkcqWGZbWCdaahaaWcbeqaaiabd2gaTbaakiabdQha6naaCaaaleqabaGaemOBa4gaaaqaaiabd6gaUjabg2da9iabicdaWaqaaiabg6HiLcqdcqGHris5aaWcbaGaemyBa0Maeyypa0JaeGymaedabaGaeyOhIukaniabggHiLdGccaWLjaGaaCzcamaabmaabaGaeGOmaiJaeGOmaidacaGLOaGaayzkaaaaaa@5878@

be the generating function for M_t, N_t. Muliplying both sides of (21) by s^m zⁿ and summing over m = l,... ∞; n = 0,...,∞ yields the partial differential equation

Ψ_t = (sz²λ - (λ + μ)z + μ)Ψ_z.     (23)

This equation was derived and solved by Kendall7, using the method of characteristics. The solution is (for λ ≠ μ)

Ψ ( s , z ; t ) = β ( s z − α ) exp ⁡ ( λ α t ) + α ( β − s z ) exp ⁡ ( λ β t ) ( s z − α ) exp ⁡ ( λ α t ) + ( β − s z ) exp ⁡ ( λ β t ) ,       ( 24 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHOoqwcqGGOaakcqWGZbWCcqGGSaalcqWG6bGEcqGG7aWocqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaGGaciab=j7aIjabcIcaOiabdohaZjabdQha6jabgkHiTiab=f7aHjabcMcaPiGbcwgaLjabcIha4jabcchaWjabcIcaOiab=T7aSjab=f7aHjabdsha0jabcMcaPiabgUcaRiab=f7aHjabcIcaOiab=j7aIjabgkHiTiabdohaZjabdQha6jabcMcaPiGbcwgaLjabcIha4jabcchaWjabcIcaOiab=T7aSjab=j7aIjabdsha0jabcMcaPaqaaiabcIcaOiabdohaZjabdQha6jabgkHiTiab=f7aHjabcMcaPiGbcwgaLjabcIha4jabcchaWjabcIcaOiab=T7aSjab=f7aHjabdsha0jabcMcaPiabgUcaRiabcIcaOiab=j7aIjabgkHiTiabdohaZjabdQha6jabcMcaPiGbcwgaLjabcIha4jabcchaWjabcIcaOiab=T7aSjab=j7aIjabdsha0jabcMcaPaaacqGGSaalcaWLjaGaaCzcamaabmaabaGaeGOmaiJaeGinaqdacaGLOaGaayzkaaaaaa@88F5@

where α = α(s), β = β(s) are the two (positive) roots of the quadratic equation

λx² - (λ + μ)x + μs = 0.     (25)

These roots are distinct for 0 ≤ s ≤ 1, except when λ = μ, where the roots are distinct for 0 ≤ s ≤ 1, but coincide for s = 1. We select β(s) to be the smaller root, so that

β ( s ) =