  We consider a general, neutral, dynamical model of biodiversity. Individuals
have i.i.d. lifetime durations, which are not necessarily exponentially
distributed, and each individual gives birth independently at constant rate
\lambda. We assume that types are clonally inherited. We consider two classes
of speciation models in this setting. In the immigration model, new individuals
of an entirely new species singly enter the population at constant rate \mu
(e.g., from the mainland into the island). In the mutation model, each
individual independently experiences point mutations in its germ line, at
constant rate \theta. We are interested in the species abundance distribution,
i.e., in the numbers, denoted I_n(k) in the immigration model and A_n(k) in the
mutation model, of species represented by k individuals, k=1,2,...,n, when
there are n individuals in the total population. In the immigration model, we
prove that the numbers (I_t(k);k\ge 1) of species represented by k individuals
at time t, are independent Poisson variables with parameters as in Fisher's
log-series. When conditioning on the total size of the population to equal n,
this results in species abundance distributions given by Ewens' sampling
formula. In particular, I_n(k) converges as n\to\infty to a Poisson r.v. with
mean \gamma /k, where \gamma:=\mu/\lambda. In the mutation model, as
n\to\infty, we obtain the almost sure convergence of n^{-1}A_n(k) to a
nonrandom explicit constant. In the case of a critical, linear birth--death
process, this constant is given by Fisher's log-series, namely n^{-1}A_n(k)
converges to \alpha^{k}/k, where \alpha :=\lambda/(\lambda+\theta). In both
models, the abundances of the most abundant species are briefly discussed.
