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Species abundance distributions in neutral models with immigration or mutation and general lifetimes. (English) Zbl 1230.92043
Summary: We consider a general, neutral, dynamical model of biodiversity. Individuals have i.i.d. life time durations, which are not necessarily exponentially distributed, and each individual gives birth independently at a constant rate $\lambda$. Thus, the population size is a homogeneous, binary Crump-Mode-Jagers process (which is not necessarily a Markov process). 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 a 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 a constant rate $\theta$. We are interested in the species abundance distribution, i.e., in the numbers, denoted ${I}_{n}\left(k\right)$ in the immigration model and ${A}_{n}\left(k\right)$ in the mutation model, of the species represented by $k$ individuals, $k=1,2,\cdots ,n$, when there are $n$ individuals in the total population. In the immigration model, we prove that the numbers $\left({I}_{t}\left(k\right);\phantom{\rule{4pt}{0ex}}k\ge 1\right)$ of the 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 W.J. Ewens’ [Theor. Popul. Biol. 3, 87–112 (1972; Zbl 0245.92009)] sampling formula. In particular, ${I}_{n}\left(k\right)$ 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 ${n}^{-1}{A}_{n}\left(k\right)$ 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 ${n}^{-1}{A}_{n}\left(k\right)$ converges to ${\alpha }^{k}/k$, where $\alpha :=\lambda /\left(\lambda +\theta \right)$. In both models, the abundances of the most abundant species are briefly discussed.
##### MSC:
 92D15 Problems related to evolution 92D40 Ecology 60J85 Applications of branching processes 92D25 Population dynamics (general) 60J80 Branching processes 60G51 Processes with independent increments; Lévy processes
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