# Species abundance distributions in neutral models with immigration or mutation and general lifetimes

Abstract : 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.
Type de document :
Pré-publication, Document de travail
16 pages, 4 figures. To appear in Journal of Mathematical Biology. The final publication is avail.. 2010
Domaine :

https://hal.archives-ouvertes.fr/hal-00514384
Contributeur : Amaury Lambert <>
Soumis le : jeudi 2 septembre 2010 - 10:13:24
Dernière modification le : mercredi 12 octobre 2016 - 01:02:27

### Identifiants

• HAL Id : hal-00514384, version 1
• ARXIV : 1009.0118

### Citation

Amaury Lambert. Species abundance distributions in neutral models with immigration or mutation and general lifetimes. 16 pages, 4 figures. To appear in Journal of Mathematical Biology. The final publication is avail.. 2010. <hal-00514384>

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