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Zbl 1230.92043
Lambert, Amaury
Species abundance distributions in neutral models with immigration or mutation and general lifetimes. (English)
[J] J. Math. Biol. 63, No. 1, 57-72 (2011). ISSN 0303-6812; ISSN 1432-1416/e

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 $I_{n}(k)$ in the immigration model and $A_{n}(k)$ in the mutation model, of the species represented by $k$ individuals, $k=1,2,\dots,n$, when there are $n$ individuals in the total population. In the immigration model, we prove that the numbers $(I_{t}(k);\ k \geq 1)$ 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 {\it W.J. Ewens}' [Theor. Popul. Biol. 3, 87--112 (1972; Zbl 0245.92009)] sampling formula. In particular, $I _{n}(k)$ converges as $n \rightarrow \infty$ to a Poisson r.v. with mean $\gamma/k$, where $\gamma := \mu/\lambda$. In the mutation model, as $n \rightarrow \infty$, 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.

MSC 2000:: *92D15 Problems related to evolution
92D40 Ecology
60J85 Appl. of branching processes
92D25 Population dynamics
60J80 Branching processes
60G51 Processes with independent increments

Keywords: Crump-Mode-Jagers process; Splitting tree; Branching process; Linear birth-death process; Immigration; Mutation; Infinitely-many alleles model; Fisher logarithmic series; Ewens sampling formula; Coalescent point process; Scale function

Citations: Zbl 0245.92009

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Zentralblatt MATH Berlin [Germany]