caa a22 4500 445329823 CHVBK 20180317142733.0 cr unu---uuuuu 170323e20110701xx s 000 0 eng 10.1007/s00285-010-0361-9 doi (NATIONALLICENCE)springer-10.1007/s00285-010-0361-9 Lambert Amaury Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 CNRS and UPMC Univ Paris 06, Case courrier 188, 4, Place Jussieu, 75252, Paris Cedex 05, France aut Species abundance distributions in neutral models with immigration or mutation and general lifetimes [Elektronische Daten] [Amaury Lambert] 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 λ. 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 constant rateμ (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 θ. 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 ≥ 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 a Poisson r.v. with mean γ/k, where γ :=μ/λ. In the mutation model, as n → ∞, 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 α k/k, where α :=λ/(λ+θ). In both models, the abundances of the most abundant species are briefly discussed. Springer-Verlag, 2010 Species abundance distribution nationallicence Crump-Mode-Jagers process nationallicence Splitting tree nationallicence Branching process nationallicence Linear birth-death process nationallicence Immigration nationallicence Mutation nationallicence Infinitely-many alleles model nationallicence Fisher logarithmic series nationallicence Ewens sampling formula nationallicence Coalescent point process nationallicence Scale function nationallicence Journal of Mathematical Biology Springer-Verlag 63/1(2011-07-01), 57-72 0303-6812 63:1<57 2011 63 285 https://doi.org/10.1007/s00285-010-0361-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00285-010-0361-9 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Lambert Amaury Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 CNRS and UPMC Univ Paris 06, Case courrier 188, 4, Place Jussieu, 75252, Paris Cedex 05, France aut NATIONALLICENCE 773 0- Journal of Mathematical Biology Springer-Verlag 63/1(2011-07-01), 57-72 0303-6812 63:1<57 2011 63 285 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer