2010/09/01 by Amaury Lambert, Lambert, Amaury
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Social Sciences · #60G51 #60G55 #60G70 (Secondary) #60J80 #60J85 #92D15 (Primary) 92D25 #92D40 #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Biological sciences #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE) #Probability (math.PR) #math.PR #msc:60G51 #msc:60G55 #msc:60G70 #msc:60J80 #msc:60J85 #msc:92D15 #msc:92D25 #msc:92D40 #q-bio.PE
paper · pdf · doi:10.48550/arxiv.1009.0118
16 pages, 4 figures. To appear in Journal of Mathematical Biology. The final publication is available at http://www.springerlink.com
arxiv created 2010/09/01 · openalex publication_date 2010/09/01 · arxiv updated 2010/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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 μ(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 In(k) in the immigration model and An(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 (It(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, In(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-1An(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-1An(k) converges to αk/k, where α:=λ/(λ+θ). In both models, the abundances of the most abundant species are briefly discussed.