Journal of Mathematical Biology

, Volume 63, Issue 1, pp 57–72

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

Article

## 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 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 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.

### Keywords

Species abundance distribution 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

### Mathematics Subject Classification (2000)

92D15 92D25 92D40 60J80 60J85 60G51 60G55 60G70