Species abundance distributions in neutral models with immigration or mutation and general lifetimes
Résumé
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. 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 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 a parts per thousand yen 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 -> a to a Poisson r.v. with mean gamma/k, where gamma : = mu/lambda. In the mutation model, as n -> a, 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.