SMILE

Stochastic Models for the Inference of Life Evolution

Bibtex

@article{lambert_species_2011,
Author = {Lambert, Amaury},
Title = {Species abundance distributions in neutral models with
immigration or mutation and general lifetimes},
Journal = {Journal of Mathematical Biology},
Volume = {63},
Number = {1},
Pages = {57--72},
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
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.},
doi = {10.1007/s00285-010-0361-9},
issn = {1432-1416},
language = {eng},
month = jul,
pmid = {20809352},
year = 2011
}