Convergence of general branching processes and functionals thereof

Stochastic Processes and their Applications

2012

32 pages, 2 figures. Companion paper in preparation "Splitting trees with neutral Poissonian mutations II: Large or old families"International audienceWe consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree, and the population counting process (N_t;t\geq 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \theta during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes and branching processes counted by random characteristics. We provide explicit formulae for the expectation of A(k,t) in a coalescent point process conditional on population size, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics. Last, we separately compute the expected homozygosity by applying a method characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations

Section: Introductionmentioning

confidence: 99%

“…. , N t .We mainly use two classes of tools: coalescent point processes, as defined in [14], and branching processes counted by random characteristics, as defined in [10,11]. We provide explicit formulae for the expectation of A(k, t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees.…”

mentioning

confidence: 99%

Splitting trees with neutral Poissonian mutations I: Small families

Champagnat

Stochastic Processes and their Applications

2012

“…Theorem 5.3 For all k ≥ 1, the following convergence holds a.s., as n → ∞ for the coalescent point process, and as t → ∞ for the splitting tree in the supercritical case and on the event of non-extinction : Remark 3 The a.s. result for coalescent point processes relies on laws of large numbers (see [21]). The a.s. result for splitting trees relies on the theory of random characteristics (see [3]) introduced in the seminal paper [15] and further developed in [16,17] and especially in [27].…”

Section: The Mutation Modelmentioning

confidence: 99%

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

2010

J. Math. Biol.

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.

“…For example, similar questions were studied for general CMJ processes, when mutations occur at birth, in the monography due to Z. Taïb [18]. These results rely heavily on the theory of branching processes counted by random characteristics, due to P. Jagers and O. Nerman [8,9,10,15], and more specifically on time dependent random characteristics as developed in [10]. Z. Taïb obtains results of convergence in distribution of the (correctly rescaled) point process of ages, similar to the results we obtain in Sections 4 and 5.…”

Section: Introductionmentioning

confidence: 98%

Splitting trees with neutral Poissonian mutations II: Largest and oldest families

Champagnat

Stochastic Processes and their Applications

2013

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn.We are interested in the sizes and ages at time t of the clonal families carrying the most abundant alleles or the oldest ones, as t → ∞, on the survival event. Intuitively, the results must depend on how the mutation rate θ and the Malthusian parameter α > 0 compare. Hereafter, N ≡ N t is the population size at time t, constants a, c are scaling constants, whereas k, k ′ are explicit positive constants which depend on the parameters of the model.When α > θ, the most abundant families are also the oldest ones, they have size cN 1−θ/α and age t − a.When α < θ, the oldest families have age (α/θ)t+a and tight sizes; the most abundant families have sizes k log(N ) − k ′ log log(N ) + c and all have age (θ − α) −1 log(t).When α = θ, the oldest families have age kt − k ′ log(t) + a and tight sizes; the most abundant families have sizes (k log(N ) − k ′ log log(N ) + c) 2 and all have age t/2.Those informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes [13,16], we are also able, when α ≤ θ, to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes. This is in deep contrast with the largest/oldest families in the standard coalescent with Poissonian mutations, which converge to some point processes after being rescaled by N [3, 4, 5].MSC 2000 subject classifications: Primary 60J80; secondary 92D10, 60J85, 60G70, 60G51, 60G55, 60K15.