Splitting trees with neutral Poissonian mutations I: Small families

We 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 [9], and the population counting process ( N t ; t ≥ 0 ) is a homogeneous, binary Crump–Mode–Jagers process. ume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called an 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 . nly use two classes of tools: coalescent point processes, as defined in [15], and branching processes counted by random characteristics, as defined in [11,13]. 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. We separately derive the a.s. limits of A ( k , t ) / N t and of A ( t ) / N t thanks to random characteristics, in the same vein as in [19]. we separately compute the expected homozygosity by applying a method introduced in [14], 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.