No abstract

We consider the compact space of pairs of nested partitions of N, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of Λ-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo Λ-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. Λ-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure ν s on (0, 1] × M 1 ([0, 1]) providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.

For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by , namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach, we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of "potential" selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the "active" selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz's Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation.

Consider a doubly infinite branching tree in varying environment. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process (A i , i ≥ 1), where A i is the coalescent time between individuals i and i + 1. In general, this process is not Markovian. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures (B i , i ≥ 1) to reconstruct the coalescent point process. We present a counterexample where we show that their process has not the Markov property.We define a vector valued Markov process with the minimal amount of information to reconstruct the genealogy of the standing population. We highlight that this process is the Markovian correction of (B i , i ≥ 1). Finally, when the offspring distributions are lineal fractional, we show that the variables (A i , i ≥ 1) are independent and identically distributed.

In this paper we study the genealogical structure of a Galton-Watson process with neutral mutations, where the initial population is large and mutation rate is small [3]. Namely, we extend in two directions the results obtained in Bertoin's work. In the critical case, we construct the version of Bertoin's model conditioned not to be extinct, and in the case with finite variance we show convergence after normalization, of allelic sub-populations towards a tree indexed CSBP with immigration. Besides, we establish the version of the limit theorems in [3], been for the unconditioned process and for the process conditioned to non-extinction, in the case where the reproduction law has infinite variance and it is in the domain of attraction of an α-stable distribution.