12 research outputs found

    Absorption and stationary times for the Λ\Lambda-Wright-Fisher process

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    We derive stationary and fixation times for the multi-type Λ\Lambda-Wright-Fisher process with and without the classic linear drift that models mutations. Our method relies on a grand coupling of the process realized through the so-called lookdown-construction. A well-known process embedded in this construction is the fixation line. We generalise the process to our setup and make use of the associated explosion times to obtain a representation of the fixation and stationary times in terms of the waiting time in a coupon collector problem

    Evolving genealogies for branching populations under selection and competition

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    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 Donnelly and Kurtz (1999), 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

    The Nested Kingman Coalescent:Speed of Coming Down from Infinity

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    The nested Kingman coalescent describes the ancestral tree of a population undergoing neutral evolution at the level of individuals and at the level of species, simultaneously. We study the speed at which the number of lineages descends from infinity in this hierarchical coalescent process and prove the existence of an early-time phase during which the number of lineages at time tt decays as 2γ/ct2 2\gamma/ct^2, where cc is the ratio of the coalescence rates at the individual and species levels, and the constant γ≈3.45\gamma\approx 3.45 is derived from a recursive distributional equation for the number of lineages contained within a species at a typical time.Comment: 24 page

    Coalescent point process of branching trees in varying environment

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    Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals in the same generation reproduce independently, forward in time, with the same offspring distribution but potentially changing among generations. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process (Ai,i≥1)(A_i, i\geq 1), where AiA_i is the coalescent time between individuals ii and i+1i+1. In general, this process is not Markov. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We present a counterexample where we show that their process does not have the Markov property. The main contribution of this work is to propose a vector valued Markov process (Bi,i≥1)(B_i,i\geq 1), that reach the goal to reconstruct the genealogy, with finite information for every ii. Additionally, when the offspring distributions are lineal fractional, we show that the variables (Ai,i≥1)(A_i, i\geq 1) are independent and identically distributed

    On branching process with rare neutral mutation

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    Non UBCUnreviewedAuthor affiliation: Centro de Investigación en MatemáticasOthe