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  • 1.
    Hössjer, Ola
    Stockholm University, Faculty of Science, Department of Mathematics.
    Spatial Autocorrelation for Subdivided Populations with Invariant Migration Schemes2014In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 16, no 4, p. 777-810Article in journal (Refereed)
    Abstract [en]

    For populations with geographic substructure and selectively neutral genetic data, the short term dynamics is a balance between migration and genetic drift. Before fixation of any allele, the system enters into a quasi equilibrium (QE) state. Hossjer and Ryman (2012) developed a general QE methodology for computing approximations of spatial autocorrelations of allele frequencies between subpopulations, subpopulation differentiation (fixation indexes) and variance effective population sizes. In this paper we treat a class of models with translationally invariant migration and use Fourier transforms for computing these quantities. We show how the QE approach is related to other methods based on conditional kinship coefficients between subpopulations under mutation-migration-drift equilibrium. We also verify that QE autocorrelations of allele frequencies are closely related to the expected value of Moran's autocorrelation function and treat limits of continuous spatial location (isolation by distance) and an infinite lattice of subpopulations. The theory is illustrated with several examples including island models, circular and torus stepping stone models, von Mises models, hierarchical island models and Gaussian models. It is well known that the fixation index contains information about the effective number of migrants. The spatial autocorrelations are complementary and typically reveal the type of migration (local or global).

  • 2.
    Petersson, Mikael
    Stockholm University, Faculty of Science, Department of Mathematics.
    Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time2017In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 19, no 4, p. 1047-1074Article in journal (Refereed)
    Abstract [en]

    We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behavior of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with a recursive algorithm for computing the coefficients in the expansions. An application to perturbed epidemic SIS models is discussed.

  • 3.
    Silvestrov, Dmitrii
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Manca, Raimondo
    Reward Algorithms for Semi-Markov Processes2017In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 19, no 4, p. 1191-1209Article in journal (Refereed)
    Abstract [en]

    New algorithms for computing power moments of hitting times and accumulated rewards of hitting type for semi-Markov processes are developed. The algorithms are based on special techniques of sequential phase space reduction and recurrence relations connecting moments of rewards. Applications are discussed as well as possible generalizations of presented results and examples.

  • 4.
    Silvestrov, Dmitrii
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Silvestrov, Sergei
    Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 12017In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713Article in journal (Refereed)
    Abstract [en]

    New algorithms for construction of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces are presented. These algorithms are based on a special technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces. Asymptotic expansions are given in two forms, without and with explicit upper bounds for remainders.

  • 5.
    Silvestrov, Dmitrii
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Silvestrov, Sergei
    Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 22017In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713Article in journal (Refereed)
    Abstract [en]

    Asymptotic expansions with explicit upper bounds for remainders are given for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces. The corresponding algorithms are based on a special technique of sequen- tial phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces. 

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