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On Two-Type Branching Random Walks and Their Applications for Genetic Modelling

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Recent Developments in Stochastic Methods and Applications (ICSM-5 2020)

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Abstract

We consider a model of the evolution of a population in the presence of epistatic lethal alleles. A model which describes the evolution of lethal and non-lethal alleles based on two-type branching random walks on multidimensional lattices is presented. We study this model in terms of subpopulations of particles generated by a single particle of each type located at every lattice point. The differential equations for the generating functions and factorial moments for the particle subpopulations are obtained. For the first moments we get explicit solutions for cases significant in the genetic context. The asymptotic behaviour for the first moments of particle distribution at lattice points is obtained for a random walk with finite variance of jumps.

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References

  1. Bulinskaya, E.V.: Spread of a catalytic branching random walk on a multidimensional lattice. Stochast. Process. Appl. 128(7), 2325–2340 (2015)

    Article  MathSciNet  Google Scholar 

  2. Dawson, D.A.: Introductory lectures on stochastic population systems. arXiv:1705.03781 (2017)

  3. Ermakova, E., Mahmutova, P., Yarovaya, E.: Branching random walks and their applications for epidemic modelling. Stoch. Model. 35(3), 300–317 (2019)

    Article  Google Scholar 

  4. di Bruno, F.: Sullo sviloppo dell Funczioni. In: Annali di Scienze Mathematiche e Fisiche, pp. 479–480 (1855). (in Italian)

    Google Scholar 

  5. Gillespie, J.H.: Population Genetics: A Concise Guide. JHU Press, Baltimore (2005)

    Google Scholar 

  6. Gluecksohn-Waelsch, S.: Lethal genes and analysis of differentiation. Science 142(3597), 1269–1276 (1963)

    Article  Google Scholar 

  7. Hartl, D.L., Clark, A.G.: Principles of Population Genetics. Sinauer Associates, Sunderland (1997)

    Google Scholar 

  8. Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes. Academic Press, Cambridge (2012)

    MATH  Google Scholar 

  9. Kolmogorov, A.N., Petrovskii, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the quality of matter, and its application to a biological problem. Bull. Moscow Univ. Math. Ser. A 1(6), 1–25 (1937). (in Russian)

    Google Scholar 

  10. Kong, A., Frigge, M.L., Masson, G., Besenbacher, S., et al.: Rate of de novo Mutations and the importance of father’s age to disease risk. Nature 488(7412), 471–475 (2012)

    Article  Google Scholar 

  11. Makarova, Y., Han, D., Molchanov, S., Yarovaya, E.: Branching random walks with immigration. Lyapunov stability. Markov Process. Relat. Fields 25(4), 683–708 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Molchanov, S.A., Yarovaya, E.B.: Large deviations for a symmetric branching random walk on a multidimensional lattice. Proc. Steklov Inst. Math. 282, 186–201 (2013). https://doi.org/10.1134/S0081543813060163

    Article  MathSciNet  MATH  Google Scholar 

  13. Sevastyanov, B.A.: Branching Processes. Nauka, Moscow (1971). (in Russian)

    Google Scholar 

  14. Yarovaya, E.B.: Branching random walks in a heterogeneous environment. Center of Applied Investigations of the Faculty of Mechanics and Mathematics of the Moscow State University, Moscow (2007). (in Russian)

    Google Scholar 

  15. Young, A.I.: Solving the missing heritability problem. PLoS Genet. 15(6), e1008222 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are grateful to Prof. S. Molchanov for useful discussions. The study was supported by the Russian Foundation for the Basic Research (RFBR), project No. 20-01-00487.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119234, Russia

    Yulia Makarova, Vladimir Kutsenko & Elena Yarovaya

Authors
  1. Yulia Makarova
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  2. Vladimir Kutsenko
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  3. Elena Yarovaya
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Corresponding author

Correspondence to Yulia Makarova .

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Editors and Affiliations

  1. Steklov Mathematical Institute of RAS, Moscow, Russia

    Albert N. Shiryaev

  2. Applied Probability & Informatics, Department of Applied Probability and Informatics, RUDN University, Moscow, Russia

    Konstantin E. Samouylov

  3. Applied Probability & Informatics, Department of Applied Probability and Informatics, RUDN University, Moscow, Russia

    Dmitry V. Kozyrev

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Makarova, Y., Kutsenko, V., Yarovaya, E. (2021). On Two-Type Branching Random Walks and Their Applications for Genetic Modelling. In: Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. (eds) Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-030-83266-7_19

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