Probability Theory and Related Fields

, Volume 73, Issue 1, pp 87–117 | Cite as

Stationary states and their stability of the stepping stone model involving mutation and selection

  • Tokuzo Shiga
  • Kohei Uchiyama


Stochastic Process Stationary State Probability Theory Mathematical Biology Step Stone 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Durrett, R.: Oriented percolation in two dimensions. Ann. Probab.12, 999–1040 (1984)Google Scholar
  2. 2.
    Ethier, S.N.: A class of infinite dimensional diffusions occurring in population genetics. Indiana Univ. Math. J.30, 925–935 (1981)Google Scholar
  3. 3.
    Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab.3, 643–663 (1975)Google Scholar
  4. 4.
    Holley, R.A., Stroock, D.: Diffusions on an infinite dimensional torus. J. Funct. Anal.42, 29–63 (1981)Google Scholar
  5. 5.
    Itatsu, S.: Equilibrium measures of the stepping stone model in population genetics. Nagoya Math. J.83, 37–51 (1981)Google Scholar
  6. 6.
    Itatsu, S.: Equilibrium measures of the stepping stone model with selection in population genetics, pp. 257–266. In: Ohta, T., Aoki, K. (eds.). Population genetics and molecular evolution. Japan Sci. Soc. Press. Berlin-Heidelberg-New York: Springer 1985Google Scholar
  7. 7.
    Kimura, M.: “Stepping stone” model of population. Ann. Rept. Nat. Inst. Genetics Japan3, 62–63 (1953)Google Scholar
  8. 8.
    Kimura, M., Weiss, G.H.: The stepping stone model of population structure and decrease of genetical correlation with distance. Genetics49, 561–576 (1964)Google Scholar
  9. 9.
    Liggett, T.M.: Interacting particle systems. Berlin-Heidelberg-New York. Springer 1985Google Scholar
  10. 10.
    Pollard, D.: Convergence of stochastic processes. Berlin-Heidelberg-New York: Springer 1984Google Scholar
  11. 11.
    Sato, K.: Limit diffusions of some stepping stone models. J. Appl. Probab.20, 460–471 (1983)Google Scholar
  12. 12.
    Sawyer, S.: Results for the stepping stone model for migration in population genetics. Ann. Probab.4, 699–726 (1976)Google Scholar
  13. 13.
    Shiga, T.: An interacting system in population genetics I, and II. J. Math. Kyoto Univ.20, 212–242 and 723–733 (1980)Google Scholar
  14. 14.
    Shiga, T., Shimizu, A.: Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ.20, 395–416 (1980)Google Scholar
  15. 15.
    Shiga, T.: Continuous time multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ.22, 1–40 (1982)Google Scholar
  16. 16.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat.36, 423–439 (1965)Google Scholar
  17. 17.
    Tweedie, R.L.: Criteria for classifying general Markov chains. Adv. Appl. Probab.8, 737–771 (1976)Google Scholar
  18. 18.
    Uchiyama, K.: Spatial growth of a branching process of particles living inR d. Anal. Probab.10, 896–918 (1982)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Tokuzo Shiga
    • 1
  • Kohei Uchiyama
    • 2
  1. 1.Department of Applied PhysicsTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of MathematicsNara Women's UniversityNaraJapan

Personalised recommendations