Advertisement

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
Article

Keywords

Stochastic Process Stationary State Probability Theory Mathematical Biology Step Stone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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