Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T15:33:43.160Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic properties of the stepping stone model

Published online by Cambridge University Press:  22 January 2016

Seiichi Itatsu
Seiichi Itatsu*
Affiliation:
Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka, 422, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to discuss some ergodic properties of the stepping stone model proposed by Kimura, M. [4] and developed by Weiss, G.H. and Kimura, M. [12]. Our model to be discussed in this paper involves selection in addition to mutation and migration which are dealt with in [4], [12]. Because of the additional factor selection, the stochastic process describing our model becomes complicated and presents particularly interesting profound structure of the random phenomena in question.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 114 , June 1989 , pp. 143 - 163
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Durrett, R., Oriented percolation in two dimensions, Ann. Probab., 12 (1984), 999–1040.Google Scholar
[2] Durrett, R. and Schonmann, R. H., Stochastic growth models, In: Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, editor) Springer IMA volume 8 (1987), 85–119.Google Scholar
[3] Itatsu, S., Equilibrium measures of the stepping stone model with selection in population genetics, In: Ohta, T. Aoki, K. (eds.) Population genetics and molecular evolution, Japan Sci. Soc. Press, Tokyo/Springer-Verlag, Berlin 1985, pp. 257–266.Google Scholar
[4] Kimura, M., ‘Stepping stone’ model of population, Ann. Rept. Nat. Inst. Genetics, Japan, 3 (1953), 62–63.Google Scholar
[5] Liggett, T. M., Attractive nearest neighbor spin systems on the integers, Ann. Probab., 6, No. 4 (1978), 629–636.Google Scholar
[6] Liggett, T. M., Interaction particle systems, Springer-Verlag, Berlin-Heidelberg-New York 1985.Google Scholar
[7] Maruyama, T., Stochastic problems in population genetics, Springer Lecture Notes in Biomath., 17, 1977.Google Scholar
[8] Nagaev, S. V., Some limit theorems for large deviations, Theory Probab. Appl., 10, No. 2 (1965), 214–235.CrossRefGoogle Scholar
[9] Nagylaki, T., Random drift in a cline, Proc. Natl. Acad. Sci. USA, (1978), 423–426.Google Scholar
[10] Nagylaki, T., The geographical structure of populations, In: Studies in Mathematics, Vol. 16: Studies in Mathematical Biology, Part II. ed. Levin, S., The Mathematical Association of America, Washington, (1978), 588–624.Google Scholar
[11] Shiga, T. and Uchiyama, K., Stationary States and their stability of the stepping stone model involving mutation and selection, Probab. Th. Rel. Fields, 73, (1986), 87–117.Google Scholar
[12] Weiss, G. H. and Kimura, M., A mathematical analysis of the stepping-stone model of genetic correlation, J. Appl. Probab., 2 (1965), 129–149.Google Scholar