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Branching diffusion processes in population genetics

Published online by Cambridge University Press:  01 July 2016

Stanley Sawyer
Stanley Sawyer*
Affiliation:
Yeshiva University, New York
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Abstract

A branching random field is considered as a model of either of two situations in genetics in which migration or dispersion plays a role. Specifically we consider the expected number of individuals NA in a (geographical) set A at time t, the covariance of NA and NB for two sets A, B, and the probability I(x, y, u) that two individuals found at locations x, y at time t are of the same genetic type if the population is subject to a selectively neutral mutation rate u. The last also leads to limit laws for the average degree of relationship of individuals in various types of branching random fields. We also find the equations that the mean and bivariate densities satisfy, and explicit formulas when the underlying migration process is Brownian motion.

Keywords

BRANCHING PROCESSBRANCHING RANDOM FIELDIDENTITY BY DESCENTGENETICSMIGRATIONPOISSON RANDOM FIELDPOINT PROCESSSTEPPING-STONE MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 659 - 689
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Athreya, K. and Ney, P. (1972) Branching Processes. Grundlehren der math. Wissenschaften, 196. Springer-Verlag, New York.CrossRefGoogle Scholar
[2] Bailey, N. (1968) Stochastic birth, death and migration processes for spatially distributed populations. Biometrika 55, 189–198.CrossRefGoogle Scholar
[3] Buhler, W. (1971) Generations and the degree of relationship in a supercritical Markov branching process. Z. Wahrscheinlichkeitsth. 18, 141–152.CrossRefGoogle Scholar
[4] Bühler, W. (1972) The distribution of generations and other aspects of the family structure of branching processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 463–480.Google Scholar
[5] Crow, J. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
[6] Dawson, D. (1972) Stochastic evolution equations. Math. Biosci. 15, 287–316, Appendix I.Google Scholar
[7] Dawson, D. (1975) Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5, 1–52.CrossRefGoogle Scholar
[8] Dynkin, E. (1965) Markov Processes, Vols. 1, 2. Academic Press and Springer-Verlag, New York.Google Scholar
[9] Ewens, W. (1969) Population Genetics. Methuen, New York.CrossRefGoogle Scholar
[10] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
[11] Felsenstein, J. (1975) A pain in the torus: some difficulties with models of isolation by distance. Amer. Naturalist 109, 359–368.CrossRefGoogle Scholar
[12] Fleischman, J. (1976) Limiting Distributions for Critical Branching Brownian Random Fields. Thesis, Yeshiva University.Google Scholar
[13] Fleming, W. (1975a) Distributed parameter stochastic systems in population biology. Proc. IRIA Symp. Control Theory, Numerical Methods and Computer Systems Models, Springer Lecture Notes in Econ. and Math. Systems, Vol. 107, 179–191.CrossRefGoogle Scholar
[14] Fleming, W. (1975b) Diffusion processes in population biology. Suppl. Adv. Appl. Prob. 7, 100–105.CrossRefGoogle Scholar
[15] Fleming, W. and Su, C.-H. (1974) Some one-dimensional migration models in population genetics theory. Theor. Pop. Biol. 5, 431–449.CrossRefGoogle ScholarPubMed
[16] Harris, T. (1963) The Theory of Branching Processes. Grundlehren der math. Wissenschaften, 119. Springer-Verlag, New York, and Prentice Hall, New Jersey.CrossRefGoogle Scholar
[17] Ikeda, N., Nagasawa, M. and Watanabe, S. (1968–69) Branching Markov processes I, II, III. J. Math. Kyoto Univ. 8, 233–278, 365–410, 9, 95–160.Google Scholar
[18] Karlin, S. (1969) A First Course in Stochastic Processes. Academic Press, New York and London.Google Scholar
[19] Kimura, M. and Weiss, G. (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 561–576.CrossRefGoogle ScholarPubMed
[20] Magnus, W., Oberhettinger, F. and Soni, R. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Grundlehren der math. Wissenschaften 52, Springer-Verlag, New York.CrossRefGoogle Scholar
[21] Malécot, G. (1948) The Mathematics of Heredity. W.H. Freeman, San Francisco (English translation (1969) with an extra appendix).Google Scholar
[22] Malecot, G. (1967) Identical loci and relationship. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 317–332.Google Scholar
[23] Maruyama, T. (1972) The rate of decrease of genetic variability in a two-dimensional population of finite size. Genetics 70, 639–651, and other papers cited in bibliography.CrossRefGoogle Scholar
[24] Moran, P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
[25] Nagylaki, T. (1974) The decay of genetic variability in geographically structured populations. Proc. Natn. Acad. Sci. U.S.A. 71, 2932–2936.CrossRefGoogle ScholarPubMed
[26] Nagylaki, T. (1976) The geographic structure of populations. MAA Studies in Mathematical Biology. To appear.Google Scholar
[27] Neyman, J. and Scott, E. (1964) A stochastic model of epidemics. In Stochastic Models in Medicine and Biology, ed. Gurland, J., Univ. of Wisconsin Press, Madison, 45–83.Google Scholar
[28] Raup, D., Gould, S., Schopf, T. and Simberloff, D. (1973) Stochastic models of phylogeny and the evolution of diversity. J. Geology 81, 525–542.CrossRefGoogle Scholar
[29] Sawyer, S. (1970a) A formula for semi-groups, with an application to branching diffusion processes. Trans. Amer. Math. Soc. 152, 1–38.CrossRefGoogle Scholar
[30] Sawyer, S. (1970b) A remark on the S-equation for branching processes. Proc. Japan Acad. 46, 427–429.Google Scholar
[31] Sawyer, S. (1974) A Fatou theorem for the general one-dimensional parabolic equation. Indiana Univ. Math. J. 24, 451–498.CrossRefGoogle Scholar
[32] Sawyer, S. (1975) An application of branching random fields to genetics. In Probabilistic Methods in Differential Equations, Springer Lecture Notes in Math. 451, Springer-Verlag, New York, 100–112.CrossRefGoogle Scholar
[33] Sawyer, S. (1976a) Results for the stepping stone model for migration in population genetics. Ann. Prob. 4, No. 5.Google Scholar
[34] Sawyer, S. (1976b) Asymptotic properties of the probability of identity in a geographically structured population. J. Appl. Prob. To appear.Google Scholar
[35] Sawyer, S. (1976c) Rates of consolidation in a selectively neutral migration model. Ann. Prob. To appear.CrossRefGoogle Scholar
[36] Skorokhod, A. (1964) Branching diffusion processes. Theor. Prob. Appl. 9, 492–497 (English translation).CrossRefGoogle Scholar
[37] Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, New York.CrossRefGoogle Scholar