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A limit theorem for patch sizes in a selectively-neutral migration model

Published online by Cambridge University Press:  14 July 2016

Stanley Sawyer
Stanley Sawyer*
Affiliation:
University of Washington
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Abstract

Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like √t in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rate u > 0 is imposed, the ‘block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u → 0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u ≈ 10–6 is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.

Keywords

POPULATION GENETICSMIGRATIONPOINT PROCESSESRATE OF CONSOLIDATIONSTEPPING STONELIMIT LAWGAMMA DISTRIBUTION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 482 - 495
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research partially supported by the National Science Foundation under grant number MCS 75–08098–A01.

∗

Present address: Purdue University, W. Lafayette, IN 47907, U.S.A.

References

Clifford, P. and Sudbury, A. (1973) A model for spatial conflict. Biometrika 60, 581–588.Google Scholar
Dawson, D. and Ivanoff, B. G. (1978) Branching diffusion and random measures. In Advances in Probability 5, ed. Ney, P. Dekker, New York, 61–104.Google Scholar
Durrett, R. T. (1979) An infinite particle system with additive interactions. Adv. Appl. Prob. 11, 355–383.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Fleischman, J. (1978) Limit theorems for critical branching random fields. Trans. Amer. Math. Soc. 239, 353–389.Google Scholar
Holley, R. and Liggett, T. (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Prob. 3, 643–663.Google Scholar
Kelly, F. P. (1977) The asymptotic behaviour of an invasion process. J. Appl. Prob. 14, 584–590.Google Scholar
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.Google Scholar
Malecot, G. (1969) The Mathematics of Heredity. W. H. Freeman, San Francisco.Google Scholar
Malecot, G. (1977) Kinship in the birth and death process of a population subdivided in finite panmictic groups. In Recent Developments in Statistics, ed. Barra, J. et al. North-Holland, Amsterdam.Google Scholar
Malecot, G. (1978) Coefficients of kinship of order n in an infinite regularly subdivided population (one locus case). Manuscript.Google Scholar
Moran, P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
Rusinek, R. (1977) Results on the Stepping Stone Model. Ph. D. Thesis, Yeshiva University, New York.Google Scholar
Sawyer, S. (1976a) Results for the stepping stone model for migration in population genetics. Ann. Prob. 4, 699–728.Google Scholar
Sawyer, S. (1976b) Branching diffusion processes in population genetics. Adv. Appl. Prob. 8, 659–689.Google Scholar
Sawyer, S. (1977a) Rates of consolidation in a selectively neutral migration model. Ann. Prob. 5, 486–493.Google Scholar
Sawyer, S. (1977b) Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv. Appl. Prob. 9, 268–282.Google Scholar
Sawyer, S. and Felsenstein, J. (1980) A continuous migration model with stable demography. To appear.Google Scholar
Sawyer, S. and Fleischman, J. (1979) The maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field. Proc. Nat. Acad. Sci. USA 76, 872–875.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, New York.Google Scholar