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Remarks on the spatial distribution of a reproducing population

Published online by Cambridge University Press:  14 July 2016

J. F. C. Kingman
J. F. C. Kingman*
Affiliation:
University of Oxford
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Abstract

Several authors have noted that simple models for the evolution of a reproducing and spatially distributed population have no limiting distribution, although a Poisson process in statistical equilibrium has sometimes been implicitly assumed. It is shown that, even when a mechanism for restricting population density is postulated, a Poisson process is usually impossible to achieve, essentially because of an assumption of independent displacements. When this assumption is abandoned, a Poisson process is possible, at least for some highly idealised models.

Keywords

POISSON PROCESSBRANCHING RANDOM WALKCOX PROCESSISOLATION BY DISTANCECLUSTER PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 577 - 583
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Besag, J. E. (1974) Spatial interaction and the statistical analysis of spatial pattern. J. R. Statist. Soc. B 36, 192–236.Google Scholar
[2] Biggins, J. D. (1976) Asymptotic Properties of the Branching Random Walk. D. Phil. Thesis, University of Oxford.Google Scholar
[3] Breiman, L. (1963) The Poisson tendency in traffic distribution. Ann. Math. Statist , 34, 308–311.Google Scholar
[4] Cliff, A. D. and Ord, J. K. (1975) Model building and the analysis of spatial pattern in human geography. J. R. Statist. Soc. B 37, 297–348.Google Scholar
[5] Cox, D. R. (1955) Some statistical methods connected with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
[6] Crump, K. S. and Gillespie, J. H. (1976) The dispersion of a neutral allele considered as a branching process. J. Appl. Prob. 13, 208–218.CrossRefGoogle Scholar
[7] Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes , ed. Lewis, P.A.W., Wiley, New York.Google Scholar
[8] Dobrushin, R. L. (1956) On Poisson laws for distributions of particles in space. Ukrain. Mat. Z. 8, 127–134.Google Scholar
[9] Felsenstein, J. (1975) A pain in the torus: some difficulties with models of isolation by distance. Amer. Nat. 109, 359–368.Google Scholar
[10] Kerstan, J., Matthes, K. and Mecke, J. (1972) Unbegrenst teilbare Punktprozesse. Akademie Verlag, Berlin.Google Scholar
[11] Kingman, J. F. C. (1976) Coherent random walks arising in some genetical problems. Proc. R. Soc. A 351, 19–31.Google Scholar
[12] Malécot, G. (1969) The Mathematics of Heredity. Freeman, San Francisco.Google Scholar
[13] Pólya, G. (1921) Über eine Aufgabe der Wahrscheinlichkeitrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann. 84, 149–160.CrossRefGoogle Scholar
[14] Preston, C. J. (1977) Spatial birth-and-death processes. Proc. 40th Session ISI 2, 371–391.Google Scholar
[15] Rogers, C. A. (1964) Packing and Covering. Cambridge University Press.Google Scholar
[16] Sawyer, S. (1976) Results for the stepping stone model for migration in population genetics. Ann. Prob. 4, 699–728.Google Scholar
[17] Sawyer, S. (1976) Branching diffusion processes in population genetics. Adv. Appl. Prob. 8, 659–689.CrossRefGoogle Scholar
[18] Sawyer, S. (1977) Asymptotic properties of the probability of identity in a geographically structured population. Adv. Appl. Prob. 9,CrossRefGoogle Scholar
[19] Sawyer, S. (1977) Rates of consolidation in a selectively neutral migration model. Ann. Prob. To appear.Google Scholar
[20] Wiener, N. (1933) The Fourier Integral and Certain of its Applications. Cambridge University Press.Google Scholar