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Stepping stone models for population growth

Published online by Cambridge University Press:  14 July 2016

E. Renshaw
E. Renshaw*
Affiliation:
University of Edinburgh
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Abstract

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours. An approximate solution is obtained for the probability structure of the population size, and exact results are derived for the process when immigration is introduced into a single colony from outside the system.

Keywords

BIRTH DEATH AND MIGRATIONIMMIGRATIONSTEPPING STONE MODELSSPATIALLY DISTRIBUTEDCONNECTED PROCESSESBESSEL FUNCTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 16 - 31
Copyright
Copyright © Applied Probability Trust 1974 

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References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. AMS 55, U. S. Department of Commerce.Google Scholar
Adke, S. R. (1964a) The generalised birth and death process and Gaussian diffusion. J. Math. Anal. Appl. 9, 336–340.CrossRefGoogle Scholar
Adke, S. R. (1964b) A stochastic population diffusing on a finite interval. J. Ind. Statist. Ass. 2, 32–40.Google Scholar
Adke, S. R. (1969) A birth, death and migration process. J. Appl. Prob. 6, 687–691.CrossRefGoogle Scholar
Adke, S. R. and Moyal, J. E. (1963) A birth, death and diffusion process. J. Math. Anal. Appl. 7, 209–224.CrossRefGoogle Scholar
Bailey, N. T. J. (1964) The Elements of Stochastic Processes. Wiley, New York.Google Scholar
Bailey, N. T. J. (1968) Stochastic birth, death and migration processes for spatially distributed populations. Biometrika 55, 189–198.CrossRefGoogle Scholar
Chiang, C. L. (1968) Introduction to Stochastic Processes in Biostatistics. Wiley, New York.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Crump, K. S. (1970) Migratory populations in branching processes. J. Appl. Prob. 7, 565–572.CrossRefGoogle Scholar
Davis, A. W. (1965) On the theory of birth, death and diffusion processes. J. Appl. Prob. 2, 293–322.CrossRefGoogle Scholar
Davis, A. W. (1967a) Branching-diffusion processes with no absorbing boundaries. I. J. Math. Anal. Appl. 18, 276–296.CrossRefGoogle Scholar
Davis, A. W. (1967b) Branching-diffusion processes with no absorbing boundaries. II. J. Math. Anal. Appl. 19, 1–25.CrossRefGoogle Scholar
Davis, A. W. (1970) Some generalisations of Bailey's birth, death and migration model. Adv. Appl. Prob. 2, 83–109.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Kimura, M. (1953) “Stepping stone” model of population. Ann. Rept. Nat. Inst. Genetics, Japan 3, 62–63.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.CrossRefGoogle ScholarPubMed
Maruyama, T. (1969) Genetic correlation in the stepping stone model with non-symmetrical migration rates. J. Appl. Prob. 6, 463–477.CrossRefGoogle Scholar
Maruyama, T. (1970) Stepping stone models of finite length. Adv. Appl. Prob. 2, 229–258.CrossRefGoogle Scholar
Matis, J. H. and Hartley, H. O. (1971) Stochastic compartmental analysis: model and least squares estimation from time series data. Biometrics 27, 77–102.CrossRefGoogle Scholar
Puri, P. S. (1968) Interconnected birth and death processes. J. Appl. Prob. 5, 334–349.CrossRefGoogle Scholar
Renshaw, E. (1972) Birth, death and migration processes. Biometrika 59, 49–60.CrossRefGoogle Scholar
Renshaw, E. (1973) Interconnected population processes. J. Appl. Prob. 10, 1–14.CrossRefGoogle Scholar
Ruben, H. (1962) Some aspects of the emigration-immigration process. Ann. Math. Statist. 33, 119–129.CrossRefGoogle Scholar
Usher, M. B. and Williamson, M. H. (1970) A deterministic matrix model for handling the birth, death, and migration processes of spatially distributed populations. Biometrics 26, 1–12.CrossRefGoogle Scholar
Weiss, G. and Kimura, M. (1965) A mathematical analysis of the stepping stone model of genetic correlation. J. Appl. Prob. 2, 129–149.CrossRefGoogle Scholar
Wiggins, A. D. (1960) On a multicompartment migration model with chronic feeding. Biometrics 16, 642–658.CrossRefGoogle Scholar