Abstract
A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.
Similar content being viewed by others
References
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Ewens, W. J.: Mathematical population genetics. New York Heidelberg Berlin Tokyo: Springer 1979
Hudson, R. R., Kaplan, N. L.: On the divergence of alleles in nested subsamples from finite populations. Genetics 113, 1057–1076 (1986)
Kaplan, N. L., Darden, T., Hudson, R. R.: The coalescent process in models with selection. Genetics 120, 819–829 (1988)
Hudson, R. R., Kaplan, N. L.: The coalescent process in models with selection and recombination. Genetics 120, 831–840 (1988)
Karlin, S., Taylor, H. M.: A second course in stochastic processes. New York: Academic Press 1981
Kingman, J. F. C: On the genealogy of large populations. J. Appl. Probab. 19, 27–43 (1982a)
Kingman, J. F. C.: The coalescent. Stochastic Processes Appl. 13, 235–248 (1982b)
Norman, M. F.: Limit theorems for stationary distributions. Adv. Appl. Probab. 7, 561–575 (1975)
Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T.: Numerical recipes in C. The art of scientific computing. Cambridge University Press 1988
Tavaré, S.: Line-of-descent and genealogical processes, and their applications in population genetic models. Theor. Popul. Biol. 26, 119–164 (1984)
Watterson, G. A.: On the number of segregating sites in genetical models without recombination. Theor. Popul. Biol. 10, 256–276 (1975)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Darden, T., Kaplan, N.L. & Hudson, R.R. A numerical method for calculating moments of coalescent times in finite populations with selection. J. Math. Biology 27, 355–368 (1989). https://doi.org/10.1007/BF00275818
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00275818