Abstract
The frequency of one of two alleles is studied as a function of position and time in a one, two, or three dimensional region. A nonlinear diffusion equation is employed. Each allele is assumed to have a selective advantage in some part of the region. An asymptotic solution is constructed for the case when the selection coefficient is large compared to the diffusion coefficient, i.e. when selection acts more rapidly than diffusion. Then as time increases, the solution tends to a cline, i.e. an equilibrium distribution in which both alleles are present everywhere, each predominating where it has the advantage. In a narrow region around the boundary where the selective advantage switches from one allele to the other, both alleles are present with comparable frequencies. Along a line normal to this boundary, the frequency varies as in a one dimensional habitat with a simple variation in selective advantage. The asymptotic solution is compared with the numerical solution for a special two dimensional case, and the agreement is found to be good.
Similar content being viewed by others
References
Haldane, J. B. S.: The theory of a cline. J. Genet. 48, 277–284 (1948)
Fisher, R. A.: Gene frequencies in a cline determined by selection and diffusion. Biometrics 6, 353–361 (1950)
Nagylaki, T.: Conditions for the existence of clines. Genetics 80, 595–615 (1975)
Fleming, W. H.: A selection-migration model in population genetics. J. Math. Biol. 2, 219–233 (1975)
Hoppensteadt, F. C.: Analysis of a stable polymorphism arising in a selection-migration model in population genetics. J. Math. Biol. 2, 235–240 (1975)
Author information
Authors and Affiliations
Additional information
Research supported by the National Science Foundation.
Rights and permissions
About this article
Cite this article
Keller, J.B. Genetic variability due to geographical inhomogeneity. J. Math. Biology 20, 223–230 (1984). https://doi.org/10.1007/BF00285348
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00285348