Abstract
The diffusion approximation is derived for migration and selection at a multiallelic locus in a dioecious population subdivided into a lattice of panmictic colonies. Generations are discrete and nonoverlapping; autosomal and X-linked loci are analyzed. The relation between juvenile and adult subpopulation numbers is very general and includes both soft and hard selection; the zygotic sex ratio is the same in every colony. All the results hold for both adult and juvenile migration. If ploidy-weighted average selection, drift, and diffusion coefficients are used, then the ploidy-weighted average allelic frequencies satisfy the corresponding partial differential equation for a monoecious population. The boundary conditions and the unidimensional transition conditions for coincident discontinuities in the carrying capacity and migration rate extend identically. The previous unidimensional formulation and analysis of symmetric, nearest-neighbor migration of a monoecious population across a geographical barrier is generalized to symmetric migration of arbitrary finite range, and the transition conditions are shown to hold for a dioecious population. Thus, the entire theory of clines and of the wave of advance of favorable alleles is applicable to dioecious populations.
Similar content being viewed by others
References
Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44, No. 285 (1983)
Brown, K. J., Furter, J. E.: A singularity theory approach to a semilinear boundary value problem. J. Math. Anal. Appl. 166, 485–506 (1992)
Brown, K. J., Hess, P.: Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem. Diff. Int. Eqs. 3, 201–207 (1990)
Brown, K. J., Lin, S. S., Tertikas, A.: Existence and nonexistence of steady-state solutions for a selection-migration model in population genetics. J. Math. Biol. 27, 91–104 (1989)
Brown, K. J., Tertikas, A.: On the bifurcation of radially symmetric steady-state solutions arising in population genetics. SIAM J. Math. Anal. 22, 400–413 (1991)
Endler, J. A.: Geographic Variation, Speciation, and Clines. Princeton: Princeton University Press 1977
Erdélyi, A.: Tables of Integral Transforms, Vol. I. New York: McGraw-Hill 1954
Ewens, W. J.: Mathematical Population Genetics. Berlin: Springer 1979
Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. New York: Wiley 1968
Fife, P. C.: Mathematical Aspects of Reacting and Diffusing Systems. Berlin: Springer 1979
Gautschi, W.: Error function and Fresnel integrals. In: Abramowitz, M., Stegun, I. A. (eds.) Handbook of Mathematical Functions, pp. 295–329. Washington: National Bureau of Standards 1964
Hess, P., Weinberger, H.: Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses. J. Math. Biol. 28, 83–98 (1990)
Karlin, S.: Classifications of selection-migration structures and conditions for a protected polymorphism. Evol. Biol. 14, 61–204 (1982)
Karlin, S., Taylor, H. M.: A Second Course in Stochastic Processes. New York: Academic Press 1981
Malécot, G.: Les mathématiques de l'hérédité. Paris: Masson 1948. Extended translation: The Mathematics of Heredity. San Francisco: Freeman 1969
Moody, M.: Polymorphism with migration and selection. J. Math. Biol. 8, 73–109 (1979)
Nagylaki, T.: Conditions for the existence of clines. Genetics 80, 595–615 (1975)
Nagylaki, T.: Clines with variable migration. Genetics 83, 867–886 (1976)
Nagylaki, T.: Selection in dioecious populations. Ann. Hum. Genet. 43, 143–150 (1979a)
Nagylaki, T.: Migration-selection polymorphism in dioecious populations. J. Math. Biol. 8, 123–131 (1979b)
Nagylaki, T.: The diffusion model for migration and selection. In: Hastings, A. (ed.) Some Mathematical Questions in Biology: Models in Population Biology. (Lectures on Mathematics in the Life Sciences, vol. 20, pp. 55–75) Providence: American Mathematical Society 1989
Nagylaki, T.: Models and approximations for random genetic drift. Theor. Popul. Biol. 37, 192–212 (1990a)
Nagylaki, T.: Gene conversion, linkage, and the evolution of repeated genes dispersed among multiple chromosomes. Genetics 126, 261–276 (1990b)
Nagylaki, T.: Introduction to Theoretical Population Genetics. Berlin: Springer 1992
Nagylaki, T.: The evolution of multilocus systems under weak selection. Genetics 134, 627–647 (1993)
Owen, R. E.: Gene frequency clines at X-linked or haplodiploid loci. Heredity 57, 209–219 (1986)
Poulsen, E. T.: Nonmonotone clines in homogeneous space cannot be stable. SIAM J. Math. Anal. 20, 148–159 (1989)
Saut, J. C., Scheurer, B.: Remarks on a non linear equation arising in population genetics. Comm. Part. Diff. Eqs. 3, 907–931 (1978)
Senn, S.: On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics. Comm. Part. Diff. Eqs. 8, 1199–1228 (1983)
Slatkin, M.: Gene flow and selection in a cline. Genetics 75, 733–756 (1973)
Slatkin, M.: Gene flow in natural populations. Ann. Rev. Ecol. Syst. 16, 393–430 (1985)
Tertikas, A.: Existence and uniqueness of solutions for a nonlinear diffusion problem arising in population genetics. Arch. Rat. Mech. Anal. 103, 289–317 (1988)
Tertikas, A., Toland, J. F.: Graph intersection and uniqueness results for some nonlinear elliptic problems. J. Diff. Eqs. 95, 154–168 (1992).
Weinberger, H. F.: Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982)
Wijsman, E. A., Cavalli-Sforza, L. L.: Migration and genetic population structure with special reference to humans. Ann. Rev. Ecol. Syst. 15, 279–301 (1984)
Author information
Authors and Affiliations
Additional information
This work was supported by National Science Foundation grant BSR-9006285
Rights and permissions
About this article
Cite this article
Nagylaki, T. The diffusion model for migration and selection in a dioecious population. J. Math. Biol. 34, 334–360 (1996). https://doi.org/10.1007/BF00160499
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00160499