Abstract
A generalization of Gillespie's SAS-CFF model for natural selection acting on multiple alleles in a randomly fluctuating environment is presented that relaxes symmetry assumptions concerning the variances and covariances of allelic effects. The stationary density for a multidimensional diffusion approximation of the model is obtained and provides approximate necessary and sufficient conditions for the existence of stable polymorphisms. These conditions have exactly the same form as those derived by Kimura and Mandel for polymorphism under multiple allele selection in a constant environment, except that the time-invariant fitnesses are replaced by the approximate geometric mean fitnesses of the genotypes over time. An example illustrates that this simple relationship between random environment and constant environment conditions for polymorphism does not hold for more general selection schemes. The implications of these results for the maintenance of multiple alleles by balancing selection are discussed.
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Chakraborty, R., Fuerst, P. A., Nei, M.: Statistical studies on protein polymorphism in natural populations. III. Distributions of allele frequencies and the number of alleles per locus. Genetics 94, 1039–1063 (1980)
Dempster, E. R.: Maintenance of genetic heterogeneity. Cold Spring Harbor Symp. Quant. Biol. 20, 25–32 (1955)
Ewens, W. J.: Mathematical population genetics. Berlin-Heidelberg-New York: Springer 1979
Fuerst, P. A., Chakraborty, R., Nei, M.: Statistical studies on protein polymorphism in natural populations. I. Distribution of single locus heterozygosity. Genetics 86, 455–483 (1977)
Gillespie, J. H.: Polymorphism in random environments. Theor. Popul. Biol. 4, 193–195 (1973)
Gillespie, J. H.: Sampling theory for alleles in a random environment. Nature 226, 443–445 (1977)
Gillespie, J. H.: A general model to account for enzyme variation in natural populations. V. The SAS-CFF model. Theor. Popul. Biol. 14, 1–45 (1978)
Gillespie, J. H.: Protein polymorphism and the SAS-CFF model. Genetics 94, 1089–1090 (1980a)
Gillespie, J. H.: The stationary distribution of an asymmetric model of selection in a random environment. Theor. Popul. Biol. 17, 129–140 (1980b)
Gillespie, J. H.: A randomized SAS-CFF model of natural selection in a random environment. Theor. Popul. Biol. in press (1981)
Haldane, J. B. S., Jayakar, S. D.: Polymorphism due to selection of varying direction. J. Genetics 58, 237–242 (1963)
Karlin, S., Liberman, U.: Random temporal variation in selection intensities: One-locus two-allele model. J. Math. Biology 2, 1–17 (1975)
Kimura, M.: Rules for testing stability of a selective polymorphism. Proc. Natl. Acad. Sci. USA 42, 336–340 (1956)
Kimura, M.: Diffusion models in population genetics. J. Appl. Probability 1, 177–232 (1964)
Kingman, J. F. C.: A mathematical problem in population genetics. Proc. Camb. Phil. Soc. 57, 574–582 (1961)
Kingman, J. F. C.: Random discrete distributions. J. Roy. Statist. Soc. Ser. B 31, 1–22 (1975)
Kingman, J. F. C.: The population structure associated with the Ewens sampling formula. Theor. Popul. Biol. 11, 274–283 (1977)
Levikson, B., Karlin, S.: Random temporal variation in selection intensities acting on infinite diploid populations: Diffusion method analysis. Theor. Popul. Biol. 8, 292–300 (1975)
Lewontin, R. C., Ginzburg, L. R., Tuljapurkar, S. D.: Heterosis as an explanation for large amounts of genic polymorphism. Genetics 88, 149–170 (1978)
Mandel, S. P. H.: The stability of a multiple allelic system. Heredity (Lond.) 13, 289–302 (1959)
Mandel, S. P. H.: The equivalence of different sets of stability conditions for multiple allelic systems. Biometrics 26, 840–845 (1970)
Nei, M: Protein polymorphism and the SAS-CFF model. Genetics 94, 1085–1087 (1980)
Norman, M. F.: An ergodic theorem for evolution in a random environment. J. Appl. Probability 12, 661–672 (1975)
Peil, H.: One-and multi-locus multi-allele selection models in a random environment. J. Math. Biol. 7, 133–148 (1979)
Shiga, T.: Diffusion processes in population genetics. J. Math. Kyoto Univ. 21, 133–151 (1981)
Turelli, M., Gillespie, J. H.: Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology. Theor. Popul. Biol. 17, 167–189 (1980)
Watterson, G. A.: Heterosis or neutrality? Genetics 85, 789–814 (1977)
Wright, S.: The distribution of gene frequencies under irreversible mutation. Proc. Natl. Acad. Sci. USA 24, 253–259 (1938)
Wright, S.: Adaptation and selection. In: Genetics, paleontology, and evolution. (G. G. Simpson, G. L. Jepson and E. Mayr, eds.) pp. 365–389. Princeton: Princeton University Press 1949
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Turelli, M. Temporally varying selection on multiple alleles: A diffusion analysis. J. Math. Biology 13, 115–129 (1981). https://doi.org/10.1007/BF00276870
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DOI: https://doi.org/10.1007/BF00276870