Abstract
Infinite-dimensional characters are those in which the phenotype of an individual is described by a function, rather than by a finite set of measurements. Examples include growth trajectories, morphological shapes, and norms of reaction. Methods are presented here that allow individual phenotypes, population means, and patterns of variance and covariance to be quantified for infinite-dimensional characters. A quantitative-genetic model is developed, and the recursion equation for the evolution of the population mean phenotype of an infinite-dimensional character is derived. The infinite-dimensional method offers three advantages over conventional finite-dimensional methods when applied to this kind of trait: (1) it describes the trait at all points rather than at a finite number of landmarks, (2) it eliminates errors in predicting the evolutionary response to selection made by conventional methods because they neglect the effects of selection on some parts of the trait, and (3) it estimates parameters of interest more efficiently.
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References
Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. New York: Dover 1965
Apostol, T. M.: Mathematical analysis, 2nd edn. Reading, Mass.: Addison-Wesley 1975
Barton, N. H., Turelli, M.: Adaptive landscapes, genetic distance and the evolution of quantitative characters. Genet. Res. 49, 157–173 (1987)
Bulmer, M. G.: The mathematical theory of quantitative genetics. Oxford: Oxford University Press 1985
Davis, M. H. A.: Linear estimation and stochastic control. London: Chapman and Hall 1977
Doob, J. L.: Stochastic processes. New York: Wiley 1953
Falconer, D. S.: Introduction to quantitative genetics, 2nd edn. New York: Longman 1981
Fisher, R. A.: The correlation between relatives on the supposition of Mendelian inheritance. Trans. Royal Soc. Edinburgh 52, 399–433 (1918)
Gould, S. J.: Ontogeny and phylogeny. Cambridge, Mass.: Belknap 1977
Huey, R. B., Hertz, P. E.: Is a jack-of-all-temperatures a master of none? Evolution 38, 441–444 (1984)
Huxley, J.: Problems of relative growth. London: MacVeagh 1932
Kimura, M.: A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Nat. Acad. Sci. 54, 731–736 (1965)
Lande, R.: Quantitative genetic analysis of multivariate evolution, applied to brain:body size allometry. Evolution 33, 402–416 (1979)
Lande, R.: The genetic covariance between characters maintained by pleiotropic mutations. Genetics 94, 203–215 (1980)
Lande, R., Arnold, S. J.: The measurement of selection on correlated characters. Evolution 37, 1210–1226 (1983)
Lyusternik, L. A., Sobolev, V. J.: Elements of functional analysis. New York: Unger 1968
Magee, W. T.: Estimating response to selection. J. Anim. Sci. 24, 242–247 (1965)
Parzen, E.: An approach to time series analysis. Ann. Math. Stat. 32, 951–989 (1962)
Rao, C. R., Mitra, S. K.: Generalized inverse of matrices and its applications. New York: Wiley 1971
Rao, C. R.: Linear statistical inference and its applications. New York: Wiley 1973
Reed, M., Simon, B. Methods of modern mathematical physics: I. Functional analysis, 2nd edn. New York: Academic Press 1980
Riska, B., Atchley, W. R., Rutledge, J. J.: A genetic analysis of targeted growth in mice. Genetics 107, 79–101 (1984)
Robertson, A.: The non-linearity of offspring-parent regression. In: Pollak, E., Kempthorne, O., Bailey, T. B. (eds.) Proceedings Int. Conferrence on Quantitative Genetics, pp. 297–306. Ames: Iowa State University Press 1987
Thompson, D. W.: On growth and form. Cambridge: Cambridge University Press 1917
Turelli, M: Effects of pleiotropy on predictions concerning mutation selection balance for polygenic traits. Genetics 111, 165–195 (1985)
Turelli, M.: Gaussian versus non-gaussian genetic analyses of polygenic mutation-selection balance. Karlin, S. Nevo, E. (eds.) Evolutionary processes and theory, pp. 607–628. New York: Academic Press 1986
Wright, S.: Evolution and the genetics of populations, vol. 1. Genetic and biometrical foundations. Chicago: University of Chicago Press 1968
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Kirkpatrick, M., Heckman, N. A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters. J. Math. Biology 27, 429–450 (1989). https://doi.org/10.1007/BF00290638
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DOI: https://doi.org/10.1007/BF00290638