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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Mathematical models, problems, and controversies of evolutionary theory

Author: Samuel Karlin
Journal: Bull. Amer. Math. Soc. 10 (1984), 221-273
MSC (1980): Primary 92A10, 92-02; Secondary 92-03
MathSciNet review: 733693
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    G. Bell, 1982. The masterpiece of nature. The evolution and genetics of sexuality, Univ. of California Press, Berkeley, Calif. F. Bernstein, 1925. Zusammenfassende Betrachtungen über die erblichen Blutstrukturen des Menschen, Z. Abst. Vereb. 37, 237-269. S. A. Boorman and P. R. Levitt, 1973. A frequency-dependent natural selection model for the evolution of social cooperation networks. Proc. Nat. Acad. Sci. USA 70, 187-189.
  • Scott A. Boorman and Paul R. Levitt, The genetics of altruism, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. MR 579182
  • J. J. Bull, 1981. Evolution of environmental sex determination from genotypic sex determination, Heredity 47, 173-184.
  • Michael George Bulmer, The mathematical theory of quantitative genetics, The Clarendon Press, Oxford University Press, New York, 1980. Oxford Science Publications. MR 587485
  • M. G. Bulmer and J. J. Bull, 1982. Models in polygenic sex determination and sex ratio evolution, Evolution 36, 13-26. B. Charlesworth, 1978. Some models of the evolution of altruistic behavior between siblings, J. Theoret. Biol. 72, 297-319. E. L. Charnov, 1982. The theory of sex allocation, Princeton Univ. Press, Princeton, N. J. B. C. Clarke, 1979. The evolution of genetic diversity, Proc. Roy. Soc. London Ser. A. 250, 543-574.
  • James F. Crow and Motoo Kimura, An introduction to population genetics theory, Harper & Row, Publishers, New York-London, 1970. MR 0274068
  • Th. Dobzhansky, 1951. Genetics and the origin of species, 3rd ed., Columbia Univ. Press, New York.
  • Monroe D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 780–783. MR 361998, DOI
  • M. Eigen, 1971. Self organization of matter and the evolution of biological macromolecules, Naturwissenschaften 58, 475-522. J. A. Endler, 1977. Geographic variation speciation and clines, Princeton Univ. Press, Princeton, N. J.
  • Ilan Eshel, On the neighbor effect and the evolution of altruistic traits, Theoret. Population Biol. 3 (1972), 258–277. MR 325176, DOI
  • I. Eshel, 1975. Selection on sex ratio and the evolution of sex determination, Heredity 34, 351-361.
  • Ilan Eshel and Marcus W. Feldman, On evolutionary genetic stability of the sex ratio, Theoret. Population Biol. 21 (1982), no. 3, 430–439. MR 667342, DOI
  • W. J. Ewens, The sampling theory of selectively neutral alleles, Theoret. Population Biol. 3 (1972). MR 325177, DOI
  • Warren J. Ewens, Mathematical population genetics, Biomathematics, vol. 9, Springer-Verlag, Berlin-New York, 1979. MR 554616
  • William Feller, Diffusion processes in genetics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 227–246. MR 0046022
  • J. Felsenstein, 1976. The theoretical population genetics of variable selection and migration, Ann. Rev. Genet. 10, 157-166. R. P. Feynman, 1964. The Feynman lectures on physics, Vols. 1-3, Addison-Wesley, Reading, Mass. R. A. Fisher, 1918. The correlation between relatives on the supposition of Mendelian inheritance, Trans. Roy. Soc. Edinburgh 52, 399-433. R. A. Fisher, 1922. On the dominance ratio, Proc. Roy. Soc. Edinburgh 56, 321-341.
  • R. A. Fisher, The genetical theory of natural selection, A complete variorum edition, Oxford University Press, Oxford, 1999. Revised reprint of the 1930 original; Edited, with a foreword and notes, by J. H. Bennett. MR 1785121
  • W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol. 2 (1975), no. 3, 219–233. MR 403720, DOI
  • S. Friedland and S. Karlin, Some inequalities for the spectral radius of non-negative matrices and applications, Duke Math. J. 42 (1975), no. 3, 459–490. MR 376717
  • D. J. Futuyma, 1979. Evolutionary biology, Sinauer, Sunderland, Mass. F. Galton, 1889. Natural inheritance, Macmillan, London. L. Ginzburg, 1979. Why are heterozygotes often superior in fitness, Theoret. Population Biol. 15, 264-267. L. Ginzburg, 1983. Mathematical evolutionary theory, Benjamin, New York. J. B. S. Haldane, 1927. A mathematical theory of natural and artificial selection. V: Selection and mutation, Proc. Cambridge Philos. Soc. 23, 235-243. J. B. S. Haldane, 1932. The causes of evolution, Harper & Row, New York. W. D. Hamilton, 1964. The genetical evolution of social behavior. I, II, J. Theoret. Biol. 7, 1-52. D. L. Hartl, 1980. Principles of population genetics, Sinauer, Sunderland, Mass. P. W. Hedrick, 1983. Genetics of populations, Van Nostrand Reinhold, New York. P. W. Hedrick, M. E. Ginevan and E. P. Ewing, 1976. Genetic polymorphism in heterogenous environments, Ann. Rev. Ecol. Syst. 7, 1-33. P. W. Hedrick, S. Jain and L. Holden, 1978. Multilocus systems in evolution, Evolutionary Biology, Vol. 11, Plenum Press, New York. S. Karlin, 1968. Total positivity, Stanford Univ. Press, Stanford, Calif.
  • Simon A. Levin (ed.), Studies in mathematical biology. Part II, MAA Studies in Mathematics, vol. 16, Mathematical Association of America, Washington, D.C., 1978. Populations and communities. MR 516495
  • S. Karlin, 1979a. Principles of polymorphism and epistasis for multilocus systems, Proc. Nat. Acad. Sci. U.S.A. 76, 541-545.
  • Samuel Karlin, Models of multifactorial inheritance. I. Multivariate formulations and basic convergence results, Theoret. Population Biol. 15 (1979), no. 3, 308–355. MR 543626, DOI
  • Samuel Karlin, Some results on optimal partitioning of variance and monotonicity with truncation level, Statistics and probability: essays in honor of C. R. Rao, North-Holland, Amsterdam-New York, 1982, pp. 375–382. MR 659490
  • S. Karlin, 1982b. Classifications of selection-migration structures and conditions for a protected polymorphism, Evolutionary Biology, Vol. 14, Plenum Press, New York, pp. 61-204.
  • Samuel Karlin and Haim Avni, Analysis of central equilibria in multilocus systems: a generalized symmetric viability regime, Theoret. Population Biol. 20 (1981), no. 2, 241–280. MR 645320, DOI
  • S. Karlin, R. Kenett and B. Bonne-Tamir, 1979. Analysis of biochemical data on Jewish populations. II: Results and interpretations of heterogeneity indices and distance measures with respect to standards, Amer. J. Human Genetics 31, 341-365.
  • Samuel Karlin and Sabin Lessard, On the optimal sex ratio, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 19, Biol. Sci., 5931–5935. MR 751169, DOI
  • Samuel Karlin and Sabin Lessard, On the optimal sex-ratio: a stability analysis based on a characterization for one-locus multiallele viability models, J. Math. Biol. 20 (1984), no. 1, 15–38. MR 758911, DOI
  • S. Karlin and C. Matessi, Kin selection and altruism, Mathematical genetics (London, 1983) Roy. Soc. London, London, 1983, pp. 107–133. MR 774783
  • Samuel Karlin and Howard M. Taylor, A second course in stochastic processes, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 611513
  • Motoo Kimura, Some problems of stochastic processes in genetics, Ann. Math. Statist. 28 (1957), 882–901. MR 94267, DOI
  • Motoo Kimura, Diffusion models in population genetics, J. Appl. Probability 1 (1964), 177–232. MR 172727, DOI
  • M. Kimura, 1983. The neutral theory of evolution, Cambridge Univ. Press, Cambridge. M. Kimura and J. F. Crow, 1964. The number of alleles that can be maintained in a finite population, Genetics 49, 725-738. M. Kimura and T. Ohta, 1973. The age of a neutral mutant persisting in a finite population, Genetics 75, 199-212.
  • J. F. C. Kingman, On an inequality in partial averages, Quart. J. Math. Oxford Ser. (2) 12 (1961), 78–80. MR 118730, DOI
  • J. F. C. Kingman, A mathematical problem in population genetics, Proc. Cambridge Philos. Soc. 57 (1961), 574–582. MR 127990, DOI
  • J. F. C. Kingman, Coherent random walks arising in some genetical models, Proc. Roy. Soc. London Ser. A 351 (1976), no. 1664, 19–31. MR 420867, DOI
  • J. F. C. Kingman, Mathematics of genetic diversity, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 34, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1980. MR 591166
  • J. F. C. Kingman, On the genealogy of large populations, J. Appl. Probab. Special Vol. 19A (1982), 27–43. Essays in statistical science. MR 633178
  • J. F. C. Kingman, The coalescent, Stochastic Process. Appl. 13 (1982), no. 3, 235–248. MR 671034, DOI
  • V. A. Kostitzin, 1937. Biologie mathématique, A. Colin, Paris. R. Lande, 1957. The maintenance of genetic variability by mutation in a polygenic character with linked loci, Genet. Res. 26, 221-235. S. Lessard, 1984. Evolutionary dynamics in frequency dependent two-phenotype models, Theoret. Population Biol. (in press). R. C. Lewontin, L. R. Ginzburg and S. D. Tuljapurkar, 1978. Heterosis as an explanation for large amounts of genie polymorphism, Genetics 88, 149-169. A. J. Lotka, 1925. Elements of physical biology, Williams and Wilkins, Baltimore, Md.
  • Yu. I. Lyubich, G. D. Maĭstrovskiĭ, and Yu. G. Ol′khovskiĭ, Selection-induced convergence to equilibrium in a single-locus autosomal population; Russian transl., Problems Inform. Transmission 16 (1980), no. 1, 93–104. MR 589840
  • R. MacArthur, 1970. Species packing and competitive equilibrium for many species, Theoret. Population Biol. 1, 1-11. G. Malécot, 1948. Les mathématiques de l’hérédité, Masson, Paris.
  • Gustave Malécot, Probabilités et hérédité, Institut National d’Études Démographiques. Travaux et Documents, Cahier No. 47, Presses Universitaires de France, Paris, 1966 (French). Introduction, in French, English, Russian, Spanish and German, by J. Sutter, M. Gillois and A. Jacquard; Preface, in English and French, by Newton E. Morton. MR 0265007
  • C. Matessi and S. D. Jayakar, Models of density-frequency dependent selection for the exploitation of resources. I. Intraspecific competition, Population genetics and ecology (Proc. Conf., Israel, 1975) Academic Press, New York, 1976, pp. 707–721. MR 0444114
  • C. Matessi and S. D. Jayakar, Conditions for the evolution of altruism under Darwinian selection, Theoret. Population Biol. 9 (1976), no. 3, 360–387. MR 414143, DOI
  • J. Maynard Smith, The theory of games and the evolution of animal conflicts, J. Theoret. Biol. 47 (1974), no. 1, 209–221. MR 444115, DOI
  • J. Maynard Smith, 1978. The evolution of sex, Cambridge Univ. Press. J. Maynard Smith, 1981. Will sexual population evolve to an ESS?, Amer. Natur. 117, 1015-1018. J. Maynard Smith, 1982. Evolution and the theory of games, Cambridge Univ. Press, Cambridge. J. Maynard Smith and G. R. Price, 1973. The logic of animal conflicts, Nature (London) 246, 15-18. E. Mayr, 1963. Animal species and evolution, Harvard Univ. Press, Cambridge, Mass. R. E. Michod, 1982. The concept of fitness in population genetics and sociobiology, King’s College Sociobiology Group: Current Problems in Sociobiology, Cambridge Univ. Press.
  • P. A. P. Moran, Wandering distributions and the electrophoretic profile, Theoret. Population Biol. 8 (1975), no. 3, 318–330. MR 418953, DOI
  • P. A. P. Moran, Wandering distributions and the electrophoretic profile, Theoret. Population Biol. 8 (1975), no. 3, 318–330. MR 418953, DOI
  • Uzi Motro, Optimal rates of dispersal. I. Haploid populations, Theoret. Population Biol. 21 (1982), no. 3, 394–411. MR 667340, DOI
  • H. J. Muller, 1929. The gene as the basis of life, Proc. Internat. Congr. Plant. Sci. 1, 897-921. T. Nagylaki, 1978. The relations between distant individuals in geographically structured populations, Studies in Mathematical Biology, Vol. 15 (S. A. Levin, ed.), Math. Assoc. Amer., Wash., D. C, pp. 588-624. M. Nei and K. Roychoudbury, 1982. Genetic relationships and evolution of races, Evolution Biology, Vol. 14, Plenum Press, New York, pp. 1-56. U. Nur, 1974. The expected changes in the frequency of alleles affecting the sex ratio, Theoret. Population Biol. 5, 143-147.
  • Tomoko Ohta and Motoo Kimura, A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population, Genet. Res. 22 (1973), no. 2, 201–204. MR 406577
  • Anthony G. Pakes and Simon Tavaré, Comments on the age distribution of Markov processes, Adv. in Appl. Probab. 13 (1981), no. 4, 681–703. MR 632956, DOI
  • K. Pearson, 1904. On a generalized theory of alternative inheritance, with special references to Mendel’s laws, Philos. Trans. Roy. Soc. Ser. A 203, 53-86.
  • Frank Proschan, Peakedness of distributions of convex combinations, Ann. Math. Statist. 36 (1965), 1703–1706. MR 187269, DOI
  • J. Roughgarden, 1979. Theory of population genetics and evolutionary ecology. An introduction, Macmillan, New York.
  • Paul A. Samuelson, Time symmetry and asymmetry in population and deterministic dynamic systems, Theoret. Population Biol. 9 (1976), no. 1, 82–122. MR 469342, DOI
  • R. F. Shaw, 1958. The theoretical genetics of the sex ratio, Genetics 93, 149-163. R. F. Shaw and J. D. Mohler, 1953. The selective advantage of the sex ratio, Amer. Natur. 87, 337-342. G. G. Simpson, 1953. The major features of evolution, Columbia Univ. Press, New York. G. L. Stebbins, 1950. Variation and evolution in plants, Columbia Univ. Press, New York.
  • Simon Tavaré, Line-of-descent and genealogical processes, and their applications in population genetics models, Theoret. Population Biol. 26 (1984), no. 2, 119–164. MR 770050, DOI
  • M. Turelli, 1984. Heritable genetic variation via mutation-selection balance. Lerch’s zeta meets the abdominal bristle, Theoret. Population Biol. 25 (to appear).
  • Marcy K. Uyenoyama and Bengt Olle Bengtsson, Towards a genetic theory for the evolution of the sex ratio, Genetics 93 (1979), no. 3, 721–736. MR 598187
  • V. Volterra, 1931. Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris. B. Wallace, 1981. Basic population genetics, Columbia Univ. Press, New York.
  • G. A. Watterson, On the number of segregating sites in genetical models without recombination, Theoret. Population Biol. 7 (1975), 256–276. MR 366430, DOI
  • G. A. Watterson, Lines of descent and the coalescent, Theoret. Population Biol. 26 (1984), no. 1, 77–92. MR 760232, DOI
  • G. C. Williams, 1975. Sex and evolution, Princeton Univ. Press, Princeton, N. J. G. C. Williams, 1979. The question of adaptive sex ratio in outcrossed vertebrates, Proc. Roy. Soc. London Ser. B 205, 567-580. E. O. Wilson, 1975. Sociobiology, the new synthesis, Belknap Press of Harvard Univ. Press, Cambridge, Mass. S. Wright, 1921. Systems of mating. I: The biométrie relations between parent and offspring, Genetics 6, 111-123. S. Wright, 1923a. Mendelian analysis of pure breeds of livestock. II: The Duchess family of shorthorns as bred by Thomas Bates, J. Hered. 14, 405-422. S. Wright, 1923b. The theory of path coefficients, Genetics 8, 239-255. S. Wright, 1931. Evolution in Mendelian populations, Genetics 16, 97-159. S. Wright, 1955. Classification of the factors of evolution, Cold Spring Harbor Sympos. Quantitative Biology 20, 16-24. S. Wright, 1968. Evolution and the genetics of populations, Vol. I: Genetic and biometric foundations, Univ. of Chicago Press, Chicago, Ill. V. C. Wynne-Edwards, 1962. Animal dispersion in relation to social behavior, Hofner, New York.
  • E. C. Zeeman, Population dynamics from game theory, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 471–497. MR 591205
  • E. C. Zeeman, Dynamics of the evolution of animal conflicts, J. Theoret. Biol. 89 (1981), no. 2, 249–270. MR 630636, DOI

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