Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Mobile Device Pairing
Bulletin of the American Mathematical Society
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
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 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.
  • S. A. Boorman and P. R. Levitt, 1980. The genetics of altruism, Academic Press, New York. MR 579182
  • J. J. Bull, 1981. Evolution of environmental sex determination from genotypic sex determination, Heredity 47, 173-184.
  • M. G. Bulmer, 1980. The mathematical theory of quantitative genetics, Clarendon Press, Oxford. 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.
  • J. F. Crow and M. Kimura, 1970. An introduction to population genetics theory, Harper and Row, New York. MR 274068
  • Th. Dobzhansky, 1951. Genetics and the origin of species, 3rd ed., Columbia Univ. Press, New York.
  • M. D. Donsker and S. R. S. Varadhan, 1975. On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Nat. Acad. Sci. U.S.A. 72, 780-783. MR 361998
  • 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.
  • I. Eshel, 1972. On the neighbor effect and the evolution of altruistic traits, Theoret. Population Biol. 3, 258-277. MR 325176
  • I. Eshel, 1975. Selection on sex ratio and the evolution of sex determination, Heredity 34, 351-361.
  • I. Eshel and M. W. Feldman, 1982. On evolutionary genetic stability of the sex ratio, Theoret. Population Biol. 21, 430-439. MR 667342
  • W. Ewens, 1972. The sampling theory of selectively neutral alleles, Theoret. Population Biol. 3, 87-112. MR 325177
  • W. Ewens, 1979. Mathematical population genetics, Springer-Verlag, Heidelberg. MR 554616
  • W. Feller, 1951. Diffusion processes in genetics, Proc. 2nd Berkeley Sympos. on Math. Statist. and Probab. (J. Neyman, ed.), Univ. of California Press, Berkeley, Calif., pp. 227-246. MR 46022
  • 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, 1930. The genetical theory of natural selection, Oxford Univ. Press, London (rev. ed., Dover). MR 1785121
  • W. H. Fleming, 1975. A selection migration model in population genetics, J. Math. Biol. 2, 219-233. MR 403720
  • S. Friedland and S. Karlin, 1975. Some inequalities for the spectral radius of nonnegative matrices and applications, Duke Math. J. 42, 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.
  • S. Karlin, 1978. Theoretical aspects of multilocus selection balance. I, Studies in Mathematical Biology, Part II: Populations and Communities (S. A. Levin, ed.), MAA Studies in Math., Vol. 16, Math. Assoc. Amer., Wash., D. C, pp. 503-587. MR 516495
  • S. Karlin, 1979a. Principles of polymorphism and epistasis for multilocus systems, Proc. Nat. Acad. Sci. U.S.A. 76, 541-545.
  • S. Karlin, 1979b. Models of multifactorial inheritance. I-IV, Theoret. Population Biol. 15, 308-438. MR 543626
  • S. Karlin, 1982a. Some results on optimal partitioning of variance and monotonicity with truncation levels, Statistics and Probability: Essays in Honor of C. R. Rao (G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh, eds.), North-Holland, Amsterdam, 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.
  • S. Karlin and H. Avni, 1981. Analysis of central equilibria in multilocus systems: A generalized symmetric viability regime, Theoret. Population Biol. 20, 241-280. MR 645320
  • 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.
  • S. Karlin and S. Lessard, 1983. On the optimal sex ratio, Proc. Nat. Acad. Sci. U.S.A. 180, 5931-5935. MR 751169
  • S. Karlin and S. Lessard, 1984. Theoretical studies of sex ratio evolution (to appear). MR 758911
  • S. Karlin and C. Matessi, 1983. Kin selection and altruism, Proc. Roy. Soc. London Ser. B 219, 327-353. MR 774783
  • S. Karlin and H. M. Taylor, 1981. A second course in stochastic processes, Academic Press, New York. MR 611513
  • M. Kimura, 1957. Some problems of stochastic processes in genetics, Ann. Math. Statist. 28, 882-901. MR 94267
  • M. Kimura, 1964. Diffusion models in population genetics, J. Appl. Probab. 1, 177-232. MR 172727
  • 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, 1961a. A matrix inequality, Quart. J. Math. Oxford Ser. 12, 78-80. MR 118730
  • J. F. C. Kingman, 1961b. A mathematical problem in population genetics, Proc. Cambridge Philos. Soc. 57, 574-582. MR 127990
  • J. F. C. Kingman, 1976. Coherent random walks arising in some genetical models, Proc. Roy. Soc. London Ser. A 351, 19-31. MR 420867
  • J. F. C. Kingman, 1980. The mathematics of genetic diversity, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 34, SIAM, Philadelphia, Pa. MR 591166
  • J. F. C. Kingman, 1982a. On the genealogy of large populations, J. Appl. Probab. 19A, 27-43. MR 633178
  • J. F. C. Kingman, 1982b. The coalescent, Stochastic Process. Appl. 13, 235-248. MR 671034
  • 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. Maistrovskii and Yu. G. Ol'khovskii, 1980. Selection-induced convergence to equilibrium in a single-locus autosomal population, Problems Inform. Transmission 16, 66-75. 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.
  • G. Malécot, 1966. Probabilités et hérédité, Presses Univ. de France. MR 265007
  • C. Matessi and S. D. Jayakar, 1976a. Models of density-frequency dependent selection for the exploitation of resources. I: Intraspecific competition, Population Genetics and Ecology (S. Karlin and E. Nevo, eds.), Academic Press, New York, pp. 707-721. MR 444114
  • C. Matessi and S. D. Jayakar, 1976b. Conditions for the evolution of altruism under Darwinian selection, Theoret. Population Biol. 9, 360-387. MR 414143
  • J. Maynard Smith, 1974. The theory of games and the evolution of animal conflicts, J. Theoret. Biol. 74, 209-221. MR 444115
  • 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, 1975. Wondering distributions and the electrophoretic profile, Theoret. Population Biol. 8, 318-330. MR 418953
  • P. A. P. Moran, 1976. Wondering distributions and the electrophoretic profile. II, Theoret. Population Biol. 10, 145-149. MR 418954
  • U. Motro, 1982. Optimal rates of dispersal. I, II, Theoret. Population Biol. 21, 394-429. MR 667340
  • 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.
  • T. Ohta and M. Kimura, 1973. A model of mutation appropriate to estimate electrophoretically detectable alleles in a finite population, Gen. Res. Camb. 22, 201-204. MR 406577
  • A. G. Pakes and S. Tavaré, 1981. Comments on the age distribution of Markov processes, Adv. in Appl. Probab. 13, 681-703. MR 632956
  • 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.
  • F. Proschan, 1965. Peakedness of distributions of convex combinations, Ann. of Math. Statist. 36, 1703-1706. MR 187269
  • J. Roughgarden, 1979. Theory of population genetics and evolutionary ecology. An introduction, Macmillan, New York.
  • P. A. Samuelson, 1976. Time symmetry and asymmetry in population and deterministic dynamic systems, Theoret. Population Biol. 9, 82-122. MR 469342
  • 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.
  • S. Tavaré, 1984. Line of descent and genealogical processes, and their applications in population genetics models, Theoret. Population Biol. 25 (to appear). MR 770050
  • M. Turelli, 1984. Heritable genetic variation via mutation-selection balance. Lerch's zeta meets the abdominal bristle, Theoret. Population Biol. 25 (to appear).
  • M. K. Uyenoyama and B. O. Bengtsson, 1979. Towards a genetic theory for the evolution of the sex ratio, Genetics 93, 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, 1975. On the number of segregating sites in genetic models without recombination, Theoret. Population Biol. 7, 256-276. MR 366430
  • G. A. Watterson, 1984. Lines of descent and the coalescent, Theoret. Population Biol. 25 (to appear). MR 760232
  • 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, 1980. Population dynamics from game theory, Global Theory of Dynamical Systems, Lecture Notes in Math., Vol. 819, Springer-Verlag, Berlin, pp. 471-497. MR 591205
  • E. C. Zeeman, 1981. Dynamics of the evolution of animal conflicts, J. Theoret. Biol. 89, 249-270R. MR 630636

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 92A10, 92-02, 92-03

Retrieve articles in all journals with MSC (1980): 92A10, 92-02, 92-03

Additional Information

PII: S 0273-0979(1984)15239-X