Speciation and the ”shifting balance” in a continuous population.

*(English)*Zbl 0614.92011Shifts between adaptive peaks, caused by sampling drift, are involved in both specification and adaptation via Wright’s ”shifting balance”. We use techniques from statistical mechanics to calculate the rate of such transitions for a population in a single panmictic deme and for a population which is continuously distributed over one- and two- dimensional regions. This calculation applies in the limit where transitions are rare.

Our results indicate that stochastic divergence is feasible despite free gene flow, provided that neighbourhood size is low enough. In two dimensions, the rate of transition depends primarily on neighbourhood size N and only weakly on selection pressure (\(\approx s^ k\exp (- cN))\), where k is a number determined by the local population structure, in contrast with the exponential dependence on selection pressure in one dimension (\(\approx \exp (-cN\sqrt{s}))\) or in a single deme (\(\approx \exp (-cNs))\). Our calculations agree with simulations of a single deme and a one-dimensional population.

Our results indicate that stochastic divergence is feasible despite free gene flow, provided that neighbourhood size is low enough. In two dimensions, the rate of transition depends primarily on neighbourhood size N and only weakly on selection pressure (\(\approx s^ k\exp (- cN))\), where k is a number determined by the local population structure, in contrast with the exponential dependence on selection pressure in one dimension (\(\approx \exp (-cN\sqrt{s}))\) or in a single deme (\(\approx \exp (-cNs))\). Our calculations agree with simulations of a single deme and a one-dimensional population.

##### Keywords:

continuous population; two-dimensional population; population genetics; Langevin equation; adaptive peaks; sampling drift; specification; adaptation; shifting balance; statistical mechanics; single panmictic deme; transitions; stochastic divergence; gene flow; rate of transition; neighbourhood size; selection pressure; one-dimensional population
PDF
BibTeX
XML
Cite

\textit{S. Rouhani} and \textit{N. Barton}, Theor. Popul. Biol. 31, 465--492 (1987; Zbl 0614.92011)

Full Text:
DOI

##### References:

[1] | Affleck, I.K., Some results on vacuum decay, () |

[2] | Barton, N.H., The dynamics of hybrid zones, Heredity, 43, 341-359, (1979) |

[3] | Barton, N.H., Spatial patterns and population structure I. theory, (1986), submitted for publication |

[4] | Barton, N.H., Drift in a continuous population, (1986), submitted for publication |

[5] | Barton, N.H.; Charlesworth, B., Genetic revolutions, founder effects, and speciation, Annu. rev. ecol. syst, 15, 133-164, (1984) |

[6] | Barton, N.H.; Hewitt, G.M., A chromosomal cline in the grasshopper podisma pedestris, Evolution, 35, 1008-1018, (1981) |

[7] | Barton, N.H.; Hewitt, G.M., Spatial patterns and population structure. II. fluctuations in a chromosomal cline, (1986), submitted for publication |

[8] | Breit, J.D.; Gupta, S.; Zaks, A., Stochastic quantisation and regularisation, Nucl. phys, 233, 61-87, (1984) |

[9] | Bulmer, M.G., The mathematical theory of quantitative genetics, (1980), Oxford Univ. Press Oxford · Zbl 0441.92007 |

[10] | Callan, C.G.; Coleman, S., Fate of the false vacuum. II. first quantum corrections, Phys. rev. D, 16, 1762-1768, (1977) |

[11] | Callaway, J., Energy band theory, (1964), Academic Press New York · Zbl 0121.23303 |

[12] | Caroli, B.; Caroli, C.; Roulet, B.; Gouyet, J.F., A WKB treatment of diffusion in a multidimensional bistable potential, J. statist. phys, 22, 515-536, (1980) |

[13] | Carson, H.L.; Templeton, A.R., Genetic revolutions in relation to speciation phenomena: the founding of new populations, Annu. rev. ecol. syst, 15, 97-131, (1984) |

[14] | Charlesworth, B.; Lande, R.; Slatkin, M., A neo-Darwinian commentary on macroevolution, Evolution, 36, 474-498, (1982) |

[15] | Coleman, S., Fate of the false vacuum: semiclassical theory, Phys. rev. D, 15, 2929-2935, (1977) |

[16] | Coleman, S., Uses of instantons, (), 805-942 |

[17] | Crow, J.F.; Kimura, M., An introduction to population genetics, (1970), Harper & Row New York · Zbl 0246.92003 |

[18] | Endler, J.A., Problems in distinguishing historical from ecological factors in biogeography, Amer. zool, 22, 441-452, (1982) |

[19] | Ewens, W.J., Mathematical population genetics, (1979), Springer-Verlag Berlin/New York · Zbl 0422.92011 |

[20] | Felsenstein, J., Excursions along the interface between disruptive selection and stabilising selection, Genetics, 93, 773-795, (1979) |

[21] | Feynman, R.; Hibbs, A.R., Quantum mechanics and path integrals, (1967), McGraw-Hill New York · Zbl 0176.54902 |

[22] | Futuyma, D.J.; Mayer, G.C., Non-allopatric speciation in animals, Syst. zool, 29, 254-271, (1980) |

[23] | Gervais, J.L.; Sakita, B., Extended particles in quantum field theory, Phys. rev. D, 11, 2943-2949, (1975) |

[24] | Graham, R., Covariant formulation of non-equilibrium statistical thermodynamics, Z. phys, 26, 397-405, (1977) |

[25] | Karlin, S., General two locus models: some objectives, results and interpretations, Theor. pop. biol, 7, 364-398, (1975) · Zbl 0315.92006 |

[26] | Keller, J., Diffraction by an aperture, J. appl. phys, 28, 426-444, (1957) · Zbl 0077.41202 |

[27] | Key, K.H.L., The concept of stasipatric speciation, Syst. zool, 17, 14-22, (1968) |

[28] | Key, K.H.L., Species, parapatry, and the morabine grasshoppers, Syst. zool, 30, 425-458, (1982) |

[29] | Kimura, M.; Ohta, T., Genetic loads at a polymorphic locus which is maintained by frequency dependent selection, Genet. res, 16, 145-150, (1970) |

[30] | Kirkpatrick, M., Quantum evolution and punctuated equilibrium in continuous genetic characters, Amer. natur, 119, 833-848, (1982) |

[31] | Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P., Optimization by simulated annealing, Science, 220, 671-680, (1983) · Zbl 1225.90162 |

[32] | Lande, R., Effective deme sizes during long term evolution estimated from rates of chromosomal rearrangement, Evolution, 33, 234-257, (1979) |

[33] | Lande, R., Genetic variation and phenotypic evolution during allopatric speciation, Amer. natur, 116, 463-479, (1980) |

[34] | Lande, R., The fixation of chromosomal rearrangement in a subdivided population with local extinction and recolonisation, Heredity, 45, 323-332, (1985) |

[35] | Lande, R., Expected time for random genetic drift of a population between stable phenotypic states, (), 7641-7645 · Zbl 0589.92010 |

[36] | Langer, J.S., Theory of the condensation point, Ann. phys. (N.Y.), 41, 108-157, (1967) |

[37] | Mackie, J., The law of the jungle: moral alternatives, Philosophy, 53, 455-464, (1978) |

[38] | Malecot, G., The mathematics of heredity, (1968), Freeman San Francisco |

[39] | Mayr, E., Systematics and the origin of species, (1942), Columbia Univ. Press New York |

[40] | Morse, T.; Feschbach, C., Mathematical methods for physicists, (1965), Academic Press New York |

[41] | Nagylaki, T., Conditions for the existence of clines, Genetics, 80, 595-615, (1975) |

[42] | Nagylaki, T., A diffusion model for geometrically structured populations, J. math. biol, 6, 375-382, (1978) · Zbl 0404.92010 |

[43] | Nagylaki, T., Random genetic drift in a cline, (), 423-426 · Zbl 0377.92008 |

[44] | Nei, M., Mathematical models of speciation and genetic distance, (), 723-765 |

[45] | Newman, C.M.; Cohen, J.E.; Kipnis, C., Neo-Darwinian evolution implies punctuated equilibria, Nature (London), 315, 400-402, (1985) |

[46] | Nichols, R.A., The ecological genetics of a hybrid zone in the alpine grasshopper, podisma pedestris, () |

[47] | Schulman, L.S., Techniques and applications of path integration, (1981), Wiley New York · Zbl 0587.28010 |

[48] | Schulman, L.S., Ray optics for diffraction: A useful paradox in a path integral context, () |

[49] | Slatkin, M., Gene flow and selection in a cline, Genetics, 75, 733-756, (1973) |

[50] | Slatkin, M., Spatial patterns in the distribution of polygenic characters, J. theor. biol, 70, 213-228, (1978) |

[51] | Slatkin, M., Fixation probabilities and fixation times in a subdivided population, Evolution, 35, 477-488, (1981) |

[52] | Slatkin, M., Gene flow in natural populations, Annu. rev. ecol. syst, 16, 393-430, (1985) |

[53] | Tier, C.; Keller, J., Asymptotic analysis of diffusion equations in population genetics, SIAM J. appl. math, 34, 549-570, (1978) · Zbl 0382.92005 |

[54] | Turner, M.S.; Wilczek, F., Is our vacuum metastable?, Nature (London), 298, 633-634, (1982) |

[55] | Van Kampen, N.E., Stochastic processes in physics and chemistry, (1980), North-Holland New York |

[56] | Walsh, J.B., Rate of accumulation of reproductive isolation by chromosome rearrangement, Amer. natur, 120, 510-523, (1982) |

[57] | Weiss, G.H.; Kimura, M., A mathematical analysis of the stepping stone model of genetic correlation, J. appl. probab, 2, 129-149, (1966) · Zbl 0151.25705 |

[58] | Weiss, U., The uses of path integrals for diffusion in bistable potentials, (), 177-187 · Zbl 0477.60077 |

[59] | White, M.J.D., Models of speciation, Science, 158, 1065-1070, (1968) |

[60] | Wright, S., Evolution in Mendelian populations, Genetics, 16, 97-159, (1931) |

[61] | Wright, S., On the probability of reciprocal translocations, Amer. natur, 75, 513-522, (1941) |

[62] | Wright, S., Isolation by distance, Genetics, 28, 114-138, (1943) |

[63] | Wright, S., Isolation by distance under diverse systems of mating, Genetics, 31, 39-59, (1946) |

[64] | Wright, S., Evolution and the genetics of populations. IV. variability within and among natural populations, (1978), Univ. of Chicago Press Chicago |

[65] | Wright, S., Genic and organismic selection, Evolution, 34, 825-843, (1980) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.