@article{4263,
  abstract     = {We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, σe; the effective population density, ρe; and a local scale, κ, at which local interactions become significant. The results are illustrated by simulations.},
  author       = {Barton, Nicholas H and Depaulis, Frantz and Etheridge, Alison},
  issn         = {0040-5809},
  journal      = {Theoretical Population Biology},
  number       = {1},
  pages        = {31 -- 48},
  publisher    = {Academic Press},
  title        = {{Neutral evolution in spatially continuous populations}},
  doi          = {10.1006/tpbi.2001.1557},
  volume       = {61},
  year         = {2002},
}

@article{4272,
  abstract     = {Analysis of multilocus evolution is usually intractable for more than n ~ 10 genes, because the frequencies of very large numbers of genotypes must be followed. An exact analysis of up to n ~ 100 loci is feasible for a symmetrical model, in which a set of unlinked loci segregate for two alleles (labeled '0' and '1') with interchangeable effects on fitness. All haploid genotypes with the same number of 1 alleles can then remain equally frequent. However, such a symmetrical solution may be unstable: for example, under stabilizing selection, populations tend to fix any one genotype which approaches the optimum. Here, we show how the 2' x 2' stability matrix can be decomposed into a set of matrices, each no larger than n x n. This allows the stability of symmetrical solutions to be determined. We apply the method to stabilizing and disruptive selection in a single deme and to selection against heterozygotes in a linear cline. (C) 2000 Academic Press.},
  author       = {Barton, Nicholas H and Shpak, Max},
  issn         = {0040-5809},
  journal      = {Theoretical Population Biology},
  number       = {3},
  pages        = {249 -- 263},
  publisher    = {Academic Press},
  title        = {{The stability of symmetrical solutions to polygenic models}},
  doi          = {10.1006/tpbi.2000.1455},
  volume       = {57},
  year         = {2000},
}

@article{3649,
  abstract     = {Selection on polygenic characters is generally analyzed by statistical methods that assume a Gaussian (normal) distribution of breeding values. We present an alternative analysis based on multilocus population genetics. We use a general representation of selection, recombination, and drift to analyze an idealized polygenic system in which all genetic effects are additive (i.e., both dominance and epistasis are absent), but no assumptions are made about the distribution of breeding values or the numbers of loci or alleles. Our analysis produces three results. First, our equations reproduce the standard recursions for the mean and additive variance if breeding values are Gaussian; but they also reveal how non-Gaussian distributions of breeding values will alter these dynamics. Second, an approximation valid for weak selection shows that even if genetic variance is attributable to an effectively infinite number of loci with only additive effects, selection will generally drive the distribution of breeding values away from a Gaussian distribution by creating multilocus linkage disequilibria. Long-term dynamics of means can depart substantially from the predictions of the standard selection recursions, but the discrepancy may often be negligible for short-term selection. Third, by including mutation, we show that, for realistic parameter values, linkage disequilibrium has little effect on the amount of additive variance maintained at an equilibrium between stabilizing selection and mutation. Each of these analytical results is supported by numerical calculations.},
  author       = {Turelli, Michael and Barton, Nicholas H},
  issn         = {0040-5809},
  journal      = {Theoretical Population Biology},
  number       = {1},
  pages        = {1 -- 57},
  publisher    = {Academic Press},
  title        = {{Dynamics of polygenic characters under selection}},
  doi          = {10.1016/0040-5809(90)90002-D},
  volume       = {38},
  year         = {1990},
}

@article{3657,
  abstract     = {Shifts between adaptive peaks, caused by sampling drift, are involved in both speciation and adaptation via Wright's “shiftingbalance.” We use techniques from statistical mechanics to calculate the rate of such transitions for apopulation in a single panmictic deme and for apopulation 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 (≈sk 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 (≈exp(− cN √s)) or in a single deme (≈exp(− cNs)). Our calculations agree with simulations of a single deme and a one-dimensional population.},
  author       = {Rouhani, Shahin and Barton, Nicholas H},
  issn         = {1096-0325},
  journal      = {Theoretical Population Biology},
  number       = {3},
  pages        = {465 -- 492},
  publisher    = {Elsevier},
  title        = {{Speciation and the "shifting balance" in a continuous population}},
  doi          = {10.1016/0040-5809(87)90016-5},
  volume       = {31},
  year         = {1987},
}

@article{3662,
  abstract     = {The evolution of the probabilities of genetic identity within and between tandemly repeated loci of a multigene family is investigated analytically and numerically. Unbiased intrachromosomal gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. Under the restriction that there is at most one crossover in the multigene family per individual per generation, the dependence on location of the probabilities of identity is treated exactly. In the “homogeneous” approximation to this “exact” model, end effects are disregarded; in the “exchangeable” approximation, to which all previous work was confined, all position dependence is neglected. Numerical results indicate that (i) the exchangeable and homogeneous models are both qualitatively correct, (ii) the exchangeable model is sometimes too inaccurate for quantitative conclusions, and (iii) the homogeneous model is always more accurate than the exchangeable one and is always sufficiently accurate for quantitative conclusions.},
  author       = {Nagylaki, Thomas and Barton, Nicholas H},
  issn         = {1096-0325},
  journal      = {Theoretical Population Biology},
  number       = {3},
  pages        = {407 -- 437},
  publisher    = {Academic Press},
  title        = {{Intrachromosomal gene conversion, linkage, and the evolution of multigene families}},
  doi          = {10.1016/0040-5809(86)90017-1},
  volume       = {29},
  year         = {1986},
}