The influence of spatial inhomogeneities on neutral models of geographical variation IV. Discontinuities in the population density and migration rate.
The equilibrium structure of the infinite, one-dimensional stepping-stone model with coincident discontinuities in the population density and migration rate is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes isotropically. The population density and the migration rate have a discontinuity at the origin, but are elsewhere uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus without selection; every allele mutates to new alleles at the same rate. The three dimensionless parameters in the theory are alpha=(rho(+)/rho(-))(2) (V(+)/V(-))(3/2), and beta(+/-)=4rho(+/-) 2uV(+/-), where rho(+) (rho(-)) and V(+) (V(-)) designate the population density and variance of gametic dispersion per generation to the right (left) of the discontinuity, respectively, and u denotes the mutation rate. The characteristic length on the right (left) is V(+)/(2u) (V(-)/(2u)). The probability of identity is continuous at the origin, but its partial derivatives have a discontinuity unless migration is conservative (rho(-) V(-)=rho(+) V(+)). At least for nonconservative migration, the probability of identity (including the expected homozygosity) can be nonmonotonic even if the migration rate is uniform and the population density is monotonic. Thus, there can be a nonmonotonic genetic response in a neutral model to a monotonic environment.
Ayati
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BP
Department of Mathematics, The University of Chicago, 1118-1132 East 58th Street, Chicago, Illinois 60637, USA.
Dupont
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TF
Nagylaki
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T
D000818:Animals; D004463:Ecology; D005787:Gene Frequency; D005828:Genetics, Population; D016014:Linear Models; D008957:Models, Genetic; D011156:Population Density; D011157:Population Dynamics