Abstract
A probability model of a population undergoing migration, mutation, and mating in a geographic continuum R is constructed, and an integrodifferential equation is derived for the probability of genetic identity. The equation is solved in one case, and asymptotic analysis done in others. Individuals at x, y ε R in the model mate with probability V(x, y) dt in any time interval (t, t + dt). In two dimensions, if V(x,y) = V(x−y) where V(x) ≈ V(x/β)/β 2 approaches a delta function, the equilibrium probability of identity vanishes as β → 0. The asymptotic rate at which this occurs is discussed for mutation rates u ≡ u o > 0 and for β ≈ Cu α, α > 0, and u → 0.
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Partially supported by NSF grant MCS79-03472
Research was partially supported by Task Agreement No. DE-AT06-76EV71005 under Contract No. DE-AM06-76RL02225 between the U.S. Dept. Energy and the University of Washington
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Sawyer, S., Felsenstein, J. A continuous migration model with stable demography. J. Math. Biology 11, 193–205 (1981). https://doi.org/10.1007/BF00275442
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DOI: https://doi.org/10.1007/BF00275442