the fisher equation gene dispersion within a population
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The Fisher Equation
Gene Dispersion Within a Population
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Sir Ronald Fisher
• 1890-1962
• Renown statistician and geneticist.
• Wrote mathematics for biologists, and biology for mathematicians.
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Simplified Behavior
• We simplify the situation to only two variables and two parameters.
• Defining f(u) = s*u*(1-u)
• u’ = v
• v’ = -f(u) + c*v
• We get an interesting model.
• http://math.rice.edu/~dfield/dfpp.html
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Slightly More Complicated View
• Assumptions:
– A population is distributed in a linear habitat.
– It is uniformly distributed.
– There are only two alleles present for the specified locus.
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Variables and parameters
• p = frequency of the mutant gene.
• q = frequency of other allele.
• m = intensity of selection in favor of p.
• x = position along the habitat.
• t = time in generations.
• k = constant of diffusion.
• Assumption: p and m are independent.
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Cases for c
• (a) c = 1
• (b) c is between 1 and sqrt(1/2)
• (c) c = sqrt(1/2)
• (d) c < sqrt(1/2)