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Frequency of A allele (p)

Figure 7.5 The relative fitness of each genotype (wxx) and the change in allele frequency (Ap) across all frequencies of the A allele under frequency-dependent natural selection. There is a stable equilibrium point at p = 0.5 in this particular case, even though the heterozygote has the lowest fitness. Two unstable equilibria at fixation and loss are marked with open circles. Here the relative fitness values are

limits. The first section of this chapter developed a natural selection model where the size of the population of any genotype one generation in the future was its population size currently multiplied by a constant (refer back to equation 6.1). This model is obviously unrealistic because no organism can grow without some eventual limits on population size. Organisms are limited by the space and resources available to them, limitations that lead to changes in the rate of growth as the density of individuals changes over time. To incorporate such limits in a model of natural selection, we can alter our basic genotype-specific population growth equations to incorporate an upper bound on the population size as well as a rate of population growth that changes with population size.

A simple model where population growth has an upper bound is called logistic growth and the upper limit is called the carrying capacity (symbolized by K). Logistic growth depends on feedback between the growth rate and the size of the population according to

Interact box 7.2 Frequency-dependent natural selection

Launch PopGeneS2 and in the Selection menu, choose Frequency dependent selection. In the model dialog you can set the frequency-sensitive relative fitness values for the three genotypes produced by one locus with two alleles. The s1, s2, and s3 values correspond to sAA, sAa, and saa in equation 7.19. First try values of s1 = 0.3, s2 = 1.0, and s3 = 0.3. Explain why the graph of genotype-specific fitness by allele frequency (the middle panel) looks the way it does.

Next enter selection coefficients of s1 = 0.7, s2 = 1.0, and s3 = 0.2. Then compare the p by delta p graph in the top panel with the p by mean fitness graph in the bottom panel to see the mean fitness at equilibrium allele frequency. Does natural selection always cause allele frequencies to reach an equilibrium that corresponds to the maximum value of the mean fitness? Through educated guesses, try to find selection coefficients where equilibrium allele frequencies do and do not correspond to the maximum mean fitness value.

where N is the population size and r is the rate of increase (X used as the growth rate earlier in Chapter 6 can be equated to r by X = 1 + r). Biologically, r represents the rate of growth in excess of the r replacement rate of one. When N = 0 then — N is 0

and the growth rate is at its maximum. But when r

N = K then — N equals r and the population replaces itself but does not change in size.

Logistic population growth can be applied to the three genotypes at a diallelic locus by defining genotype-specific carrying capacities and rates of increase to obtain absolute fitness values for each genotype:

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