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Interact box 7.3 Density-dependent natural selection

Launch Populus. In the Model menu, choose Natural Selection and then Density-Dependent

Selection w/ Genetic Variation. In the options dialog you can set the genotype-specific carrying capacity and growth rates. Click on the radio button for Nine-Frequency to display results for nine initial allele frequencies (the Single Frequency button shows the results in terms of the total population size N). The N text box sets the initial population size. Press the View button to see the simulation results.

Parameter values to simulate:

Kaa = 8000, KAa = 8000, and Kaa = 10,000; rAA = 0.4, rAa = 0.4, and raa = 0.3; generations = 100 Km = 8000, KAa = 10,000, and Kaa = 8000; rM = 0.4, rka = 0.3, and raa = 0.35; generations = 100 Kaa = 8000, Kka = 6000, and Kaa = 9000; rAA = 0.5, rAa = 0.3, and raa = 0.4; generations = 100

fitnesses. As N for each genotype approaches zero r the — N term will also approach zero, leaving each K

absolute fitness increasingly determined by the genotype-specific growth rate. The genotype with the highest growth rate should then increase fastest and dictate genotype and allele frequencies at equilibrium (if rAA is highest A fixes, if raa is highest a fixes, if rAa is highest both alleles segregate, and if rAa is lowest then initial allele frequencies dictate fixation of either A or a). This effect can be seen in Fig. 7.6 where allele frequencies change toward a lower frequency of p when the population is small. Despite the fact that the population is expected to fix for p at carrying capacity, the growth rate of the aa genotype is the greatest and so it has the greatest impact on allele frequencies when the population is small.

7.3 Combining natural selection with other processes

• Natural selection and genetic drift acting simultaneously.

• The balance between natural selection and mutation.

Natural selection takes place at the same time that other processes are also operating and having an impact on allele frequencies. These other processes may work in concert with natural selection and work toward the same equilibrium allele frequencies, or they may work against natural selection toward alternative equilibrium allele frequencies. Since many population genetic processes are likely to be acting simultaneously in actual biological populations, it is important to put natural selection into the context of other processes that impact allele frequencies. This section first considers allele frequencies when natural selection and genetic drift are in opposition and then natural selection and mutation acting in opposition. When natural selection and other processes act in concert, this simply shortens the number of generations required to reach equilibrium and does not alter equilibrium allele frequencies.

Natural selection and genetic drift acting simultaneously

Wright (1931) showed the probability that a population has a given allele frequency when exposed to the simultaneous processes of natural selection, genetic drift, and mutation as given by

where ^ (pronounced "phi") means a probability density, p and q are the allele frequencies, Ne is the effective population size, || and v are the forward and backward mutation rates, s is the selection coefficient, and C is a constant used to adjust the total probability across all allele frequencies to sum to 1.0 for each value of Nes. This equation is a probability density function for an ensemble population like those discussed in Chapter 3 for genetic drift. It describes the chance that one of many replicate populations will reach any allele frequency between zero and one at equilibrium given values of the effective population size and the selection coefficient.

This equation is most easily understood by examining its predictions in graphical form (Fig. 7.7). When Nes is near zero, either genetic drift is very

Figure 7.7 The expected distribution of allele frequencies for a very large number of replicate finite populations under natural selection where there is overdominance for fitness (wAA = waa = 1 - s and wAa = 1). In an infinite population the expected allele frequency at equilibrium is 0.5. However, in finite populations the equilibrium allele frequency will depend on the balance of natural selection and genetic drift. This balance is determined by the product of the effective population size and the selection coefficient (Nes). Low values of Nes mean that selection is very weak compared to drift and each population reaches fixation or loss. High values of Nes mean that selection is strong compared to drift and most populations reach an equilibrium allele frequency near 0.5. Here forward and backward mutation rates are equal (|i = v = 0.00001).

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