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The Biology Project > Biomath > Polynomials > Applications > Frequency-Dependent Selection

Polynomial Applications

Frequency-Dependent Selection

At first glance, population geneticists typically assume that genotypes have constant fitness values, which are a measure of their reproductive success. In some case this is not a reasonable assumption. For example, the fitness of a genotype may depend on time, environment, or on the frequency-dependent selection. Positive frequency-dependent selection occurs when the fitness of a genotype increases as it becomes more common in a population. In contrast, negative frequency-dependent selection favors rare alleles by assigning the highest fitness to the rarest allele. Negative frequency-dependent selection can only give rise to balanced polymorphisms in a population because common genotypes will be selected against, limiting their chance of fixation. Examples of negative-frequency dependent selection included: self-compatibility alleles in plants, search images in predator-prey systems, mating preferences in Drosophila, and jaw handedness in cichlid fish.



Mating preferences in Drosophila melanogaster is an example of negative-frequency dependent selection. Photo credit: Steve A. Kay, The Scripps Research Institute, Courtesy of the National Science Foundation.


Consider a two allele system (A1 and A2) and let p and q represent the frequency of A1 and A2, respectively. Let wij represent the fitness of genotype AiAj (assume wij = wji for ij). Therefore, for this system w11 is the fitness of allele A1A1, w12 is the fitness of allele A1A2, and w22 is the fitness of allele A2A2. We can express the new frequency of A1 after one generation of selection using the rational function,

where is the mean fitness of the population. We are interested in finding equilibrium values of p, in other words values of p such that p' = p, indicating no change in allele frequency in the next generation. Setting p' = p gives,

Assuming p ≠ 0 we can cancel p on both sides of the above equation as,

Bringing all terms to the right-hand side of the equation and substituting q = 1 - p gives the polynomial,

Therefore, we have deduced that solutions to the above equations are equilibrium values of p (we are only concerned with biologically reasonable equilibria). Using fitness values:

where c > 0 is a constant, answer the following questions.

Answer the following questions:

Given fitness values, find the equation that defines the equilibrium value of A1.

Find the values of A1 that are roots for a given equation describing the equilibrium values of A1.

Find the mean fitness given a specific value of A1.

Describe the graph of the function representing the equilibrium value of A1.

Find the approximate frequency of A1 after one generation of selection.

Given the frequency-dependent fitness of a genotype, determine if is subject to negative or positive frequency-dependent selection.


The Biology Project > Biomath > Polynomials > Applications > Frequency-Dependent Selection

The Biology Project
Department of Biochemistry and Molecular Biophysics

The University of Arizona

March 2007
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