One-Locus Selection Models

• WA = relative viability of A individuals
• Wa = relative viability of a individuals
• p[t] = frequency of A individuals at generation t
• q[t] = 1-p[t] =frequency of a individuals

Let the adults produce haploid juvenile offspring that are subject to viability selection. After selection, the ratio of A alleles to a alleles becomes

p[t] WA : q[t] Wa

With these ratios, the frequency of A alleles is

and the frequency of a alleles is

(The denominator in these equations is the mean RELATIVE fitness in the population.) Calculate p[t+2].

Example: For p[0] to change from 0.01 to 0.99 with WA=1.01 and Wa=1 (allele A has a 1% selective advantage) takes 924 generations.

The mean RELATIVE fitness of a population is calculated as the frequency of each type in the population times its fitness. For the haploid viability model:

Is W[t+1] always greater than or equal to W[t]? First, note that:

From this,

Fundamental Theorem of Natural Selection.

"The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time."

-- Fisher (1930) The Genetical Theory of Natural Selection

Remark 2: Variance in fitness will be highest (and mean fitness increases greatest) when gene frequencies are near 1/2 and relative fitness differences are high.

Problem: While mean RELATIVE fitness may increase over time, this does not mean that the mean ABSOLUTE fitness of the population is increasing.

In fact, as Fisher (1930) noted, the mean absolute fitness of a population cannot increase indefinitely since the world would soon become overrun with a species that was growing exponentially.

He argued, that in the long-term, the mean absolute fitness of a population must hover around one (one offspring per parent). This was achieved, in his view, by a balance between the "progress" of natural selection and a "deterioration" of the environment:

"Against the rate of progress in fitness must be set off, if the organism is, properly speaking, highly adapted to its place in nature, deterioration due to undirected changes either in the organism [mutations], or in its environment [geological, climatological, or organic]."

-- Fisher (1930) The Genetical Theory of Natural Selection

We showed last time that with either random union of gametes in a gamete pool or with random mating of individuals, the diploid offspring are in Hardy-Weinberg proportions: p^2[t] : 2 p[t] q[t] : q^2[t], where p[t] is the frequency of the A allele among the mating individuals in generation t.

If the offspring are then subject to viability selection, their proportions become:

p^2[t] WAA : 2 p[t] q[t] WAa : q^2[t] Waa

Among the adults after selection, the frequency of the A is therefore:

There are a number of approaches to analysing this recursion equation. We will:

• Start with a graphical approach
• Find the equilibria of the system
• Analyse the stability of the equilibria

When p[t+1]=p[t] (i.e. the allele frequencies remain constant)

EQUILIBRIUM

To determine the equilibria for the diploid model, we must find the solution to:

Try solving for p yourself.

If you get stuck, follow these steps:

From this algebra we learn that p[t+1] will only equal p[t] in the diploid model if p=0, p=1, or

For the polymorphic equilibrium to be meaningful, p must fall between 0 and 1.

Challenge: Show that for the polymorphic equilibrium to be between 0 and 1, there must be either overdominance or underdominance.

Even when this holds, is it stable? Are the fixation states stable?

A stability analysis must be performed to address these questions.

Write the recursion equation (p[t+1]) as f(p[t]) since it is a function of p[t].

Global stability: Converges to an equilibrium from every starting position.

Local stability: Converges to an equilibrium from nearby starting positions.

We will only examine local stability: If we perturb the population a little (p[t]=p*+eps), does it return to p*?

Taylor approximation:

Note that f(p*) is p*. Therefore, p[t+1] will be close to p* than p[t] only if the derivative is less than one.

When the derivative is less than one, the equilibrium is locally stable.

• If WA > Wa, population converges to fixation on A
• If WA < Wa, population converges to fixation on a
• Mean relative fitness always increases or stays the same

Diploid Model

• If WAA > WAa > Waa, population converges to fixation on A
• If WAA < WAa < Waa, population converges to fixation on a
• If WAA < WAa > Waa, population converges to polymorphic equilibrium
• If WAA > WAa < Waa, population converges to fixation on A or on a
• Mean relative fitness always increases or stays the same (Challenge).