It has been a while since the last EvoMath. In this installment I am going to begin to discuss classical selection theory. Selection occurs when certain alleles are likely to transmit more copies of themselves to the next generation than other alleles at the same locus. The simplest way to think of this is in terms of the viabilitity of individuals. If an individual dies before it can reproduce, then it is not able to transmit its genes. If such a death was influenced by the genes it carried then selection can occur. Classical selection theory assumes that there exists viability selection and that it is constant, i.e. independent of allele or genotype frequencies. There is also theory behind frequency-dependent selection, but it beyond the scope of this article.
First up, we are going to look at selection on a haploid locus with two alleles. These are the assumptions we are going to make, so that we can develop the model.
- Infinite population size
- No mutation or migration
- Constant viability selection
- Discrete, non-overlapping generations
- Asexual reproduction
Haploids only carry one copy of each gene, thus we are concerned with two genotypes: A and a. A proportion of each of these genotypes will survive long enough to reproduce. This proportion is called the absolute, viability fitness (AVF). Let the AVFs of A and a be and respectively. To demonstrate how this selection works, we are going to temporarilly assume that each generation begins with a constant number of individuals, , where is the number of individuals of genotype , and where and are the proportions of A and a individuals respectively. Through selection, only a proportion of these genotypes survive to adulthood. Therefore, the number of adults is
where is the average fitness of the population. The proportions of individuals amongst the adults are
Because the N’s cancel out, clearly
And because the number of offspring in the next generation is infinite, the genetic makeup of the offspring will be equivalent to that of the adults. Thus if subscript represents the adult generation,
Now that we have determined the equation of evolution for our model. We can ask some questions about it. The first question usually asked is when will it stop, i.e. . The population is considered to be at equilibrium if this relationship holds true. What are the equilibria in our model? If , then at equilibrium
This implies that three are two trivial equilibria, and which correspond to the situations where only one allele remains in the population. This demonstrates an important point about evolution: no genetic diversity, no evolution. Furthermore, if , the population will always be at equilibrium regardless of the allele frequency. This demonstrates another important point, that selection produces evolution only when there are differences in fitness.
The stability of equilibria is very important. An equilibrium is considered stable if there exists a neighborhood of allele frequencies around it for which the population will end up at that equilibrium. There is a very simple way to determine if an equilibrium is stable or not. Let . If then the equilibrium, is stable. If it is unstable. And finally, if it is neutrally stable. For this model,
which implies that the equilibrium, , will be stable if and only if . Since ,
which implies that the equilibrium, will be stable if and only if .
Now the next question usually asked is what is the direction of evolution, i.e. when is positive or negative? Since, , , and are always positive, the sign of is then determined by . Thus will be positive if and only if and will be negative if and only if . From this you can see that A increase in frequency if it is favored, and will decrease in frequency if it is disfavored.
We can look at the average fitness, as the population fitness. How does evolution change the population fitness? First note that since is a linear function, its maximums and minimums will be found at the boundaries, . Therefore, will be maximized at , if , and maximized at if .
Putting this all together produces the following conclusions, if :
|Condition||Behavior of Alleles||Behavior of Population Fitness|
|A becomes fixed and a extinct.||The population fitness becomes maximized at .|
|a becomes fixed and A extinct.||The population fitness becomes maximized at .|
|Allele frequencies do not change.||The population fitness does not change.|
Since given an intital population that is genetically diverse, the population will evolve to a maximal fitness, this is sometimes refered to as a “hill-climbing” process.
In the next installment, I’ll analysize a diploid model.