A simple butterfly model

The first generation of a certain butterfly has two phenotypes, which are brown color and white color. There is a gene can control this color and if either allel is \(A\) the color will be brown. In other words, \(A\) is the dominant gene and \(a\) is the recessive gene. If we do not allow any genetic mutation and there is no difference of the living condition for two colors, which means white color butterflies are not caught by predators more easily, the only variation through generations is the distribution of each phenotype and genotype by random mating. This process is call genetic drift.

Simulation

Let us run a simulation to see how sexual reproduction works in changing the population of brown and white butterflies. We define \( x_{11} \) as the number of butterflies that have \(AA \) genotype, \(x_{12}\) as the number of those have \(Aa\) and \(x_{22}\) as for \(aa\). We will simulate the following process: suppose there are \(N=x_{11}+x_{12}+x_{22}\) butterflies in the first generation and we want to reproduce the next generation of \(N\) butterflies. Their parents are chosen randomly from the previous generation and randomly inherit one allel from each parent. We run this recursively to see how the ratio of each genotype evolves through \(T\) generations. Try different parameters below to see how 2 phenotypes and 3 genotypes vary.

Fixation and loss

You might find that there are two common results of the above simulation in the long term. One is that $$x_{12}=2x_{11}=2x_{22}=\frac{N}{2},$$ which is a steady steady in the sense of genotype frequency. This steady state is call fixation since the gene distribution is stable and the average change of distribution will be 0 eventhough none of the genotype is 0.
Another is there is only one phenotye, either \(AA\) or \(aa\). It is also a steady state but called loss, which means we lose some genotype. If loss happens there is no way to retrieve the loss gene. In other words it works like a trap. Therefore, if \(T\rightarrow\infty \), genetic drift will go to the loss.

Genetic diversity

There is no way to avoid genetic loss (without no mutation), but the time to loss can be adjusted by the number of population. You might find that it is harder to lose gene if \(N\) is larger. In other words, population size is crucial to maintain genetic diversity. If somehow the white butterflies are easier to be found by predators and there is few brown butterflies left, they will probably be eliminated from the earth soon. This is part of the reason why we want to keep endangered animals above certain population size.