In the simplest model I previously developed, each organism in a large population had one of two possible phenotypes, A or B (the binary system). Reproduction was exclusively asexual and there was no mutation. This means that all organisms of a given phenotype could give rise only to offspring of the same phenotype. I recommend reviewing my previous article, but here I will summarize the main results:
The equilibrium behavior of this type of system is easy to predict: as long as one of the phenotypes confers greater fitness, it will eventually entirely out-compete the other. That is, the equilibrium state will always be a population composed entirely of the more fit phenotype. While I examined some affects of mutation rate on evolution in my previous article, I did not investigate its affect on the equilibrium state of the population. Thus, in this article I will elucidate this relationship.
A model for population evolution with mutation
The relationship between mutation rate and population equilibrium state
The rates of change in the relative frequency of phenotype A and B are given by:
Examining the terms in this equation, notice that mutation rate (M) always occurs as (1-M)/M. It is interesting to consider the meaning and significance of this. This term gives the ratio of unmutated to mutated offspring a parent will have. For instance, if the mutation rate is 10%, 9 unmutated offspring are produced for every mutated one. Correspondingly, we have (1-M)/M=9. I therefore define this term as conservation (C) since it quantifies how conserved a trait is from parent to offspring.
The conservation term allows us to rewrite the equilibrium equation in a simpler form.
Limiting cases of the mutation-equilibrium equation
pB = 0
Suppose a mutant phenotype is nonlethal but renders an organism entirely unable to reproduce. In my equations that would be indicated by pB = 0. In this case the equilibrium ratio simplifies greatly to the unmutated to mutated ratio (by definition, the conservation).
This extreme case represents the maximum mutation rate, where offspring phenotype is entirely unrelated to that of the parents. As we intuitively expect, this gives rise to 1:1 phenotypic ratio at equilibrium, independent of their relative fitness values.
When no mutation is possible, the situation simplifies to the binary model given in my previous article. Here we obtain a homogeneous population of the more fit phenotype at equilibrium, with the phenotypic ratio tending toward infinity.