I am working on a assignment to program a genetic drift simulator and include population growth. My program works fine as far as I can tell, and I have the not unexpected result that in the time I am willing to let my program run, some number of simulations neither allele ever fixes in the population. I believe this is because the rate of compound growth outpaces the rate of drift.
I am trying to come up with a direct mathematical model to show that this is the case (which is well beyond the scope of the assignment).
Population growth equation is $P_t = P_o(1+g)^t$, the well known compound interest formula.
However, modeling genetic drift in terms of population size is more complicated. The most common genetic drift equations assume constant population:
$$H_t = H_o(1-(1/2N)^t$$
Where N is population size and H is the heterozygosity of the population. Heterozygosity can also be thought of as the proportion of the population that is heterozygous. If Hardy-Weinberg equilibrium is assumed (which I am not sure we can in this case) then $H=2p(1-p)$.
So my first approach is to try and relate $H_t$ to $P_t$ by saying that $P_{ht}=H_tP_o(1+g)^t$ or that the heterozygous population at time $t$ is the population multiplied by $H_t$. Then if you substitute this into the equation you get:
$$P_{ht}=H_o(1-1/(2P_{ht}))^tP_o(1+g)^t$$
But I have no idea how I would solve this equation for $P_{ht}$ or if this even makes sense.
Does anyone have any suggestions as to how I could combine these two equations or otherwise model the combination of genetic drift with compound growth?