Ancestral inference from demographic data

Question

The question is inspired by this one, in the History community:
We consider a population, whereas the data we have the number of migrants arriving every year, as well as the initial and the current population size. We would like to determine what fraction of the population descends from the migrants that came in a particular year.

Naive approach
Let $n_t$ be the number of migrants arriving in year $t$ from the beginning of observation whereas $n_0$ is the number of individuals already present at year $t=0$, $T$ is the number of years. Let $s$ be the rate of increase of the population which we assume constant, so that the number of individuals descending from the immigrants from year $t$ is $$N_t=n_t s^{T-t}.$$ One can now estimate the parameter $s$ from the total size of the population at $t=T$: $$ \sum_{t=0}^Tn_ts^{T-t}=N, $$ and we obtain the proportions of interest as $$w_t=\frac{N_t}{N}$$.

Q1: is this a reasonable approach or is there a better way?

Haploid population
The approach above is correct for averages, if we assume haploid population. This is the case when we look only at Y-chromosome, i.e., only at the paternal descent, as suggested in the cited question. There is however demographic stochasticity (or genetic drift - depending on whether we prefer population ecology or population genetics language), which means that the fluctuations in $N_t$ may make our estimates obsolete. I could probably obtain estimates using birth-and-death process ($s=1+b-d$) or a colascent. I am wondering however, if there is a more canonical way of doing it. (this is Q2)

Diploid population
For diploid organism most currently living individuals will have many ancestors from different years. One can thus ask a somewhat different question: how many of the currently living individuals at least one ancestor that arrived in year $t$?

Q3: how could one perform such an estimate?

Answer 1

As mentioned in the OP, one can perform an estimate using birth-and-death process. Let us consider an population from a particular year with the initial size $N_0$ - then the probability that the population has size $N$ obeys the following difference equation: $$p^N_{t+1} = (1-b-d)p^N_t + b(N-1)p^{N-1}_t + d(N+1)p^{N+1}_t,$$ where $b$ and d are the birth and the death rates (per generation or per year - depending on how we choose the time scale). Rather than solving for probability, we limit consideration to the mean and the variance: $$ N_t=\sum_{N=0}^{+\infty}Np^N_t,\\ V_t=\sum_{N=0}^{+\infty}N^2p^N_t-N_t^2.$$ After some algebra we wrrive at: $$N_t=N_0(1+b-d)^t,\\ V_t=N_0^2\left[(1+2b-2d)^t-(1+b-d)^{2t}\right]+ \frac{b+d}{b-d}N_0\left[(1+2b-2d)^t-(1+b-d)^t\right] $$ The following analysis is simplified, if we assume slow population change, so that $|b-d|\ll 1$, so that we can switch to continuous time: $$ N_t=N_0 e^{(b-d)t},\\ V_t= \frac{b+d}{b-d}N_0\left[e^{2(b-d)t}-e^{(b-d)t}\right] $$

Constant population
Let us first consider the case of a constant total population size $b=d$. The equations above then reduce to $$N_t=N_0,\\ V_t=N_0(b+d)t.$$ The relative error is then $$ \frac{\sqrt{V_t}}{N_t}=\sqrt{\frac{(b+d)t}{N_0}} $$ This error grows with time and eventually becomes large, which is the reflection of the fact that genetic/ecological drift will eventually lead to the extinction or fixation of the allele/population. However, as long as we look at times much shorter than $t\ll N_0/{b+d}$, the error is small and we can make a reliable estimate using the naive approach outlined in the beginning. Indeed, if we take birth/death rate as 1 per generation, and consider the time scales of up to 20 generations (Mayflower came to the US in 1620, 400 years ago) we have the relative error behaving as $\sqrt{40/N_0}$. Mayflower brought about a hundred of passangers, so that the error is relatively big, but for later and much larger imigration waves the error can bemuch smaller.

Growing population
If we assume growing population, $b>d$, and assuming in addition that we look at long time periods, $t\gg|b-d|^{-1}$, we can write the relative error as: $$ \frac{\sqrt{V_t}}{N_t}=\sqrt{\frac{1}{N_0}\frac{b+d}{b-d}} $$ This error is small, if $N_0\gg \frac{b+d}{b-d}$. With a birth rate of $3$ children per family, i.e., $b=1.5$ per person, this condition becomes $N_0\gg 5$, i.e., even small immigration waves shoudl be discernable, provided the community grows fast.