Given two sister populations that diverged some time ago, say pop2 migrates to pop1 and breeds there, giving offspring with sequence segments similar to those from pop1 but not in pop2. There are stats to find how similar two populations are; for instane, mean dxy takes the average of number of pairwise sequence differences between the two sister populations. Say if there is no migration, mean dxy at a locus of length 10000 is 100. Then with migration, say mean dxy at that locus is 300. Detecting a dxy that's unlikely under the null hypothesis of no migration indicates migration and interbreeding occurred.
I've been thinking for a while on if there are summary statistics which can distinguish between migration from pop2 to pop1 vs. migration from pop1 to pop2 (assuming only one of these events occurs). Do such statistics exist? Without loss of generality assume pop2 members migrate to pop1. Then pop1 will have members that are more similar to pop2 compared to if no migration occurred, but pop2 will be as if no migration occurred.
If we run simulations of a population w/ no migration, we can expect to see some value of pi and # of segregating sites. These measurements may be different in pop1 when pop2 migrates into pop1; some members of pop1 will have SNPs that pop2 do not have, and thus there would be more genetic variation (higher pi and ss) in pop1 compared to its expected pi and ss under no migration. So dxy can say that there's some similarity between pop1 and pop2 that's not expected under null hypothesis, and then pi or ss can say which of the two has a higher genetic variation than expected. Can some statistic that depends on dxy and pi distinguish between migration from pop1 to pop2 vs migration from pop2 to pop1?