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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?

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Yes, migrants affect patterns of genetic diveristy

Immigrants yield to higher effective population size and therefore higher genetic diversity. They will also affect the site frequency spectrum and I would expect that if the migrants are numerous enough, they would yield to positive Tajima's D.

Therefore, unidirectional migration will definitely yield to differences in patterns of genetic variance among populations (that can be measured by any summary statistic of the site frequency spectrum).

Existing methods for detecting directional selection

I have never had to do such inference so I am not able to do a good review of the existing technics.

You should probably read Slatkin and Maddison 1998 and more importantly Beerli and Felsenstein 2001. Beerli and Felsenstein 2001 has been cited 1,341 times and you should probably quickly see whether some of these papers suggest more recent methods. John Novembre has some very interesting work on how to estimate migration rate between any two populations (e.g. Novembre and Stephens 2008). He might have done something of interest to your problem.

You can always use Approximate Bayesian Computation (ABC;see for example Jeffrey Jensen's work for example of inferences with ABC in population genetics) to figure out what pattern of migration is most likely. With a simple enough model, a good ABC algorithm is not that long (with good computing ressources). The question then is what statistic(s) to use for the rejection algorithm. Again for that, you should have a look at the existing literature to come up with a set of potentially useful statistics.

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