There is an ongoing debate in the field of human evolution about the possible 
contribution of Neanderthals to the modern human gene pool. To study how the 
Neanderthal private alleles may have spread over the genes of Homo sapiens, we 
propose a deterministic model based on recursive equations and ordinary 
differential equations. If the Neanderthal population was large compared to the 
Homo sapiens population at the beginning of the contact period, we show that 
genetic introgression should have been fast and complete meaning that most of 
the Neanderthal private alleles should be found in the modern human gene pool in 
case of ancient admixture. In order to test/reject ancient admixture from 
genome-wide data, we incorporate the model of genetic introgression into a 
statistical hypothesis-testing framework. We show that the power to reject 
ancient admixture increases as the ratio, at the time of putative admixture, of 
the population size of Homo sapiens over that of Neanderthal decreases. We find 
that the power to reject ancient admixture might be particularly low if the 
population size of Homo sapiens was comparable to the Neanderthal population 
size.