One of Turchin's most prominent fields of research is his study of the hypothesis that population pressure causes increased warfare. Turchin, in collaboration with Korotayev, has shown that such negative results do not falsify the population-warfare hypothesis. Population and warfare are dynamical variables. If their interaction causes sustained oscillations, then we do not in general expect to find strong correlation between the two variables measured at the same time (that is, unlagged). Turchin and Korotayev have explored mathematically what the dynamical patterns of interaction between population and warfare (focusing on internal warfare) might be in stateless and state societies. Next, they tested the model predictions in several empirical case studies: early modern England, Han and Tang China, and the Roman Empire. Their empirical results have supported the population-warfare theory: Turchin and Korotayev have found that there is a tendency for population numbers and internal warfare intensity to oscillate with the same period but shifted in phase (with warfare peaks following population peaks). Furthermore, they have demonstrated that the rates of change of the two variables behave precisely as predicted by the theory: population rate of change is negatively affected by warfare intensity, while warfare rate of change is positively affected by population density.