The works of [Cha-DunAlvInoNieCarFieLaw,Cha-Dun] describe upward sweeps in populations of city-states and attempt to characterize such phenomenon. The model proposed in both [TurKor,Tur] describes how the population, state resources and internal conflict influence each other over time. We show that one can obtain an upward sweep in the population by altering particular parameters of the system of differential equations constituting the model given in [TurKor,Tur]. Moreover, we show that such a system has a unstable critical point and propose an approach for determining bifurcation points in the parameter space for the model.