Globalization and global climate change will probably be accompanied by rapid social and biophysical changes that may be caused by external forcing or internal nonlinear dynamics. These changes often subject residing populations (human or otherwise) to harsh environments and force them to respond. Research efforts have mostly focused on the underlying mechanisms that drive these changes and the characteristics of new equilibria towards which populations would adapt. However, the transient dynamics of how populations respond under these new regimes is equally, if not more, important, and systematic analysis of such dynamics has received less attention. Here, we investigate this problem under the framework of replicator dynamics with fixed reward kernels. We show that at least two types of population responses are possible--cohesive and population-dividing transitions--and demonstrate that the critical transition between the two, as well as other important properties, can be expressed in simple relationships between the shape of reward structure, shift magnitude and initial strategy diversity. Importantly, these relationships are derived from a simple, yet powerful and versatile, method. As many important phenomena, from political polarization to the evolution of distinct ecological traits, may be cast in terms of division of populations, we expect our findings and method to be useful and applicable for understanding population responses to change in a wide range of contexts.