Discrete time matrix models for the dynamics of structured populations provide one way to study the dynamic consequences of different life history strategies. One fundamental strategy is semelparity. Mathematically, semelparity can be associated with a high co-dimensional bifurcation at R0 = 1 which results in a dynamic dichotomy between persistence equilibrium states (lying in the interior of the positive cone) and synchronous cycles and cycle chains (lying on the boundary of the cone). Biologically, the dynamic alternative is between equilibration with overlapping generations and periodic oscillations with non-overlapping generations. I will describe what has been proved about the bifurcation at R0 = 1 for lower dimensional models. It remains a difficult mathematical challenge to describe the nature of the bifurcation at R0 = 1 for higher dimensional models. Time permitting I will discuss the bifurcation at R0 = 1 for matrix models extended to an evolutionary setting (by evolutionary game theory).