Despite that outbreaks had been observed for hundreds of years for many populations, it took until the 1920s before the first mechanisms that did not involve human interference were suggested. Just a few mechanisms were included in the first models, and the question whether the inclusion of other, very plausible, mechanisms could alter the predictions remained. In this chapter, we follow the development of models that have been proposed to explain oscillatory population dynamics from the early models suggested by Lotka (Elements of physical biology. Williams and Wilkins, Baltimore, 1925) and Volterra (Mem R Accad Natl Lincei 6:31–113, 1926) until global dynamical questions that are still open for models incorporating explicit resource dynamics, like the chemostat, cf. Kuang (SIAM J Appl Math 49:1759–1767, 1989).