Many models of animal populations show complex yet predictable patterns of density changes under simple and plausible assumptions. Yet one previous attempt to determine the extent and importance of complex dynamics concluded that they were likely only in some laboratory populations, but not in field populations. Ecologists have treated changes more complex than a return to a simple equilibrium, such as the cyclical changes in populations of lynx and voles in the arctic, as special cases. Highly variable populations, such as insects, are usually thought to be driven by unpredictable changes in the weather. Here, we assemble 71 populations counted for over 50 years, and suggest that complex yet predictable population changes are more common than previously thought. © 1990 Chapman and Hall Ltd.