J. theor. Biol. (1975) 51, 511-524 Biological Populations Obeying Difference Equations : Stable Points, Stable Cycles, and Chaos ROBWT M. MAY Biology Department, Princeton University, Princeton, N.J. 08540, U.S.A. (Received 28 June 1974) For biological populations with nonoverlapping generations, population growth takes place in discretetime stepsand is described by difference equations. Someof the simplest such nonlinear differenceequationscan exhibit a remarkrible spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillationsbetweentwo population points, to stable cycles with four points, then eight, 16,etc., points, through to a chaotic regimein which (depending on the initial population value) cycles of any period, or even totally aperiodic but bounded population fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearizedstability analyses; its existence in the simplest and fully deterministic nonlinear (“density dependent”) difference equations is a fact of considerable mathematical and ecological interest. 1. Introduction In some biological situations (such asman), population growth is a continuous process and generations overlap; the appropriate mathematical description involves nonlinear differential equations. In other biological situations (such as 13 year periodical cicadas), population growth takes place at discrete intervals of time and generations are completely nonoverlapping; the appro- priate mathematical description is in terms of nonlinear difference equations. For a single species,the simplest such differential equations, with no time- delays, lead to very simple dynamics: a familiar example is the logistic, dN/dt = rN(1 -N/K), with a globally stable equilibrium point at N = K for all r > 0. But the corresponding simplest difference equations, with their built-in time lag in the operation of regulatory mechanisms, can have a complicated dynamical structure, the great richness of which is not commonly appreciated either in the ecological literature, or in elementary mathematical discussions of difference equations. For a single species, the difference equations arising in population biology are usually discussed as having either a stable equilibrium point or unstable, growing oscillations. In fact, some of the most elementary of these nonlinear T.B. 511 33