Biological populations obeying difference equations: stable points, stable cycles, and chaos.

For biological populations with nonoverlapping generations, population growth takes place in discrete time steps and is described by difference equations. Some of the simplest such nonlinear difference equations can exhibit a remarkrible spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between two population points, to stable cycles with four points, then eight, 16, etc., points, through to a chaotic regime in 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 linearized stability analyses; its existence in the simplest and fully deterministic nonlinear (“density dependent”) difference equations is a fact of considerable mathematical and ecological interest.