A simple one-dimensional model of single-species populations is studied by means of computer simulations. Although the model has a rich spectrum of dynamics including chaotic behavior, the introduction of survival thresholds makes the chaotic region so small that it can be hardly observed. Stochastic fluctuations further reduce the chaotic region because they accidentally lead populations to extinction. The model thus naturally explains the observation that the majority of natural populations do not show chaotic behavior but a monotonic return to a stable equilibrium point following a disturbance. 1. Introduction. It is now well known that certain systems, though entirely deterministic and describable by difference or differential equations, sometimes yield irregular and apparently "chaotic" fluctuations (see for example, Lorenz, 1963; May, 1974;R6ssler, 1976;Schuster, 1988). The generation of such unpredictable motions is quite remarkable because the equations entirely lack stochastic elements. These chaotic patterns can arise in ecological singlespecies populations with discrete, non-overlapping generations, described by nonlinear one-dimensional maps (May, 1974;May and Oster, 1976).
This approach, however, entails obvious risks because there are few, if any, single-species systems in nature. Nonlinear one-dimensional maps thus may not be regarded as a literal description of any real world ecological systems (Nisbet and Gurney, 1982). Lots of elaborate models and techniques therefore have been invented in order to study field data (for example, Pimm and Redfearn, 1988; Sugihara and May, 1990; Turchin et al., 1991). Nevertheless, this analysis is very important because it is sometimes possible to extract a onedimensional mapping even from a complex higher order continuous system (Schaffer and Kot, 1985, 1986).
In the sense mentioned above, laboratory studies have the advantage that there really is a good approximation to a single-species situation with discrete dynamics. In fact, laboratory experiments on insect populations exhibit a rich spectrum of dynamical behavior, ranging from a stable equilibrium point, through stable cycles, to apparently chaotic population fluctuations (Hassell et al., 1976).