NONLINEAR TIME-SERIES ANALYSIS OF EMPIRICAL POPULATION-DYNAMICS

The classical approach towards time series analysis of fluctuating phenomena is based on linear stochastic processes in the basin of attraction of globally stable equilibria (ARMA models). However, there are deductive reasons derived from nonequilibrium statistical mechanics that many self-generated population fluctuations should be interpreted as stochastic processes influenced by nonequilibrium attractors of nonlinear dynamical systems. The incidence pattern of measles epidemics in New York City is used to demonstrate that highly nonlinear autoregressive models can serve as a new semi-phenomenological level of description for complex self-generated fluctuations in biological or ecological systems. The nature of the unpredictability of the incidence pattern of measles is characterized by a subtle interaction between a chaotic nonlinear determinism and stochastic fluctuations. The sensitive dependence on initial conditions, characteristic for chaos, is not limited to chaotic attractors but can also occur on so-called chaotic transients. The dynamics of recurrent outbreaks of measles in New York City turns out to be close to a so-called boundary crisis which converts a stable chaotic attractor to metastable chaotic transients leading finally to a period-3 attractor. However, the demographic noise destabilizes the periodic orbit as well, and creates a situation of intermittent jumps between episodic periodicities and longer chaotic transients. A simple autoregressive model is used to achieve a plausible geometric understanding of the noise-induced intermittency switching between episodic periodicity and transient chaos.