STABILITY SWITCHES IN DISCRETE FOOD-CHAIN PROBLEMS

Abstract

In an already classical paper, May [1976] pointed out that simple nonlinear models in ecology can possess complicated dynamics. A long debate concerning the possible existence of chaotic ecologies has followed the appearance of this paper. Morris [1990] stressed that several problems exist when making conclusions regarding chaotic or nonchaotic dynamics as models are fitted to ecological data. Takens' theorem [1981] should, in principle, provide tools for solving some of these problems, but in reality essential assertions are most often violated. First, limits regarding the length of the used time-series must be assumed. Second, the theorem can be applied for evaluating positive Lyapunov exponents only, and is hence not applicable for distinguishing nonchaotic dynamics from chaotic. We use a deterministic food-chain model that possesses a wide range of different dynamical patterns to demonstrate the existence of cases that are misclassified with respect to chaotic or nonchaotic motion as models are fitted to data in the various dynamical regimes. Our results are valid regardless of what finite data size is assumed. The results are best understood for high-periodic cases in the noise-free limit. If the relevant phase-space is not sufficiently well-populated with data in the vicinity of the periodic orbit, then sufficient complexity of the models are not supported by standard model selection criteria like AIC or GCV. Thus, the fitted models only in the best cases contain information concerning both the location of the periodic orbit and the eigenvalues of the Jacobians evaluated along the periodic orbit. If the space is moderately well populated with data, then the data is often described by an unstable periodic orbit of the fitted model. The attractor that was to be described by the fitted model was destabilized and is now a repeller. We call this phenomenon stability switches. It reminds about the noise-induced phenomenon reported by Rand and Wilson [1992], but we point out that the problem reported here is caused by the fitting procedure itself, not by the added noise. The repeller that has been created by the model-fitting procedure can be located in the basins of attraction of some fixed point, the infinity, or some periodic or chaotic attractors of the fitted models. The situation seems similar when periodic attractors are not well enough populated with data for describing their location and when chaotic attractors are to be described by data. Similar stability switches occur and the dynamics of models fitted to data may differ or coincide with the dynamics of the attractor that generated the data in an unpredictable manner.