Can noise induce chaos?

Abstract

Can noise induce chaos? -Oikos 102: 329-339.An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that ''chaos'' should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

The limits to prediction in ecological systems

“…Chaotic dynamics have been identified in some experimental manipulations of populations (Constantino et al 1997), but most evidence of chaos has come from models of ecological processes (e.g., Maquet et al 2007, Upadhyay 2009, Gassmann et al 2005. The presence of chaos in models of ecological systems, however, does not necessarily mean that the ecological systems themselves are, in reality, chaotic (Rai 2009), and chaos is likely to be difficult to definitely detect in ecological data (e.g., Dennis et al 2003, Ellner and Turchin 2005, Scheuring and Domokos 2007, Upadhyay 2009). But if chaos is common in ecological systems, then ecologists have two problems in prediction: First, identifying the precise form of the dynamical process governing the evolution of a given ecological system and, second, establishing with arbitrary precision the system state at some initial time.…”

Dynamics of resource production and utilisation in two-component biosphere-human and terrestrial carbon systems

“…There is an ongoing debate about whether external noise can induce chaos in ecological systems with otherwise stable equilibrium points. Dennis et al (2003) argued that this is not possible, while Ellner and Turchin (2005) argued that the boundary between deterministic and noise-induced chaos is more subtle, exhibiting regions of "noisy stability", "noisy chaos", "quasi-chaos" and "noise-domination" depending on the noise level and the dominant Lyapunov exponent (the real part of the fastestgrowing eigenvalue). The debate appears to depend on the precise definition of "chaos".…”

Computational ecology as an emerging science

“…Other and possibly more common sources are poor accuracy of ecological data and their transient nature. Noise that is inevitably present in ecosystems can significantly change the properties of an ecological model, both in the wellmixed 'non-spatial' case [88,89] and in the spatially explicit case [90,91], and this fundamental uncertainty affects the accuracy of ecological data.…”