Introduction - Noise-induced Phenomena in the Environmental Sciences - page 3

The purpose of this topic is to provide conceptual and mathematical tools that
allow environmental scientists to familiarize themselves with the notion of noise-
induced phenomena and with the idea that environmental noise (e.g., random climate
fluctuations) is not necessarily a mere source of disturbance in environmental systems.
The existence of a more fundamental role of noise should also be recognized in that it
could have a crucial role in the way these systems respond to changes in environmental
variability.
An example of the relevance of these constructive effects of noise can be found
in the study of ecosystems' response to climate variability. Research in the field of
ecosystem and population ecology has been investigating the effect of climate and
land-use changes on ecosystem structure and function. In most cases the focus has
been on how ecosystems respond to changes in the mean values of environmental
parameters (e.g., mean annual precipitation or temperature) whereas the impact of
changes in the variance has seldom been studied. However, recent climate-change
studies indicate that, in addition to trends in the mean values of climate variables,
interannual variability is also increasing ( Katz and Brown , 1992 ; Easterling et al. ,
2000a , 2000b ). It becomes therefore important to understand how this increase in the
variance of environmental parameters will affect the dynamics of natural systems.

1.2 Time scales and noise models
The dynamics investigated in this topic include four major components, namely,
(i) a dynamical system, (ii) the external environment, (iii) a stochastic forcing, and
(iv) some possible feedbacks between the state of the system and its environment or
stochastic drivers. The first element is the deterministic dynamical system of interest.
This system is a conceptually separate “entity” from a much more complex dynamical
system, called the environment . The dynamical system generally involves a limited
number of physical variables. We model it with a minimalist approach, which captures
only the fundamental features of the dynamics. The physical variables representing
the state of the system (e.g., plant biomass, soil moisture, soil thickness) are usually
referred to as state variables . In this topic we focus on systems that we can investigate
by considering only one state variable, though we also consider the case of noise-
induced phenomena that can emerge only in multivariate systems. The focus on
univariate dynamics is motivated by their conceptual simplicity, the possibility of
investigating them with analytical mathematical models, and the fact that their study
allows us to show how noise-induced behaviors may emerge even without invoking
complex interactions among a number of environmental variables. We express these
univariate dynamics by using a first-order differential model,

d d t =

f ( φ ) , (1.1)