Geoscience Reference
In-Depth Information
parts, and non-linear because such small changes can lead to large changes in other
parts (Phillips, 2003). The population model is a particularly interesting example,
because it is only sensitive to changes in
over a small range of values. So long as
its value remains between one and three, a small change in
α
α
has a correspondingly
minor impact on
X
. Similarly, any change in
, when it is less than one or greater
than four, has no impact in the sense that
X
drops to zero or becomes infi nite
regardless of the size of change in
α
α
. In contrast to those ranges, the system is very
sensitive to changes in
α
when it takes a value between three and four. Any small
shift in the value of
in this range results in large shifts in
X
among multiple attrac-
tors, which are equivalent to population boom-bust cycles in real-world populations
(fi gure 5.1). The sudden shifts that occur due to this sensitivity are termed bifurca-
tions or catastrophic changes (May, 1976; Feigenbaum, 1980; Brown, 1995). The
term butterfl y effect, which metaphorically suggests that the fl apping of a butterfl y's
wings may cause severe weather elsewhere (Lorenz, 1973), also describes sensitivity,
particularly in the initial values of a model. As discussed below, the potential for
sensitivity to initial conditions raises fundamental questions about our ability to
model complex systems and predict their behaviour, as well as, more broadly, about
the nature of equilibrium and change.
Many mathematical systems have stable attractors, but those describing deter-
ministically complex systems can also have strange attractors, or sets of values
towards which the system tends, but never quite reaches. In our population model,
the population size
X
seemingly becomes completely random when
α
3.8 (fi gure
5.1, inset). In terms of deterministic complexity, however, this system is not truly
chaotic or unknowable because we can model it with a single equation and know
exactly which value of
α
=
generates the seemingly random values of
X
. Moreover,
the values of
X
will generally cluster around a certain set of values that defi ne the
strange attractor. A system that exhibits these two characteristics - being modelled
with equations and having attractors - is termed deterministically chaotic as opposed
to truly chaotic (Leiber, 1999).
One kind of strange attractor that has garnered much attention is the fractal.
This term refers to a pattern that remains unchanged over the spatial or temporal
scale of observation. Trees and river systems, for example, are fractal in the sense
that they appear to have a branching structure at scales ranging from the very small,
such as veins in leaves or the smallest stream branches, to the very large, such as
the branching structure of the entire tree or river system (Pecknold et al., 1997). So
too is the general branching structure of the population system in fi gure 5.1, which
is mirrored at the small scale in the fi gure inset. Systems that exhibit fractal patterns
are interesting because the appearance of similar patterns at different scales implies
that similar underlying processes may exist. Thus, understanding the processes
operating at one scale may lead to understanding of the processes operating at
others. As examined below, however, actually using the discovery of fractal patterns
at one scale to understand or predict the behaviour of a system at other scale is
fraught with diffi culty.
α
Aggregate complexity
Aggregate complexity examines many of the same features of systems as algorithmic
and deterministic complexity, such as feedback or non-linearity. However, aggre-
gate complexity is more concerned with how systems are created by the simple and
