Non-Linear Dynamics, Self-Organization and Cellular Automata Models - Environmental Modelling: Finding Simplicity in Complexity - page 53

expression of a theory. Thus a CA model may be regarded
as an expression of the notion of self-organization and
emergence in complex systems.
In any developing area of science, it is inevitable that
terminology is somewhat fluid and unsettled. Thus some
workers (for example, Coulthard et al ., 2002 eschew the
phrase 'CA model' altogether, preferring 'cellular model'.
The distinction made appears to be the nature of the local
rules applied to each automaton, with 'cellular model'
used if these rules are relatively complicated, and 'CA
model' used only if the local rules are relatively simple.
Notwithstanding its imprecision, this is a usage that may
or may not become generally accepted in time. Here, the
two expressions are used interchangeably.
However, it is certainly undesirable to focus only on
the methodology (CA or cellular model) to the neglect of
the underlying theory (self-organization and emergence).
To do so is tantamount to describing the operation
of an internal combustion engine without considering
the expanding gases that drive it. Cellular automaton
models, when used to represent some real-world system,
are just tools for reproducing the self-organization and
emergence which is assumed to also manifest in the real-
world system. It is self-organization and emergence that
are the deeper concepts, and which thus best deserve
our attention.

the landscape, the simple rules gives rise to a complex,
fluvially sculpted landscape.
Werner and Hallet (1993) also used the CA approach to
investigate the formation of sorted stone stripes by needle
ice, and found that the formation of stone stripes and
similar features, such as stone polygons and nets, reflects
self-organization resulting from local feedback between
stone concentration and needle ice growth rather than
from an externally applied, large-scale template. Werner
and Fink (1993) similarly simulated the formation of
beach cusps as self-organized features: the CA model here
was based on the interaction of water flow, sediment
transport, and morphological change. Werner (1995)
again used the approach to study the formation of aeolian
dunes, and Werner (1999) reviewed earlier work and
pointed out research directions.
The CA model of Murray and Paola (1994, 1996, 1997)
was developed to investigate the formation of braided
channel patterns. Their model replicated the typical
dynamics of braided rivers, with lateral channel migra-
tion, bar erosion and formation, and channel splitting,
using local transport rules describing sediment transport
between neighbouring cells. Thomas and Nicholas (2002)
also tackled a similar problem.
Simulation of rill networks formed by soil erosion by
water was the goal of the plot-scale RillGrow 1 model
constructed by Favis-Mortlock (1996, 1998b). The model
successfully crated apparently realistic rill networks, and
was also able to reproduce other 'global' responses such as
the relationship between total soil loss and slope gradient.
In addition to focusing on the pattern-forming aspects
of self-organization, there has also been a flurry of
interest in modelling the more statistical aspects of self-
organization such as SOC. Hergarten and Neugebauer
(1998, 1999) investigated the magnitude-frequency dis-
tribution of landslides with a cellular model (see Chapter
16). Landslides were also the focus of a CA modelling
study by Clerici and Perego (2000).
More recently, while Hergarten et al . (2000) and Her-
garten and Neugebauer (2001) have shifted the focus
of their magnitude-frequency CA modelling work to
drainage networks. Modelled and observed drainage
basins and their channel networks were also the focus
of the work of Rodriguez-Iturbe and co-workers (for
example, Rodrıguez-Iturbe and Rinaldo, 1997). This
investigation concentrated on the fractal and multifractal
properties of channel networks. The models developed
by Rodrıguez-Iturbe and co-workers are based on the
continuity equations for flow and sediment transport
which are solved on a two-dimensional, rectangular grid.

4.3.6 Geomorphological applicationsofCA
models
Geomorphological modellers have eagerly embraced self-
organization. A number of studies have made use of CA
models from the late 1980s.

4.3.6.1 Early geomorphological CA modelling
studies

Amongst the first geomorphologically focused CA studies
were those of Anderson and Haff (1988), who observed
self-organization in simulations of aeolian saltation;
Smith (1991), who used a CA model to reproduce
two-dimensional erosional features; and Chase (1992),
who used a cellular model to investigate the evolution
of fluvially eroded landscapes over long periods of time
and at large spatial scales. In this model, rainfall occurs
on the simulated landscape in single increments called
'precipitons'. After being dropped at a random position,
a precipiton runs down a slope and starts eroding and
depositing sediment, depending on the conditions in
the cell. Through time, as numerous precipitons modify