Non-Linear Dynamics, Self-Organization and Cellular Automata Models - Environmental Modelling: Finding Simplicity in Complexity - page 52

gives a magisterial overview of CA models, which should
be consulted for further details to those presented here.
A simple CA model is the 'Game of Life' devised by
mathematician John Conway (Wolfram, 1982), in which
a limited number of individual cells start out 'alive'. Each
timestep, these cells either 'live' (i.e. persist) or 'die' (i.e.
are removed), depending on the number of surrounding
cells. As simple as this model may appear, when applied
iteratively some very complex patterns can result. Overall,
the grid may evolve to a steady state, to some iterating
pattern, or to an apparently disordered configuration (cf.
the chaotic systems described previously). This is a binary
system (with just two states for each cell): CA models may
have more discrete states, or indeed continuously valued
states (Wolfram, 2002). Note that whereas the rules in
the 'Game of Life' CA model are deterministic, it is not
necessarily the case, particularly in more complicated
CA models, such as the rill growth models described
below. Also, the original configuration of the CA grid
(at the beginning of the simulation) may be reached by
deterministic or stochastic means 15 .
Despite its simplicity, research into the original 'Game
of Life' is ongoing. New strategies for replication are still
being discovered (for example, Aron, 2010). In a slightly
more sophisticated variant, Szab o and Hauert (2002)
found evidence for emergent co-operation and other
social phenomena. A variant of the simple CA model
which is suggested to be better suited to representing
continuum systems such as fluvial flow is the 'lattice-gas
automata' model (Wolfram, 1986; Garcia-Sanchez et al .,
1996; Pilotti and Menduni, 1997; see also Succi, 2001).
A still more recent variant is termed the 'Boltzmann
automata' model (for example, Geier et al ., 2006).

is to use a parallel computer (Bar-Yam, 1997: 488), on
which a number of instruction pipelines can be processed
simultaneously. To implement a CA model on a parallel
computer is not, however, as simple as it might appear in
all cases, 16 and remains an area of active development.

4.3.4 Modellingself-organization: theproblem
of context andboundaries

'If everything in the universe depends on everything else
in a fundamental way, it may be impossible to get close
to a full solution by investigating parts of the problem
in isolation' (Hawking, 1988: 11). As Stephen Hawking
acknowledges, it is increasingly recognized that reduc-
tionism, i.e. breaking apart many systems - including
those of interest to geographers - into smaller, more
tractable units, poses a risk for full understanding. The
reductionist's concentration on the components of a
self-organizing system, away from the context in which
such components interact and give rise to emergent
self-organization, will miss vital points about the way
the system works. Even with a more holistic focus, the
imposition of an artificial boundary between the system's
components will constrain the interactions between com-
ponents in the region of the boundary, with potentially
strong effects on emergent responses of the system. 17
This constraint is notably the case for CA models of
self-organizing systems, where the 'problem of bound-
ary conditions' may be severe. We must set boundaries,
but doing so conceptually breaks some of the model's
connections to the 'outside world' and so can result in
a distorted model (Bar-Yam, 1997: 8). So, in the final
analysis, we must accept that even the best possible model
of a self-organizing system remains incomplete.

4.3.3 Computational constraints toCA
modelling

4.3.5 Terminology: self-organization
andcellular automata

One constraint to the application of CA models is com-
putational. Since the CA model's rules/relationships are
implemented on a per-cell basis rather than upon the
entire grid, any calculations specified by the local rules
may well need to be carried out a very large number of
times during a simulation of any length. Since the major-
ity of present-day computers are fundamentally 'serial',
i.e. processing user instructions on a strictly sequential
basis (or at least programmed in that way), very long run
times can be the result for large grids. An obvious solution

The relationship between model and theory is the sub-
ject of much debate among philosophers of science and
logicians. Harvey, during an extensive discussion of the
model-theory dichotomy (1969: Chapters 10 to 12), sug-
gests (p. 145) that a model may be regarded as a formalized

16 For example, what is the best way of splitting up the cellular
grid between processors if cells must be processed in a random
sequence (cf. Watts, 1999, p205)? What about local interactions
between cells allocated to different processors?
17 'Biologists cannot adopt a reductionist approach when work-
ing with a living organism, or it dies': Jack Cohen, personal
communication, 2000.

15 See www.santafe.edu/

hag/class/class.html for a hierarchical
classification of cellular automata.

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