Non-Linear Dynamics, Self-Organization and Cellular Automata Models - Environmental Modelling: Finding Simplicity in Complexity - page 51

of thermodynamics defines the current against which all
living things successfully swim while they are alive.
All self-organizing systems, living or nonliving, are
alike in being thermodynamically open. This is not a new
insight for many earth scientists: for example, Leopold
et al . (1964: 267) note that dynamic equilibrium of fluvial
systems refers 'to an open system in which there is a con-
tinuous inflow of materials, but within which the form or
character of the system remains unchanged. A biological
cell is such a system. The river channel at a particular
location over a period of time similarly receives an inflow
of sediment and water, which is discharged downstream,
while the channel itself remains essentially unchanged.'
Thus each non-living dissipative system must, as must
any living system, maintain a flow of matter and energy
through itself in order to retain its integrity. Without this
maintenance, it becomes a thermodynamically closed sys-
tem, with bland equilibration as its only future. While
our intuition correctly distinguishes between closed and
dissipative systems when the dissipative system is living,
it does not reliably distinguish between closed and dissi-
pative systems when the dissipative system is nonliving.
Nonliving dissipative systems that organize themselves
are therefore a surprise to us. 11

4.3.1 Self-organizationonacellulargrid

Cellular automaton models were first proposed by math-
ematician John von Neumann (1966), in an attempt to
learn more about a biological problem: the nature of self-
replication. Such models discretize continuous space 12
into a series of cells. These cells are usually part of a
regular square or rectangular grid, but can also be, for
example, hexagonal. This spatial discretization 13 is neces-
sary if the model is to be computationally tractable. The
rules and relationships that comprise the model are then
applied at the scale of individual cells.
These rules and relationships may be viewed as positive
and negative feedbacks, and thus each cell may be seen
as an 'automaton' with its behaviour controlled by the
positive and negative feedbacks to which it is subject.
Interactions are usually (but not always: see e.g. Li, 1997)
between adjacent or nearby cells: thus the model's inter-
actions are all 'local'. If the CA model then self-organizes
and gives rise to larger-scale responses, these will mani-
fest as patterns on the cellular grid (e.g. Mahnke, 1999;
Wooton, 2001; Wolfram, 2002).
Bar-Yam (1997: 490) suggests that 'The idea of a
cellular automaton is to think about simulating the space
rather than the objects that are in it', and (p. 139)
that 'Cellular automata are an alternative to differential
equations for the modelling of physical systems' (see also
Tucker and Bradley, 2010). While the same real-world
system may be described either by a CA model, or by a
more conventional approach (e.g. a fluid dynamics model
for a fluvial system), it appears to be the case that the CA
approach usually brings out different characteristics of
the system when compared with the more conventional
representation. 14

4.2.6 Recentwork: complexsystemsscience

Recent work has tended to group the scientific study of
many kinds of self-organizing system under the blanket
heading of 'complex systems science' (see for example
Bar-Yam, 1997). Within complexity science, areas of
particular interest to the environmental modeller are
agent-based modelling (see Chapter 18) and cellular
automaton (CA) modelling. The latter is the focus of
the remainder of this chapter.

4.3.2 KindsofCAmodels
Watts (1999: 181 et seq .) gives a good overview of the his-
tory and development of CA models. They are now widely
used in the study of self-organization in a wide range of
fields (for example, Bar-Yam, 1997). Wolfram (2002)

4.3 Cellular automaton models

If a real system is presumed to manifest self-organization
and emergence, then one way to study this is by means of
a model which is also capable of manifesting these phe-
nomena. The main tool for modelling self-organization
in spatial systems is the 'cellular automaton' (CA) model.

12 Just as almost all models that operate in a true time dimension
discretize continuous time into distinct 'timesteps'.
13 Discretization appears to be a fundamental operation for humans
attempting to make sense of the world: see e.g. Chapter 1 of Bohm
(1980).
14 This is related to the concepts of 'trivial duality' and 'nontrivial
duality' in physics (Greene, 1999: 297 et seq .).Ifeachoftwo
different descriptions of the same system tell us something that the
other does not, this is 'nontrivial duality'.

11 Yet from still another point of view, they certainly shouldn't be.
For if nonliving systems do not organize themselves, how would
they get organized in the first place? Unless we invoke an organizer
(i.e. teleology), then from this perspective 'if self-organization did
not exist, it would be necessary to invent it.'