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without any boundary conditions. This is clearer in the deterministic models, but it is also apparent
in the probabilistic where the thresholds are greater than ρ
= 0.5. In this example, two waves pulsate
through the structure where the age threshold is set at 15 years. In terms of the long-term age trajec-
tory of the system measured as the average age of cells, the age profile increases at an increasing
rate until the first wave of redevelopment kicks in and then it builds up again until the second wave.
The relative drop in average age associated with these pulses is gradually ironed out as the waves of
redevelopment become the dominant feature of the development.
These waves are in fact
emergent phenomena
in that they are a result of delayed local actions.
In a sense, the growth around the central seed site is one of increasing waves which result from
local actions acting immediately on the geometry, and the waves of redevelopment are essentially
these same waves which are delayed through the age threshold. We refer to waves of change in real
cities, but as in these examples, although they depend upon the interactions associated with the
local dynamics, they cannot be predicted from just a knowledge of these local rules. In this sense,
they are formed in the same way a wave of laughter or excitement ripples through an audience
which depends on how one's neighbours are reacting, not upon any macro-property of the system
at large. It is thus necessary to observe the system in motion before such emergent phenomena can
be explained and understood.
2.8 MORE COMPLICATED GROWTH REGIMES
The diffusion model that is intrinsic to all CA can be constrained in various ways. In Figure 2.6,
we show how this can be done by masking the neighbourhood in terms of the cells within the
Moore neighbourhood that are eligible for development. Orientation and directions of growth
can be established in this manner, while disconnected automata can also be fashioned if the
neighbourhood is enlarged and elements of it which are not connected to the seed site are allowed
to be developed. However, the simplest and probably the archetypal growth model which uses a
generalisation of CA is the diffusion-limited aggregation (DLA) model. This model begins from
a seed site and activates a set of agents in the space around it who wander in their neighbourhood
making a random move to one of their cells at each time period. This process continues until an
agent reaches a cell where there is a developed cell in its neighbourhood. Then the agent makes a
development decision and essentially turns the cell on and becomes settled. In this sense, this is
like a wandering migrant who finds a site already developed in its neighbourhood and decides to
pitch camp
so to speak. Of course, the first site is the seed site and thus the development occurs
from that seed outwards.
As the agent must be in a cell that is physically connected to the site that has been developed, the
entire development is connected, although this can be relaxed in terms of letting the agent develop
a site within some distance of an already developed site. It turns out that the growing cluster in fact
develops with tree-like branches diffusing out from the seed. It does not generate an amorphous
mass but a highly structured form that is entirely due to the fact that once a site is settled, its neigh-
bours become very slightly more likely to be developed for the agent is more likely to discover them
as the cluster is growing outwards and the agent is wandering across the entire space. The kinds
of structure that can be developed under different variants of this process are shown in Figure 2.7,
and it is immediately clear that there are very close parallels with the way mono-centric cities
grow around their central business districts. In fact, the DLA model is the example
par excellence
of fractal growth which has been exploited in many urban applications (Batty and Longley, 1994;
Batty, 2005, 2013).
The extension of these CA to many seeds is straightforward. The immediate interest is in
the kinds of morphologies that are generated when the different patterns around each of the
seeds begin to overlap. If the automata are based on the simplest morphologies of diffusion as
we noted earlier, and if the different seeds generate patterns that are not synchronised spatially,
then interesting overlapping patterns in the form of waves can occur. It is difficult, however,
