N. Boccara et al., ROUTE TO CHAOS FOR A GLOBAL VARIABLE OF A 2-DIMENSIONAL GAME-OF-LIFE TYPE AUTOMATA NETWORK, Journal of physics. A, mathematical and general, 27(24), 1994, pp. 8039-8047
We consider a two-dimensional cellular automaton whose rule consists o
f two subrules. The first, applied synchronously, is a local rule insp
ired from the 'game of life', with a larger neighbourhood. The second,
applied sequentially, describes the motion of a fraction rn of indivi
duals. Such rules appear to be useful for modelling complex systems in
ecology, such as natural populations of animals, in which the motion
of the individuals is believed to play an important role. If the motio
n is long-range, the density of individuals exhibits a sequence of per
iod-doubling bifurcations and behaves chaotically when m is large enou
gh. If the motion is short-range (i.e. restricted to first neighbours)
, patterns become inhomogeneous. Spatial correlations decay with a fin
ite correlation length xi of the order of root m. We observe the forma
tion of domains, of mean width xi, with a chaotic behaviour of the loc
al density of individuals, but the collective behaviour is stationary
(the global density tends to a fixed value when the lattice size is mu
ch larger than xi x xi).