A Model for Urban Growth Processes with Continuum State Cellular Automata and Related Differential Equations

A new kind of cellular automaton (CA) for the study of the dynamics of urban systems is proposed. The state of a cell is not described using a finite set, but by means of continuum variables. The time evolution is given by Poisson distributed stochastic jumps affecting the dynamical variables, with intensities depending on the configuration of the system in a suitable set of neighbourhoods. The intensities of the Poisson processes are given in term of a set of potentials evaluated applying fuzzy logic rules. Even if the CA automata describes in detail the micro-dynamics of the urban system, the use of a continuum state space enables one to prove that it also verifies a system of differential equations for the time evolution, and thus to study the CA and the urban system from a continuum dynamical systems theory perspective. This makes it possible, in particular, to look systematically for bifurcations and phase transitions. This modelling ideas can be generalized to other complex systems; this permits to connect simplicity of CA approach to differential equations description