Formalism for discrete multidimensional dynamic systems

The purpose of this paper is to suggest a formalism given by an equation suitable for simulating discrete systems with space-time variation in addition to other change variables. With such formalism, multidimensional dynamical models of discrete complex systems, such as the social systems and ecosystems, can be built.

This formalism is named as discrete multidimensional dynamic system (DMDS). The DMDS provides a way to consider the variation of the density of a state variable with regard to the variables of the change space as a function of multidimensional rates. Multidimensional rates describe this evolution as a consequence of the relation of each multidimensional-point with a given set of other points of the change space. This relation contains the accessibility domains (sets of space points with which each space point is related).

This equation is compared with both the reaction-diffusion equation written in its finite difference form and the cellular-automata model, demonstrating its compatibility with them and an increase in generality, widening the scope of application. The steps to construct models of systems with multidimensional variation based on the equation that defines the DMDS are specified and tested.

Through the DMDS and a well-stated methodology, an application case is provided in order to describe the multidimensional demographic dynamics of an urban system. In this case, the numerical evolution of the population density by districts and cohorts is determined by the DMDS based on some hypothesis about functions of population diffusion between the different districts of the system.

The scope of application of the space-time dynamic system (STDS), given by the authors in a previous work, has been extended to discrete and multidimensional systems. STDS model produces better results than the reaction-diffusion model in validation.