Spatial synchronization of cellular automata in evolutionary processes simulation tasks

Abstract

Many applied tasks are simulated by difference equations that describe the vector of system states evolution in time. However it is required to take into account the spatial structure of simulated processes or systems in some tasks. In paper the possibility of a spatio-temporal processes simulation by cellular automata is considered. The brief review of two-dimensional cellular automata properties is provided. The principle of the most famous two-dimensional cellular automata “Game of Life” is described. Also the general way to set these automata in an analytical form by Reaction-Diffusion equation is considered. Concrete forms of the Reaction equation and Diffusion equation are constructed and invariant sets for this system are defined. The generalization of analytical cellular automata representation in total is provided. As an example, the model of population development is considered. It utilizes the classic Ferhulst equation, in which the spatial structure is taken into account having form of the cumulative neighbors’ impact on population changes rate. As per using of analytical form of cellular automata, different schemas of system spatio-temporal characteristics control are suggested. These schemas are based on feedback: delayed feedback (that is one that uses previous system states) and predictive feedback (that is one that uses predicted system states). As a result there is managed to synchronize spatial configuration of cellular automata and it can be interpreted as stable population development. Particularly, cellular automata could work in cycle with cycle length set earlier. For cellular automata evolution visualization the algorithms and their computer implementation are developed. Discrepancy function is suggested, due to which it is possible to evaluate the synchronization accuracy. Research results and examples of received configurations are presented.