Modeliranje bioloških in kemijskih sistemov s celičnimi avtomati

Extended abstract. Cellular automata were used as a tool for natural systems modelling. Four properties characterise a cellular automaton. The first is the geometry of the array of cells. The most frequently used geometries for the two-dimensional cellular automata are regular meshes of equilateral shapes which are shown in Figure 1. The second property, close related to the geometry of the array of cells, is the neighbourhood. It comprises cells which determine next state of the cell and are usually close to the corresponding cell. In the two-dimensional rectilinear array two neighbourhoods have been given much attention: von Neumann and Moore, shown in Figure 2. The third property is the number of states per cell, which is usually not higher than 4. This property is highly related to the last property of the cellular automaton, the rule for determining the future state of the cell. In general, it can be described with equation (1) where i and j are determined with the definition of the neighbourhood, ax,y represents the state of the cell, t represents time and l represents the influence of the previous generations of cells. This property is the primary source of variety in the field of cellular automata. We further examined the properties of the best known cellular automaton, the “game of life” invented in 1970 by John Horton Conway and shown in Figure 3. The next aim of our study was to explore the potential usefulness of cellular automata as a tool for modelling of biological and chemical systems. Different growth models were built and the influence of rules and neighbourhoods on the development of cellular automata was examined. In the model of unconstrained growth we examined the influence of different neighbourhoods on growing patterns which is shown in Figure 4. Constrained and competitive growth were also modelled using cellular automata with appropriate rules. On the left side of Figure 5 a development of the model of constrained growth resembling lichens is shown while on the right side, a model of competitive growth is presented. Oscillating chemical reaction Belousov-Zhabotinsky, shown in Figure 6 was also modelled by cellular automata with hexagonal mesh of cells. A rule introduced by J. M. Greenberg and S. P. Hastings and shown in Figure 7 was used. The development of cellular automata modelling Belousov-Zhabotinsky reaction is shown in Figure 8 where typical structures, observed in the chemical reaction can be seen. Different initial conditions which