S. El Yacoubi, B. Chopard, and S. Bandini (Eds.): ACRI 2006, LNCS 4173, pp. 657 – 666, 2006. © Springer-Verlag Berlin Heidelberg 2006 In Search of Cellular Automata Reproducing Chaotic Dynamics Described by Logistic Formula Witold Dzwinel AGH University of Science and Technology, Institute of Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland dzwinel@agh.edu.pl Abstract. Two-dimensional cellular automata (CA) systems are widely used for modeling spatio-temporal dynamics of evolving populations. Conversely, the logistic equation is a 1-D model describing non-spatial evolution. Both clustering of individuals on CA lattice and inherent limitations of the CA model inhibit the chaotic fluctuations of average population density. We show that crude mean-field approximation of stochastic 2-D CA, assuming untied, random “collisions” of individuals, reproduces full logistic map (2≤r≤4) only if infinite neighborhood is considered. Whereas, the value of the growth rate parameter r obtained for this CA system with the Moore neighborhood is at most equal to 3.6. It is interesting that this type of behavior can be observed for diversity of microscopic CA rules. We show that chaotic dynamics of population density predicted by the logistic formula is restrained by the motion ability of individuals, dispersal and competitions radiuses and is rather exception than the rule in evolution of this type of populations. We conclude that the logistic equation is very unreliable in predicting a variety of evolution scenarios generated by the spatially extended systems. Keywords: spatially extended systems, cellular automata, logistic equation, chaotic dynamics. 1 Introduction The individuals from evolving populations are usually distributed in space. This generates at least two potentially important consequences for their dynamics. First, individuals interact more frequently with neighbors than with more distant individuals creating rich collection of spatial structures. Second, individuals at different locations may experience different environmental interactions influencing their birth and death rates. Therefore, creation of spatial patterns and heterogeneity of the environment have been suggested in [1] as an explanation of substantial differences between behavior generated by simple ecological models e.g., by the logistic equation, and real evolution of many organisms. Of course, there are other reasons why the logistic equation may fail to give an adequate description of population growth [2]. The per capita effect of density on population growth may not increase linearly with density or there may be a time delay