In Search of Cellular Automata Reproducing Chaotic Dynamics Described by Logistic Formula

摘要

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.