Killer Geometries in Competing Species Dynamics
  We discuss a cellular automata model to study the competition between an emergent better fitted species against an existing majority species. The model implement local fights among small group of individual and a synchronous random walk on a 2D lattice. The faith of the system, i.e. the spreading or disappearance of the species is determined by their initial density and fight frequency. The initial density of the emergent species has to be higher than a critical threshold for total spreading but this value depends in a non-trivial way of the fight frequency. Below the threshold any better adapted species disappears showing that a qualitative advantage is not enough for a minority to win. No strategy is involved but spatial organization turns out to be crucial. For instance at minority densities of zero measure some very rare local geometries which occur by chance are found to be killer geometries. Once set they lead with high probability to the total destruction of the preexisting majority species. The occurrence rate of these killer geometries is function of the system size. This model may apply to a large spectrum of competing groups like smoker-non smoker, opinion forming, diffusion of innovation setting of industrial standards, species evolution, epidemic spreading and cancer growth. 
Serge Galam, Bastien Chopard and Michel Droz