Threshold Phenomena versus Killer Clusters in Bimodal Competion for Standards

Abstract

Given an individually used standard on a territory we study the conditions for total spreading of a new emergent better fitted competing standard. The associated dynamics is monitored by local competing updating which occurs at random among a few individuals. The analysis is done using a cellular automata model within a two-dimensional lattice with synchronous random walk. Starting from an initial density of the new standard the associated density evolution is studied using groups of four individuals each. For each local update the outcome goes along the local majority within the group. However in case of a tie, the better fitted standard wins. Updates may happen at each diffusive step according to some fixed probability. For every value of that probability a critical threshold, in the initial new emergent standard density, is found to determine its total either disappearance or spreading making the process a threshold phenomenon. Nevertheless it turns out that even at a zero density measure of the new emergent standard there exits some peculiar killer clusters of it which have a non zero probability to grow and invade the whole system. At the same time the occurrence of such killer clusters is a very rare event and is a function of the system size. Application of the model to a large spectrum of competing dynamics is discussed. It includes the smoker-non smoker fight, opinion forming, diffusion of innovation, species evolution, epidemic spreading and cancer growth.