Spatial Effects in Stochastic Microscopic Models - Case Study and Analysis

In this paper we are going to present techniques to investigate the theoretical background of so called microscopic models (i.e. models consisting of a large number of individual but yet cooperating actors). We will lay special emphasis on the analysis of so called aggregated numbers, hereby speaking of usually scalar, summarising variables, dependent on all actors simultaneously, which are typically some kind of sums or statistics. We are going to analyse the behaviour of those quantities in case of a very large number, respectively in the limit case, an infinite number of individual actors. We will especially focus on the influence of spatial relationships between the actors on the aggregated number. Stochastic methods are going to play the key role in this theory. Furthermore we will apply the results of the theoretical research on three different microscopic models, each of them chosen to particularly point onto an important observation. The first model, a simplified epidemics model, is going validate the theory and demonstrates how to use it. The second model, based on famous Game of Life by John H. Conway, will reveal the limits of the method and finally, the third model, an extension of the second one, will show the benefits and applicability of the analytically derived theory.