Coordination Game on Stepping Stone Model

By João da Gama Batista (09.10.2013)

**Spacial distribution and frequency-dependent dynamics play a very significant role in evolution. An interesting situation is that of a population consisting of two different species where individuals benefit from aligning their actions, i.e., behaving similarly. This so-called coordination game model is thoroughly analyzed in this thesis.**

There are numerous applications of such a model in a wide range of scales. At a macroscopic level, human societies provide some notably interesting examples regarding critical mass phenomena observed in the markets. At a microscopic level, although specific and clear examples of coordination game situations are still lacking in the literature, some experiments have shown cases of microbia clustering into two discrete states by random switching mechanisms. The fact that there are two different strategies clustering into different subpopulations indicates that it may be possible to describe this biological situation as a coordination game with non-homogeneous spacial distribution and migration.

Thus, as mentioned above, a relevant aspect concerning a population undergoing an evolutionary process is the spacial distribution of its individuals. In nature, individuals are usually distributed rather discontinuously, giving rise to numerous subpopulations between which migration is often observed. Therefore, it would be of interest to study the changes induced by this spacial structure in a particular system.

In the special case of a coordination game, which is the focus of this thesis, the average extinction time is one of the most significant quantities of interest, provided the observed bistability at the population level in such a strategic game.

When the general multiple-island case is considered, the dependence of the average extinction time on the migration rate gives rise to a crossover between two distinct regimes, one for high and another for low migration rates. While high migration rates correspond to the well-mixed scenario, where spacial structure naturally has no significant impact on the global evolution of the system, low migration rates represent situations in which spacial structure and migration strongly influence the competition processes between individuals as well as the average extinction time, mainly because it promotes spacial heterogeneity.

The aspects mentioned above are consequently analyzed, theoretically and numerically, for the twoisland case, which is found to be representative of the general multipleisland situation.

**2006-2009**- B.Sc. in Technological Physics Engineering at the Instituto Superior Técnico, Universidade Técnica de Lisboa
**2009-2011**- M.Sc. in Theoretical and Mathematical Physics at the Ludwig-Maximilians-Universität and Technische Universität München
**since 2012**- Ph.D. in Applied Mathematics at the Laboratoire de Mathématiques Appliquées aux Systèmes, École Centrale Paris

**2008-2009**- Research scholarship about Cosmology from the Portuguese Government Foundation for Science and Technology
**2011-2012**- Research scholarship about Statistical Physics and Evolutionary Game Theory from the Alexander von Humboldt Foundation
**since 2012**- Ph.D. scholarship about Applied Mathematics from the Portuguese Government Foundation for Science and Technology