BioND — Dynamics of Biological Networks

Cooperation and the adverse reaction to social ties

Andreas Kämpgen
Bachelor thesis in physics, Julius Maximilian Universität Würzburg, 2009.


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In this thesis I have investigated the dynamics of a Prisoner's Dilemma game on an adaptive network, which has been recently proposed by Segbroeck et al. In this game, the authors equip individuals with the capacity to treat unfavorable interactions di fferently. Additionally they introduce diversity of individual behavioral classes and investigate how this diversity in individual responses to adverse social ties influences the evolution of cooperation.

In a previous publication, Pacheco et al. have studied the game by using the active linking approximation. This approximation assumes topological dynamics to proceed faster than the evolution of strategies, so that strategy updates occurs under stationary, but rewired network topology. In this regime, the evolution of strategies can be mapped onto a different game in a well mixed population, where everybody interacts equally likely with everybody else. By this means van Segbroeck et al. show that diversity in individual responses promotes the evolution of cooperation.

In order to gain more detailed insights in the dynamical interplay of state and topology I analyzed the system with a different approximation scheme: the moment closure approximation. In this approximation the system is described by a system of ordinary differential equations (ODEs), which capture the dynamics of the moments of the system.

The moment equations are truncated on a certain level to obtain a closed model. My investigations revealed that moment closure approximation provides a convenient tool for the investigation of the model under consideration. By means of numerical integration we verified the emergence of long-term cooperation in consistence with the results of van Segbroeck et al. Moreover, we showed that investigations using moment closure approximation can give insights that are difficult or even impossible to find by means of simulations. By applying the Netwon method, we found unstable steady states with coexistent populations of cooperators and defectors. The existence of these states is remarkable as the Prisoner's Dilemma in a well-mixed population cannot support stationary coexistence, even in unstable states.