Pattni, K. (2017). Evolution in finite structured populations with group interactions. (Unpublished Doctoral thesis, City, University of London)
Abstract
The study of an evolutionary process has traditionally considered a population with a homogeneous structure where each pair of individuals is equally likely to interact with one another. Later studies have considered heterogeneous structures implemented using evolutionary graph theory, and other studies have considered group interactions of fixed size. This work builds upon these later studies by implementing a set of evolutionary dynamics that can be used to study more complex evolutionary processes consisting of a population with a heterogeneous structure where individuals interact in groups of varying size.
This research begins by analytically studying simple evolutionary processes using a set of standard evolutionary dynamics. Results are derived that identify the structures for which an evolutionary process is identical to a Moran process, which has a homogeneous population structure, for each of the evolutionary dynamics. These results form a basis for the work that follows by providing a better understanding of evolutionary dynamics.
Before considering more complex evolutionary processes, a class of multiplayer games called social dilemmas are defined for variable group sizes. The two main types of social dilemmas are identified, namely public goods dilemmas and commons dilemmas, and examples of each type of dilemma are given whose characteristics are visually illustrated.
More complex evolutionary processes are then studied based on the framework of BroomRychtář that provides the mathematical tools to model group interactions in mobile individuals. First, the evolutionary dynamics that can be used within this framework are developed. The updated version of the framework is then used to demonstrate how it can applied to study various kinds of behaviour in an evolutionary setting.
The first application is the territorial raider model. It considers territorial behaviour where each individual has their own territory that overlaps with those of other individuals. Interactions take place between groups of individuals when they meet in the overlapping parts of their territories. Two kinds of social dilemmas are studied in this model: a multiplayer hawkdove game and a multiplayer public goods game. It is shown that the temperature, which measures how often an individual is likely to be replaced, plays an important role in determining the success of a given strategy.
A generalized version of the territorial raider model is also considered where subpopulations rather than individuals share the same territory. A multiplayer public goods game is used to study the evolution of cooperation, which is a suboptimal strategy at the individual level but an optimal strategy at the group level. The structure and dynamics are shown to be critical in the evolution of cooperation where an extension of the temperature, called the subpopulation temperature, dictates the relative success of cooperators.
Finally, a model where individual move base upon their previous interactions is considered called the Markov movement model. A multiplayer public goods game is used to study the evolution of cooperation. It is shown that cooperators can benefit by staying with one another provided that there is a movement cost that slows down their competitors, the defectors. In this case, the dynamics play a less critical role in the evolution of cooperation.
Publication Type:  Thesis (Doctoral) 

Subjects:  Q Science > QA Mathematics 
Departments:  School of Mathematics, Computer Science & Engineering > Mathematics Doctoral Theses Doctoral Theses > School of Mathematics, Computer Science and Engineering 
URI:  http://openaccess.city.ac.uk/id/eprint/18246 

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