Replicator-Mutator Dynamics On Evolutionary Networks

Evolution has always been the hot topic in natural and social fields.Replicator–mutator dynamics is a mathematical framework of the evolution ofrelated individuals and their behaviors in the population. All the individualscontact with each other in the classic circumstances. However, these individu-als and their interactions are described by networks and the evolution of theirbehaviors is studied in a structured population, which makes the replicator–mutator dynamics more extensive, various and complex.Evolutionary behavior networks is the combination of the replicator-mutator dynamics and the general networks, and also the key model to study thedynamics on the networks. With the prosperity of evolutionary game theory—another important application field of replicator–mutator dynamics, the net-worked evolutionary game has been an effective method to study the dynamics.The dissertation surveys a wide range of related works and current situationof replicator–mutator dynamics on evolutionary networks, and introduces thelatest works of the evolutionary behavior networks and networked evolutionarygame theory. Our investigations focus on the roles of small-world effect, het-erogeneity of degrees and degree-mixing pattern to the behavioral diversity ofevolutionary behavior networks. We also study the evolutionary cooperation ofsnowdrift game with variable memory lengths on complex networks.The main content and contributions of this dissertation are summarized asfollows: 1. We study the effect of the small-world and the heterogeneity of degreeson evolutionary behavior networks. We find that, with the small-worldeffect the collective behaviors are prone to the greatly higher diversity onhomogeneous networks while to no obvious movement on heterogeneousnetworks. And the heterogeneity of degrees promotes the behavioral di-versity a little on homogeneous networks but inhibit it on heterogeneousnetworks. The explanations of the mechanism of these differences areprovided.2. We investigate the degree-mixing pattern's effect to the diversity on evo-lutionary behavior networks and observe that while a scale-free networkbecomes more degree-mixed assortative, its hub vertices will be clusteredmore closely with each other, which therefore inhibits the evolution ofbehavioral states. However, there're no corresponding conclusions on dis-assortative scale-free networks. The explanations of our conclusions aregiven by the clustering of hubs with mutual rewards.3. Prisoner's delimma game and snowdrift game are most widely acceptedmodels of evolutionary game theory on complex networks. We intro-duce a tunable parameter of memory length and present a new memory-based snowdrift game, where individuals have variable individual memorylengths, to investigate the individuals'dynamic behaviors and cooperationstability on heterogeneous scale-free networks. We find that, when thecost to benefit ratio r < 0.5 (r > 0.5), with the increase (decrease) of it,a scale-free network having long-memory-length hubs (small degree ver-tices) yields the higher frequency of cooperation and of pure cooperation,and finally lead to better cooperation stability to enhance the strategy ofcooperation.