Spatio-Temporal Patterns, Correlations, and Disorder in Evolutionary Game Theory
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Evolutionary game theory originated from the application of mathematical game theory to biological studies. Well-known examples in evolutionary game theory are the prisoner's dilemma, predator-prey models, the rock-paper-scissors game, etc. Recently, such well-known models have attracted increased interest in population dynamics to understand the emergence of biodiversity and species coexistence. Meanwhile, it has been realized that techniques from statistical physics can aid us to gain novel insights into this interdisciplinary field. In our research, we mainly employ individual-based Monte Carlo simulations to study emerging spatio-temporal patterns, spatial correlations, and the influence of quenched spatial disorder in rock-paper-scissors systems either with or without conserved total population number. In balanced rock-paper-scissors systems far away from the ``corner'' of configuration space, it is shown that quenched spatial disorder in the reaction rates has only minor effects on the co-evolutionary dynamics. However, in model variants with strongly asymmetric rates (i.e., ``corner'' rock-paper-scissors systems), we find that spatial rate variability can greatly enhance the fitness of both minor species in``corner'' systems, a phenomenon already observed in two-species Lotka-Volterra predator-prey models. Moreover, we numerically study the influence of either pure hopping processes or exchange processes on the emergence of spiral patterns in spatial rock-paper-scissors systems without conservation law (i.e., May-Leonard model). We also observe distinct extinction features for small spatial May-Leonard systems when the mobility rate crosses the critical threshold which separates the active coexistence state from an inactive absorbing state. In addition, through Monte Carlo simulation on a heterogeneous interacting agents model, we investigate the universal scaling properties in financial markets such as the fat-tail distributions in return and trading volume, the volatility clustering, and the long-range correlation in volatility. It is demonstrated that the long-tail feature in trading volume distribution results in the fat-tail distribution of asset return, and furthermore it is shown that the long tail in trading volume distribution is caused by the heterogeneity in traders' sensitivities to market risk.
- Doctoral Dissertations