京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19476号 / 理博第4136号 / 新制||理||1595(附属図書館) / 32512 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 石田 憲二, 教授 高橋 義朗 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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. / Ph. D.

Nesta tese estudamos os efeitos da desordem espacial e temporal na transição de fase entre a sobrevivência e extinção de populações biológicas. Na primeira parte estudamos um modelo epidemiológico com quatro estados. Apesar deste modelo não conter desordem, concluímos que seu comportamento crítico é o mesmo do processo de contato com desordem (espacial) quenched. Na segunda parte estudamos o movimento Browniano fracionário refletido, onde vimos que a combinação dos efeitos do ruído com correlações de longo alcance e a parede refletora cria uma singularidade em lei de potência na densidade de probabilidade da posição do caminhante. Por fim, estudamos a equação logística com desordem temporal através do mapeamento no movimento Browniano fracionário refletido. Neste último estudo vimos como as correlações de longo alcance mudam o comportamento crítico deste sistema. / We have studied the effects of spatial and temporal disorder at the phase transition between survival and extinction of biological populations. In the first part we studied a four states biological population model. Despite having no disorder, we have seen that its critical behavior is the same of the contact process with (spatial) quenched disorder. In the second part, we studied the reflected fractional Brownian motion, where the interplay between the correlated noise and the reflecting wall results in a power-law singularity in the probability density of the position of the walker. Finally, we deduced the critical properties of the logistic equation with temporal disorder by mapping it onto the reflected fractional Brownian motion. This mapping allow us to understand how long-range correlations change the critical behavior of this system.