Mathematical Approach to Nonlinear-Non-equilibrium Phenomena - Understanding of Transient Spatio-temporal Patterns-

在各种各样的非线性现象中，我们研究了建模、分析以及数学和数值方法的发展，以了解瞬变过程中的非线性-非平衡现象。(1)利用数学模型研究趋化性细菌中菌落模式的多样性。(2)利用反应扩散假说和载体假说下的两种反应-扩散模型来理解树叶脉络形成的机制。(3)为了理解实验中观察到的合并和分离过程，提出并模拟了描述黏菌的数学模型。这是细胞智能理论研究的第一步：(4)作为瞬变过程中出现的问题，描述非均匀介质中生物个体空间弥散的概率模型的提出和分析，以及个体间相互作用对空间弥散速度的影响分析。(5)反应扩散系统中前沿和斑点相互作用的研究(6)用奇异极限分析研究交叉扩散系统中空间分离模式的动力学。

Among diverse nonlinear phenomena, we have studied modeling, analysis and the development of mathematical and complementarily numerical methods in order to understand nonlinear-non-equilibrium phenomena in the transient process.(1)Investigation of diversity of colonial patterns in chemotactic bacteria by using mathematical models.(2)Understanding of mechanism of venation formation of leafs by using two reaction-diffusion models under reaction-diffusion hypothesis and carrier hypothesis.(3)Proposal and simulation of mathematical models describing slime mold in order to understand merge and separation processes observed in experiments. This is the first step of theoretical study of cell-intelligence(4)As a problem arising in transient processes, proposal and analysis of probabilistic models describing spatial dispersion of biological individuals in heterogeneous medium, and analysis of influence of inter-specific interaction between individuals on velocity in spatial dispersion.(5)Study of interaction of fronts and spots arising in reaction-diffusion systems(6)Investigation of dynamics of spatially segregated patterns in cross-diffusion systems by singular limit analysis.