Mathematical studies on spatial and temporal patterns in reaction-diffusion systems

关于上述研究项目，我们得到了以下结果。(1)对斜梯度系统进行了研究，该系统是一种广义的活化剂-抑制剂系统。当稳态被表征为能量泛函的最小最大值点时，我们证明了稳态是稳定的，无论参数值如何。我们还明确了与所谓的图灵不稳定性的关系。(2)对一个模型方程进行了数值模拟，该模型方程描述了在某些细菌菌落中观察到的空间格局形成。研究表明，这种空间格局表现为这一过程的历史。(3)两物种竞争系统是数学生态学理论中的一个主要问题。在竞争率无限大的奇异极限下，给出了系统简化为Stephen型自由边值问题的数学严密验证。(4) Gierer-Meinhardt系统是数学生物学中形态发生的数学模型。研究了尖形平稳图形的稳定性，给出了稳定和不稳定的判据。

Concerning the above research project, we obtained the following results.(1) We carried out the study on skew-gradient systems, which are a generalized activator-inhibitor systems. When a steady state is characterized as a min-max point of an energy functional, we showed that the steady state is stable regardless of parameter values. We also make clear the relation with the so-called Turing instability.(2) We carried out numerical simulations on a model equation which describes a spatial pattern formation observed In a colony of some bacteria. It is shown that such spatial patterns appear as a history of the process.(3) Two-species competition system is a main problem in the theory of mathematical ecology. We gave a mathematically rigorous verification for the fact that the system reduces to a Stephen -type free boundary value problem in a singular limit where the competition rate is infinitely large.(4) The Gierer-Meinhardt system is a mathematical model for the morphogenesis in Mathematical biology. We studied stability of spiky stationary patterns, and gave some criteria for the stability and Instability.