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

针对上述研究项目，我们取得了以下研究成果：(1)开展了斜梯度系统的研究，它是一种广义的激活剂-抑制剂系统。当稳态被刻画为能量泛函的极小极大点时，我们证明了无论参数值如何，稳态都是稳定的。我们还明确了与所谓的图灵不稳定性的关系。(2)我们对一个描述在一些细菌群体中观察到的空间格局形成的模型方程进行了数值模拟。两种群竞争系统是数学生态理论中的一个主要问题。对于竞争速率无穷大的奇异极限问题，我们给出了严格的数学证明。(4)Gierer-Meinhardt系统是数学生物学中形态发生的数学模型。我们研究了尖峰平稳模式的稳定性，给出了一些稳定性和不稳定性的判据。

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.