1999 Fiscal Year Final Research Report Summary
Mathematical studies on spatial and temporal patterns in reaction-diffusion systems
| Project/Area Number |
09440075
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Tokyo |
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Principal Investigator |
INABA Hisashi Graduate School of Mathematical Sciences, University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (80282531)
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| Co-Investigator(Kenkyū-buntansha) |
TAKAHASHI Katsuo Graduate School of Mathematical Sciences, University of Tokyo, Research Assistant, 大学院・数理科学研究科, 助手 (90114529)
YAMADA Michio Graduate School of Mathematical Sciences, University of Tokyo, Professor, 大学院・数理科学研究科, 教授 (90166736)
YANAGIDA Eiji Graduate School of Mathematical Sciences, University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (80174548)
NINOMIYA Hirokazu Department of Applied Mathematics and Informatics, Ryukoku University, Assistant Professor, 理工学部, 講師 (90251610)
MIZOGUCHI Noriko Department of Mathematics, Tokyo Gakugei University, Associate Professor, 教育学部, 助教授 (00251570)
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| Project Period (FY) |
1997 – 1999
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| Keywords | reaction-diffusion system / nonlinear partial differential equation / stability analysis / activator-inhibitor system / singular limit analysis / blowup of solutions / free boundary / modeling |
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| Research Abstract |
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.
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