Project/Area Number  09640252 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Toyama University 
Principal Investigator 
IKEDA Hideo Toyama University Science Associated Professor, 理学部, 助教授 (60115128)

CoInvestigator(Kenkyūbuntansha) 
FUJITA Yasuhiro Toyama University Science Associated Professor, 理学部, 助教授 (10209067)
KOBAYASHI Kusuo Toyama University Science Professor, 理学部, 教授 (70033925)
AZUKAWA Kazuo Toyama University Science Professor, 理学部, 教授 (20018998)
WATANABE Yoshiyuki Toyama University Science Professor, 理学部, 教授 (50018991)
YOSHIDA Norio Toyama University Science Professor, 理学部, 教授 (80033934)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  reactiondiffusion systems / singular perturbation method / traveling pulses / standing pulses / bifurcation phenomena / stability property / 反応拡散方程式 / 特異摂動法 / 進行パルス / 定常パルス / 分岐現象 / 安定性 / 反応拡散方程式系 / 分岐問題 / nonlocal 
Research Abstract 
Bifurcation problem of traveling spot patterns in 2dimensional domain is studied. Our model equations are 2 component reactiondiffusion systems including a nonlocal term. . Numerically they exhibit traveling spot patterns which is bifurcated from standing spot patterns stably. Theoretically we know that 2component reactiondiffusion systems without a nonlocal term do not exhibit such phenomena. Then first we considered that such phenomena came from the nonlocal term. But later we could succeed in rewriting the 2component systems with the nonlocal term into the 3component reactiondiffusion systems without a nonlocal term. We conclude that such phenomena come from the delicate balance of the ratio of diffusions, the ratio of reactions and the nonlinear terms. This enables us to analyze them mathematically. In 1dimensional case, we showed the existence and the stability of traveling front and back solutions under the assumption of bistability. Now we try to construct standing pulse solutions connecting these traveling front and back solutions. For the stability property, we will show that standing pulse solutions destabilize under the outofphase (asymmetric) modes first and the inphase (symmetric) modes secondly when some parameter is changed. At this time, the informatin of the stability of traveling front and back solutions and the connecting manner of these two solutions will help us to analyze the stability property. These new bifurcated solutions are stable traveling pulse solutions. In 2dimensional case, it is confirmed numerically that these systems have stable 2dim traveling spot patterns. For this case, we will show the existence of radial symmetric standing pulse solutions and then catch traveling spot patterns as the destabilization of radial symmetric standing pulse solutions.
