2003 Fiscal Year Final Research Report Summary
On the existence and stability of singularly perturbed solutions for oneactivator and twoinhibitors reactiondiffusion models
Project/Area Number 
13640107

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Toyama University 
Principal Investigator 
IKEDA Hideo Toyama University, Sciences, Professor > 富山大学, 理学部, 教授 (60115128)

CoInvestigator(Kenkyūbuntansha) 
FUJITA Yasuhiro Toyama University, Sciences, Associated Professor, 理学部, 助教授 (10209067)
YOSHIDA Norio Toyama University, Sciences, Professor, 理学部, 教授 (80033934)

Project Period (FY) 
2001 – 2003

Keywords  singular perturbation method / transition layer / stability / reactiondiffusion system / travelling wave / dynamical system / three component system 
Research Abstract 
Our purpose of this project is to find a stationary solution of a threecomponent reactiondiffusion system and examine the stability of the solution by applying a singular perturbation method. The most difficult problem in these process is to solve a reduced problem, that is, to find a C^1'class solution of the pair of elliptic boundary value problems with discontinuous nonlinearities and to know the dependency of the solution on physical parameters. With a dynamical system view point, the reduced problem is rewritten as a fourdimensional dynamical system with discontinuous nonlinearities. There are two hyperbolic equilibrium points, both have twodimensional stable manifolds and twodimensional unstable manifolds, in R^4. The above problem corresponds to find a continuous trajectroy on the restricted twodimensional space R^2. Under an artificial condition, we succeed in finding such trajectroy. And using this, we show the existence of a solution of the original problem applying a singular perturbation method. Next we consider the same problem on the twodimensional region R^2. Under a similar condition to the above, we can find a radialy symmetric solution. Finally we study the linearized eigenvalue problem around the solution and trace the, critical eigenvalue by numerical simulation. We check the appearance of instability modes, that is, the 0th mode and the first mode. The former case corresponds to the bifurcation phenomena of a travelling wave solution, that is, a traveling spot, from a standing wave solution and the latter case does to the bifurcation of an asymmetric solution from a radialy symmetric standing wave solution. But we cannot find a parameter on which the Hopf bifurcation occurs.

Research Products
(12 results)