On the existence and stability of singularly perturbed solutions for one-activator and two-inhibitors reaction-diffusion models

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Our purpose of this project is to find a stationary solution of a three-component reaction-diffusion 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 four-dimensional dynamical system with discontinuous nonlinearities. There are two hyperbolic equilibrium points, both have two-dimensional stable manifolds and two-dimensional unstable manifolds, in R^4. The above problem corresponds to find a continuous trajectroy on the restricted two-dimensional 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 two-dimensional 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 0-th 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.