08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
Browse
1 results
Search Results
Item Open Access Concentrated patterns in biological systems(2003) Winter, Matthias; Mielke, Alexander (Prof.)We study pattern formation for reaction-diffusion systems of mathematical biology in the case of the Gierer-Meinhardt system. In this thesis we show that there is a critical growth rate of the inhibitor such that the position of boundary spikes is given by a linear combination of the boundary curvature and a Green function. There are two main results. The first one concerns the existence of boundary spikes for the activator. It says that the solutions are such that in the neighborhood of a boundary point for which the linear combination mentioned above possesses a nondegenerate critical point in tangential direction there is a spike (i.e. a peak whose spatial extension contracts but which after rescaling has a limit profile). Outside this boundary point the solutions are constant in first approximation. The proof uses Liapunov-Schmidt reduction, fixed point theorems and asymptotic analysis. The second main result concerns stability and says that the stability of this boundary spike depends on the parameters of the system. We assume that the linear combination from above possesses a nondegenerate local maximum at that boundary point. Then the stability depends on the size of a time relaxation constant. The proof studies small eigenvalues (i.e. they converge to zero) using asymptotic analysis. These small eigenvalue are connected with the second tangential derivatives of this linear combination. Large eigenvalues are explored using nonlocal eigenvalue problems.