Disc Assemblies and Spike Assemblies in Inhibitory Systems
Open AccessIn this thesis we will study the self-organizing phenomena observed in two biological inhibitorysystems. The first system originates from the modeling of membrane phase separation on lipidvesicles. We introduce a reduced energy functional which includes two parts: the local part measuring the size of the interface and the nonlocal part measuring the interaction between different phases. We find disc assemblies as local minimizers of this reduced energy functional. The second system is a special example of activator-inhibitor type reaction-diffusion equations called the Gierer-Meinhardt system. The system was first introduced to explain morphogenetic pattern formation of hydra. We are able to prove the existence of assemblies of interior and boundary spikes as the steady states of the system. The proofs of both results involve a Lyapunov-Schimidt type reduction procedure. In the proof of the first result, we choose an area variable instead of a radius variable to cast the problem in a Hilbert space. The advantage of taking the area variable is that one can write the area constraint into a linear constraint. Moreover, we observe the existence of a single circular domain as a local minimizer of the energy functional and the location of the domain may depend on both the local property of the membrane, i.e. the Gauss curvature and the global geometry of the membrane, i.e. the regular part of the Green’s function. However, the locations of disc assemblies only depend on the global properties of the membrane surface under a slightly different parameter range. Similar conclusion is also drawn in the second system. Locations of interior and boundary spikes depend on an interaction between the global properties of the domain boundary via the Green’s function and the local properties, i.e. the mean curvature of the domain boundary.
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