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Frontiers in Mathematical Oncology

Pattern formation – on the modeling of multi-constituent inhibitory systems

Chong Wang

George Washington University


Skin pigmentation, animal coats and block copolymers, which can be considered as multi-constituent inhibitory systems, are all around us. Theoretical analysis and numerical simulation of multi-constituent inhibitory systems will be provided here. An inhibitory system is studied as a nonlocal geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. We establish that in different parameter ranges there are corresponding assemblies of certain patterns that exist as the stationary sets of the free energy functional. Numerically, a diffusive interface model is proposed and many self-assembly processes, which form various patterns, are vividly showed here. Different numerical schemes are compared and a new technique is introduced to be consistent with the Euler-Lagrange equation in the sharp interface model.