In an attempt to describe how patterns emerge in biological systems, Alan Turing proposed a mathematical model encapsulating the properties of such processes. It details a partial differential equation governing the dynamics of two or more substances, called morphogens, reacting and diffusing in a specific manner, in turn generating what has now come to be denoted as Turing patterns. In recent years, evidence has accumulated to support Turing's claim and it has been proposed that it is responsible for the dynamical characteristics of phenomena such as skin pigmentation and branching of lungs in vertebrates. The aim of this paper is to study how the choice of model parameters and reaction kinetics influence the nature of patterns generated, as well as explore how boundary control can be employed to generate pre-defined patterns and the efficiency of this procedure. To simulate the patterns, the differential equation is solved in Python by means of a spectral method using discretized space and time domains. The model parameters were then studied to try to gain insight in their effects on the patterns yielded. The boundary control was implemented in MATLAB using a difference method. The metric used for efficiency was taken to be the energy expenditure of the boundary cells. The complex dynamics of the studied systems make it difficult to draw valuable conclusions on the influence of the parameters, but the results support the expected characteristics of the models used. The efficiency of the pattern generation is deemed to be closely related to the amount of boundary control utilized.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:kth-315130 |
| Date | January 2022 |
| Creators | Forsström, Oskar, Falgén Nikula, Oskar |
| Publisher | KTH, Fysik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | TRITA-SCI-GRU ; 2022:066 |
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