Keywords and Phrases
Anomalous Diffusion; Gray-Scott Equations; Pattern Formation; Schnakenberg Equations; Turing Space; Two-Layer Reaction-Diffusion Equations
“Pattern formation and selection is an important topic in many physical, chemical, and biological fields. In 1952, Alan Turing showed that a system of chemical substances could produce spatially stable patterns by the interplay of diffusion and reactions. Since then, pattern formations have been widely studied via the reaction-diffusion models. So far, patterns in the single-component system with normal diffusion have been well understood. Motivated by the experimental observations, more recent attention has been focused on the reaction-diffusion systems with anomalous diffusion as well as coupled multi-component systems. The objectives of this dissertation are to study the effects of superdiffusion on pattern formations and to compare them with the effects of normal diffusion in one-, and multi-component reaction-diffusion systems. Our studies show that the model parameters, including diffusion coefficients, ratio of diffusion powers, and coupling strength between components play an important role on the pattern formation. Both theoretical analysis and numerical simulations are carried out to understand the pattern formation in different parameter regimes. Starting with the linear stability analysis, the theoretical studies predict the space of Turing instability. To further study pattern selection in this space, weakly nonlinear analysis is carried out to obtain the regimes for different patterns. On the other hand, numerical simulations are carried out to fully investigate the interplay of diffusion and nonlinear reactions on pattern formations. To this end, the reaction-diffusion systems are solved by the Fourier pseudo-spectral method. Numerical results show that superdiffusion may substantially change the patterns in a reaction-diffusion system. Different superdiffusive exponents of the activator and inhibitor could cause both qualitative and quantitative changes in emergent spatial patterns. Comparing to single-component systems, the patterns observed in multi-component systems are more complex”--Abstract, page iv.
Singler, John R.
Balakrishnan, S. N.
Mathematics and Statistics
Ph. D. in Computational and Applied Mathematics
Missouri University of Science and Technology
Journal article titles appearing in thesis/dissertation
- Pattern selection in the Schnakenberg equations: From normal to anomalous diffusion
- Analysis and simulations of Turing patterns in two-layer reaction-diffusion systems
- Complex patterns in the fractional Gray-Scott system
xi, 87 pages
© 2020 Hatim Kareem Khudhair, All rights reserved.
Dissertation - Open Access
Khudhair, Hatim K., "Pattern selection models: From normal to anomalous diffusion" (2020). Doctoral Dissertations. 3039.