Doctoral Dissertations

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.


Zhang, Yanzhi

Committee Member(s)

Singler, John R.
He, Xiaoming
Hu, Wenqing
Balakrishnan, S. N.


Mathematics and Statistics

Degree Name

Ph. D. in Computational and Applied Mathematics


This work was supported by the US National Science Foundation under Grant Number DMS-1620465.


Missouri University of Science and Technology

Publication Date

Spring 2020

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

Note about bibliography

Includes bibliographic references.


© 2020 Hatim Kareem Khudhair, All rights reserved.

Document Type

Dissertation - Open Access

File Type




Thesis Number

T 11878