Mathematik | Informatik
Maxie Victoria Brüderlin, 2005 | Rheinfelden, AG
This study looks at the effect of initial conditions on the formation of Turing patterns. It focuses on symmetric initial states to which different levels of noise are added. To do so, the corresponding reaction-diffusion equations are numerically simulated in MATLAB using a finite difference approximation. The aim is to provide a better understanding of how different initial states affect the symmetry of patterns.
Introduction
Turing patterns can explain spatial organisations in nature and are applied in many areas from synthetic biology to engineering. This paper tries to answer the following questions:
How do distinct initial conditions qualitatively affect Turing pattern formation in the Gray–Scott model? How does the addition of noise to symmetric initial states affect the development of asymmetries in patterns?
Thus, this work analyses the effect that different initial conditions have on the symmetry and arrangement of Turing patterns. It specifically looks at symmetric initial states and the extent to which they can form symmetric patterns. Furthermore, it analyses the effect that noise added to the initial states has on the symmetry in patterns.
Methods
The formation of Turing patterns was simulated in MATLAB to see which patterns form for which initial states. The code for this simulation solves the partial differential Turing equations numerically, using a finite difference approximation, more precisely the explicit Euler method.
Results
The results show that the symmetry in an initial condition is directly reflected in the resulting pattern. However, all patterns develop slight asymmetries that grow over time and are too small to propagate to the visible state. Many patterns require an activator concentration below 0.4 in order to form. Furthermore, the addition of noise beyond a certain threshold in the initial state makes the pattern with parameter values f = 0.050, k = 0.064 develop asymmetries for all initial conditions tested. Patterns with parameter values f = 0.035, k = 0.062 and f = 0.015, k = 0.056 react similarly. All of their asymmetries converge to values in the range between 0.5 and 0.8. In contrast, patterns with parameter values 0.095, k = 0.056 and f = 0.1, k = 0.056 show only small asymmetries below the level of noise added to them.
Discussion
The first hypothesis that symmetric initial conditions would form symmetric patterns was proven correct. The asymmetries that arise nevertheless can be explained with the non-associativity of the floating number addition in the discretized Laplacian.
The second hypothesis stating that an activator concentration below 0.5 in the initial state will not form stable patterns was shown to be incorrect. Thus, patterns may have different concentration requirements, depending on their growth mechanisms.
The third hypothesis that noise added to symmetric initial states beyond a threshold level causes patterns to develop asymmetries was shown to be true. Noise acts as a perturbation, causing a large change in the final pattern. The values of the asymmetries converge to match the asymmetries the patterns develop with non-symmetric initial conditions, indicating a maximum asymmetry each pattern can reach. Furthermore, patterns showing asymmetries below the noise level can retrieve their symmetry over time, possibly due to their less complex structure.
To obtain more accurate results, it would have been advisable to collect more data and work around the asymmetries caused by the numerical procedure.
Conclusions
In conclusion, symmetric initial conditions form symmetric patterns that show different arrangements if the size and distribution of the activator patches are changed. Furthermore, all patterns develop slight asymmetries due to the non-associativity of the floating number addition in the discretized Laplacian. A perturbation like the addition of noise to the initial states causes the development of visible asymmetries in most patterns. They converge to a value between 0.5 and 0.8 representing the maximal asymmetry of the patterns. Other patterns with more simplistic structures can restore their symmetries over time. For further research, it would be useful to perform an extensive bifurcation analysis of the patterns with all the initial conditions to generate more precise quantitative data. Also, the initial conditions should be tested on more patterns as well as for different domains and boundary conditions to see if the findings in this work are consistent and can be used for applications in material design or synthetic biology.
Würdigung durch den Experten
Alexander Mordvintsev
Maxie Victoria Brüederlin untersucht Reaktions-Diffusions-PDE-Modelle mit Fokus auf dem Gray-Scott-System. Sie präsentiert effiziente numerische Methoden und eine MATLAB-Implementierung zur Musterbildungsanalyse. Systematisch erforscht sie den Einfluss von Anfangskonfigurationen und Rauschpegeln auf Mustersymmetrien und beobachtet Symmetriebrüche durch numerische Berechnungen. Ihre Arbeit verknüpft Computermathematik mit physikalischer Modellierung und zeigt deren Wert für komplexe Systemanalysen sowie die Bedeutung präziser Detailbetrachtung.
Prädikat:
gut
Gymnasium am Münsterplatz, Basel
Lehrer: Dr. Ramon Gonzalez