Turing Instability and Pattern Formation in an Activator-Inhibitor System with Nonlinear Diffusion

In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; %favors the mechanism of pattern formation with respect to the classical linear diffusion case; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.Comment: Accepted for publication in Acta Applicandae Mathematica

Pattern selection models: From normal to anomalous diffusion

“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

Turing pattern formation in the Brusselator system with nonlinear diffusion

In this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in 1D and 2D spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover we consider traveling patterning waves: when the domain size is large, the pattern forms sequentially and traveling wavefronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and through a matching procedure we construct the outer amplitude equation and the inner core solution.Comment: Physical Review E, 201

Stochastic fluctuations and quasi-pattern formation in reaction-diffusion systems with anomalous transport

Many approaches to modelling reaction-diffusion systems with anomalous transport rely on deterministic equations and ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasi-patterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the formation of both deterministic and stochastic patterns. Our analysis shows that the combination of subdiffusion and intrinsic stochasticity can reduce the threshold ratio of the effective diffusion coefficients required for pattern formation to a greater degree than either effect on its own.Comment: 26 pages, 5 figure

Stochastic reaction-diffusion models in biology

Every cell contains several millions of diffusing and reacting biological molecules. The interactions between these molecules ultimately manifest themselves in all aspects of life, from the smallest bacterium to the largest whale. One of the greatest open scientific challenges is to understand how the microscopic chemistry determines the macroscopic biology. Key to this challenge is the development of mathematical and computational models of biochemistry with molecule-level detail, but which are sufficiently coarse to enable the study of large systems at the cell or organism scale. Two such models are in common usage: the reaction-diffusion master equation, and Brownian dynamics. These models are utterly different in both their history and in their approaches to chemical reactions and diffusion, but they both seek to address the same reaction-diffusion question. Here we make an in-depth study into the physical validity of these models under various biological conditions, determining when they can reliably be used. Taking each model in turn, we propose modifications to the models to better model the realities of the cellular environment, and to enable more efficient computational implementations. We use the models to make predictions about how and why cells behave the way they do, from mechanisms of self-organisation to noise reduction. We conclude that both models are extremely powerful tools for clarifying the details of the mysterious relationship between chemistry and biology