Diffusion, Advection and Pattern Formation

Abstract

Patterns are ubiquitous in nature and can arise in reaction-diffusion systems with differential diffusions. The existence and stability of (in)homogeneous steady state are the classical topics in the dynamics of reaction-diffusion systems. In this thesis we study the influences of anomalous diffusion and advection upon the patterns.

In the first part of this thesis we consider the impact of subdiffusive process on the instability of homogeneous states in three types of reaction-subdiffusion systems. The modelling of linear and nonlinear reaction-subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time-fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results by defining generalised spectra through dispersion relations, which allows us to examine the onset of instability and in particular inspect Turing-type instabilities. These results are numerically illustrated. Moreover, we prove expansions that imply for one class of reaction-subdiffusion equations algebraic decay for stable spectrum, whereas for another class this is exponential. We also study the linearisation of a nonlinear reaction subdiffusion equation in a nonzero homogeneous state. Here the spectrum cannot be analysed directly by Fourier-Laplace transform, so we provide an energy estimate, existence, uniqueness and dynamics of Fourier modes of such a linearisation.

It is well known that for reaction-diffusion systems with differential isotropic diffusions, a Turing instability yields striped solutions. In the second part of this thesis we study the impact of weak anisotropy by directional advection on the stability of such solutions, and the role of quadratic nonlinearities. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using Lyapunov-Schmidt reduction, Floquet-Bloch decomposition and centre manifold reduction we derive rigorous parameter expansions for existence, stability against large-wavelength and lattice modes, respectively. This provides detailed formulae for the loci of bifurcations and stability boundaries under the influences of the advection and quadratic terms. In particular, while destabilisation of the background state is through modes perpendicular to the advection (Squire-theorem), we show that stripes can bifurcate zigzag unstably. The well known destabilising effect of quadratic terms can be counterbalanced by advection, which leads to intriguing arrangements of stability boundaries. We illustrate these results numerically by an example. Finally, we show numerical computations of these stability boundaries in the extended Klausmeier model for vegetation patterns and show stripes bifurcate stably in the presence of advection.