Pattern formation driven by cross--diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction-diffusion system with nonlinear diffusion terms and the competitive Lotka-Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, hexagonal patterns

Instabilities and Patterns in Coupled Reaction-Diffusion Layers

We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the inter-layer coupling. For systems of $n$-component layers and non-identical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a $\sqrt{2}$:1 length scale ratio produces an unusual steady square pattern.Comment: 13 pages, 5 figures, accepted for publication in Phys. Rev.

Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure

Homogenization induced by chaotic mixing and diffusion in an oscillatory chemical reaction

A model for an imperfectly mixed batch reactor with the chlorine dioxide-iodine-malonic acid (CDIMA) reaction, with the mixing being modelled by chaotic advection, is considered. The reactor is assumed to be operating in oscillatory mode and the way in which an initial spatial perturbation becomes homogenized is examined. When the kinetics are such that the only stable homogeneous state is oscillatory then the perturbation is always entrained into these oscillations. The rate at which this occurs is relatively insensitive to the chemical effects, measured by the Damkohler number, and is comparable to the rate of homogenization of a passive contaminant. When both steady and oscillatory states are stable, spatially homogeneous states, two possibilities can occur. For the smaller Damkohler numbers, a localized perturbation at the steady state is homogenized within the background oscillations. For larger Damkohler numbers, regions of both oscillatory and steady behavior can co-exist for relatively long times before the system collapses to having the steady state everywhere. An interpretation of this behavior is provided by the one-dimensional Lagrangian filament model, which is analyzed in detail

Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples

We implement a computer-assisted approach that, under appropriate conditions, allows the bifurcation analysis of the coarse dynamic behavior of microscopic simulators without requiring the explicit derivation of closed macroscopic equations for this behavior. The approach is inspired by the so-called time-step per based numerical bifurcation theory. We illustrate the approach through the computation of both stable and unstable coarsely invariant states for Kinetic Monte Carlo models of three simple surface reaction schemes. We quantify the linearized stability of these coarsely invariant states, perform pseudo-arclength continuation, detect coarse limit point and coarse Hopf bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit