Front propagation and segregation in a reaction–diffusion model with cross-diffusion (original) (raw)
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Pattern formation for a type of reaction diffusion system with cross diffusion
International Journal of ADVANCED AND APPLIED SCIENCES
In this paper, pattern formation for a Schnakenberg model is studied in one and two dimensions. The model has been studied when the diffusion is nonlinear and so called cross diffusion. The conditions of diffusion driven instability are applied to this model and shown that this model can formulate patterns, and the existence of bifurcation for specific parameters are shown and for different values of wave number k. The use of COMSOL Multiphysics finite element package in simulation shows nice graphs of pattern formations in two dimensions.
Cross-diffusion and pattern formation in reaction–diffusion systems
Phys. Chem. Chem. Phys., 2009
Cross-diffusion, the phenomenon in which a gradient in the concentration of one species induces a flux of another chemical species, has generally been neglected in the study of reaction-diffusion systems. We summarize experiments that demonstrate that cross-diffusion coefficients can be quite significant, even exceeding ''normal,'' diagonal diffusion coefficients in magnitude in systems that involve ions, micelles, complex formation, excluded volume effects (e.g., surface or polymer reactions) and other phenomena commonly encountered in situations of interest to chemists. We then demonstrate with a series of model calculations that cross-diffusion can lead to spatial and spatiotemporal pattern formation, even in relatively simple systems. We also show that, in the absence of cross-diffusion among the reacting species, introduction of a nonreactive species that induces appropriate cross-diffusive fluxes with reactive species can lead to pattern formation.
Pattern Formations Dynamics in a Reaction-Diffusion Model
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Propagating reaction-diffusion fronts constitute one of the paradigms in the realm of the nonlinear chemical phenomena. In this paper different situations are considered. Firstly, we discuss the problem of front propagation in a two-variable chemical system exhibiting multiple stationary states. Emphasis is put on the question of velocity selection. In section 3 of our contribution, the question of front propagating in spatially modulated and noisy media is addressed. Finally, we also briefly comment on the problem of reaction-diffusion fronts in non-quiescent media. In this particular scenario we simply aim at introducing the two basic propagation modes, i.e., thin versus distributed reaction fronts, that are identified in our numerical simulations.
Cross-Diffusion Driven Instability in a Predator-Prey System with Cross-Diffusion
2014
In this work we investigate the process of pattern formation induced by nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra predator-prey kinetics. We show that the cross-diffusion term is responsible of the destabilizing mechanism that leads to the emergence of spatial patterns. Near marginal stability we perform a weakly nonlinear analysis to predict the amplitude and the form of the pattern, deriving the Stuart-Landau amplitude equations. Moreover, in a large portion of the subcritical zone, numerical simulations show the emergence of oscillating patterns, which cannot be predicted by the weakly nonlinear analysis. Finally, when the pattern invades the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude equation which is able to describe the shape and the speed of the wave.
Pattern formation driven by cross-diffusion in a 2D domain
Nonlinear Analysis: Real World Applications, 2013
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.
Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations
Nonlinearity, 1994
Patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical wavelength or domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally develop in bistable and excitable systems, but may also appear far beyond Hopf and Turing bifurcations. The global behavior of domain patterns strongly depends on the fronts' inner structures. In this paper we study a symmetry breaking front bifurcation expected to occur in a wide class of reaction-diffusion systems, and the effects it has on pattern formation and pattern dynamics. We extend previous works on this type of front bifurcation and clarify the relations among them. We show that the appearance of front multiplicity beyond the bifurcation point allows the formation of persistent patterns rather than transient ones. In a different parameter regime, we find that the front bifurcation outlines a transition from oscillating (or breathing) patterns to traveling ones. Near a boundary we find that fronts beyond the bifurcation can reflect, while those below it either bind to the boundary or disappear.
Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems
IMA Journal of Applied Mathematics, 2021
Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation be...
Mathematics and Computers in Simulation, 2012
In this work we investigate the phenomena of pattern formation and wave propagation for a reaction-diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart-Landau equation respectively.
Reaction–diffusion fronts with inhomogeneous initial conditions
Journal of Physics: Condensed Matter, 2007
Properties of reaction zones resulting from A + B → C type reaction-diffusion processes are investigated by analytical and numerical methods. The reagents A and B are separated initially and, in addition, there is an initial macroscopic inhomogeneity in the distribution of the B species. For simple two-dimensional geometries, exact analytical results are presented for the time evolution of the geometric shape of the front. We also show using cellular automata simulations that the fluctuations can be neglected both in the shape and in the width of the front.