93,787 research outputs found
Turing Instability in a Boundary-fed System
The formation of localized structures in the chlorine dioxide-idodine-malonic
acid (CDIMA) reaction-diffusion system is investigated numerically using a
realistic model of this system. We analyze the one-dimensional patterns formed
along the gradients imposed by boundary feeds, and study their linear stability
to symmetry-breaking perturbations (Turing instability) in the plane transverse
to these gradients. We establish that an often-invoked simple local linear
analysis which neglects longitudinal diffusion is inappropriate for predicting
the linear stability of these patterns. Using a fully nonuniform analysis, we
investigate the structure of the patterns formed along the gradients and their
stability to transverse Turing pattern formation as a function of the values of
two control parameters: the malonic acid feed concentration and the size of the
reactor in the dimension along the gradients. The results from this
investigation are compared with existing experiments.Comment: 41 pages, 18 figures, to be published in Physical Review
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
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
Master stability functions reveal diffusion-driven pattern formation in networks
We study diffusion-driven pattern-formation in networks of networks, a class
of multilayer systems, where different layers have the same topology, but
different internal dynamics. Agents are assumed to disperse within a layer by
undergoing random walks, while they can be created or destroyed by reactions
between or within a layer. We show that the stability of homogeneous steady
states can be analyzed with a master stability function approach that reveals a
deep analogy between pattern formation in networks and pattern formation in
continuous space.For illustration we consider a generalized model of ecological
meta-foodwebs. This fairly complex model describes the dispersal of many
different species across a region consisting of a network of individual
habitats while subject to realistic, nonlinear predator-prey interactions. In
this example the method reveals the intricate dependence of the dynamics on the
spatial structure. The ability of the proposed approach to deal with this
fairly complex system highlights it as a promising tool for ecology and other
applications.Comment: 20 pages, 5 figures, to appear in Phys. Rev. E (2018
Zoology of a non-local cross-diffusion model for two species
We study a non-local two species cross-interaction model with
cross-diffusion. We propose a positivity preserving finite volume scheme based
on the numerical method introduced in Ref. [15] and explore this new model
numerically in terms of its long-time behaviours. Using the so gained insights,
we compute analytical stationary states and travelling pulse solutions for a
particular model in the case of attractive-attractive/attractive-repulsive
cross-interactions. We show that, as the strength of the cross-diffusivity
decreases, there is a transition from adjacent solutions to completely
segregated densities, and we compute the threshold analytically for
attractive-repulsive cross-interactions. Other bifurcating stationary states
with various coexistence components of the support are analysed in the
attractive-attractive case. We find a strong agreement between the numerically
and the analytically computed steady states in these particular cases, whose
main qualitative features are also present for more general potentials
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