Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion

Gambino, G.; Lombardo, Maria Carmela; Sammartino, M.

doi:10.1016/j.matcom.2011.11.004

Cited by 98 publications

(90 citation statements)

References 39 publications

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“…In order to accurately predict the spatial features of the expected Turing patterns, non-linear bifurcation analysis and the amplitude equations formalism must be used, see i.e. [19,22,29,30]. Nevertheless, this kind of analysis is beyond the scope of the present paper, for this reason we resorted to numerical investigation of pattern selection issues.…”

Section: Numerical Investigationsmentioning

confidence: 99%

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

2014

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In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013 J. Solid State Electr. 17, 467-479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimensiontwo Turing-Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing-Hopf dynamics.

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Section: Numerical Investigationsmentioning

confidence: 99%

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

2014

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“…Following the approach based on the multiple scales method adopted by [5,9,10], we set a small control parameter η 2 = (χ −χ c )/χ c , which gives the dimensionless distance from the bifurcation value of χ . Upon translation of the equilibrium P * to the origin, the system (1) can be written as:…”

Section: Traveling Wavefront Equationsmentioning

confidence: 99%

“…Thus, by substituting the above expansions (9) and (6)- (8) into (2) and collecting the terms at each order of η, we obtain the following systems:…”

Section: Traveling Wavefront Equationsmentioning

confidence: 99%

“…Conversely, when the chemotaxis is weak, a wavefront connecting the diseasefree equilibrium to the stable equilibrium corresponding to a homogeneous plaque is found. The issue of existence of traveling wave solutions for reaction-diffusion models has been addressed in many papers, which show that a small perturbation near a homogeneous steady state may lead to a wavefront invasion with the consequent pattern formation [4,9,11,27].…”

Section: Introductionmentioning

confidence: 99%

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Wavefront invasion for a chemotaxis model of Multiple Sclerosis

et al. 2016

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In this work we study wavefront propagation for a chemotaxis reactiondiffusion system describing the demyelination in Multiple Sclerosis. Through a weakly non linear analysis, we obtain the Ginzburg-Landau equation governing the evolution of the amplitude of the pattern. We validate the analytical findings through numerical simulations. We show the existence of traveling wavefronts connecting two different steady solutions of the equations. The proposed model reproduces the progression of the disease as a wave: for values of the chemotactic parameter below threshold, the wave leaves behind a homogeneous plaque of apoptotic oligodendrocytes. For values of the chemotactic coefficient above threshold, the model reproduces the formation of propagating concentric rings of demyelinated zones, typical of Baló's sclerosis.

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“…Shi et al [25] showed that cross-diffusion can destabilize or stabilize a uniform equilibrium in a reaction-diffusion system. Recently, cross-diffusion driven Turing instability has been investigated in [26][27][28]. In addition to these theoretical aspects, an important interest, especially for physicists and biologists, lies in the behavior of numerical approximations exhibiting spatial patterns.…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

Ruiz–Baier

Tian

2013

Nonlinear Analysis: Real World Applications

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a b s t r a c tCross-diffusion driven instabilities have gained a considerable attention in the field of population dynamics, mainly due to their ability to predict some important features in the study of the spatial distribution of species in ecological systems. This paper is concerned with some mathematical and numerical aspects of a particular reaction-diffusion system with cross-diffusion, modeling the effect of allelopathy on two plankton species. Based on a stability analysis and a series of numerical simulations performed with a finite volume scheme, we show that the cross-diffusion coefficient plays a important role on the pattern selection.

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Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion

Cited by 98 publications

References 39 publications

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

Wavefront invasion for a chemotaxis model of Multiple Sclerosis

Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

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