scholarly journals From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ

2020 ◽  
Vol 17 (162) ◽  
pp. 20190621 ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Thomas E. Woolley ◽  
Eamonn A. Gaffney
Keyword(s):  
Symmetry Breaking ◽  
Heterogeneous Media ◽  
Heterogeneous Reaction ◽  
Reaction Diffusion ◽  
Spatial Patterning ◽  
Instability Mechanism ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Pattern Forming

Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction–diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions. We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction–diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing’s original thesis to a far wider and more realistic class of systems.

CHEMICAL PATTERNS IN SIMPLE FLOW SYSTEMS

2003 ◽  
Vol 06 (01) ◽  
pp. 155-162 ◽  
Author(s):  
ANNETTE TAYLOR
Keyword(s):  
Chemical Reaction ◽  
Reaction Diffusion ◽  
Stationary Concentration ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Differential Flow ◽  
Chemical Waves ◽  
Pattern Forming ◽  
Simple Flow ◽  
Concentration Patterns

The addition of flow to chemical reaction-diffusion systems provides robust pattern-forming mechanisms which are expected to occur in a wide variety of natural and artificial systems. Experiments demonstrating some of these mechanisms are presented here, including the differential-flow-induced chemical instability (DIFICI), which gives rise to traveling chemical waves, and flow-distributed oscillations (FDO), which produce stationary concentration patterns.

scholarly journals Multi-spike solutions of a hybrid reaction–transport model

2021 ◽  
Vol 477 (2247) ◽  
pp. 20200829
Author(s):  
P. C. Bressloff
Keyword(s):  
Transport Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Strongly Nonlinear ◽  
Diffusion Systems ◽  
Spike Solution ◽  
Classical Pattern ◽  
Final Steady State ◽  
Reaction Transport ◽  
Pattern Forming

Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit D a  ≪  D h , where D a and D h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate α between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans . We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α  → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α  → ∞ with a graphical construction for finite α .

scholarly journals The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guido Schneider ◽  
Matthias Winter
Keyword(s):  
Numerical Simulation ◽  
Analytical Approach ◽  
Reaction Diffusion ◽  
Short Wave ◽  
Reaction Diffusion Systems ◽  
Long Wave ◽  
Diffusion Systems ◽  
Schnakenberg Model ◽  
Pattern Forming ◽  
Amplitude System

<p style='text-indent:20px;'>We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.</p>

INSTABILITIES OF WAVE TRAINS AND TURING PATTERNS IN LARGE DOMAINS

2007 ◽  
Vol 17 (08) ◽  
pp. 2679-2691 ◽  
Author(s):  
JENS D. M. RADEMACHER ◽  
ARND SCHEEL
Keyword(s):  
Period Doubling ◽  
Reaction Diffusion ◽  
Turing Patterns ◽  
Instability Mechanism ◽  
Reaction Diffusion Systems ◽  
System Parameters ◽  
Diffusion Systems ◽  
Special Cases ◽  
Wave Trains ◽  
Spatio Temporal

We classify generic instabilities of wave trains in reaction–diffusion systems on the real line as the wavenumber and system parameters are varied. We find three types of robust instabilities: Hopf with nonzero modulational wavenumber, sideband and spatio-temporal period-doubling. Near a fold, the only other robust instability mechanism, we show that all wave trains are necessarily unstable. We also discuss the special cases of homogeneous oscillations and reflection symmetric, stationary Turing patterns.

ON TURING-HOPF INSTABILITIES IN REACTION-DIFFUSION SYSTEMS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 257-274 ◽  
Author(s):  
MARIANO RODRÍGUEZ RICARD
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Hopf Bifurcations ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Spatially Homogeneous

We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.

EXAMPLES OF FORCED SYMMETRY-BREAKING TO HETEROCLINIC CYCLES AND NETWORKS IN THREE-DIMENSIONAL EUCLIDEAN-INVARIANT SYSTEMS

2009 ◽  
Vol 19 (05) ◽  
pp. 1655-1678
Author(s):  
M. J. PARKER ◽  
M. G. M. GOMES ◽  
I. N. STEWART
Keyword(s):  
Symmetry Breaking ◽  
Nonlinear Optical ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Three Dimensions ◽  
Optical Systems ◽  
Heteroclinic Cycles ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Face Centered

In [Parker et al., 2008a] group theory was employed to prove the existence of homoclinic cycles in forced symmetry-breaking of simple (SC), face-centered (FCC), and body-centered (BCC) cubic planforms. In this paper we extend this classification demonstrating that more elaborate heteroclinic cycles and networks can arise through the same process. Our methods naturally generate graphs that represent possible heteroclinic cycles and networks. The results do not depend on the representation of the symmetry group and are thus quite general. This study is motivated by pattern formation in three dimensions which occur in reaction–diffusion systems, certain nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction. This work extends previous work by Parker et al. [2006, 2008a, 2008b] and Hou and Golubitsky [1997].

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