Dissipative structures and oscillations in reaction-diffusion models with or without time-delay

Author(s):  
Manuel G. Velarde
2017 ◽  
Vol 43 (2) ◽  
pp. 247-264 ◽  
Author(s):  
Aldo Ledesma-Durán ◽  
Lorenzo-Héctor Juárez-Valencia ◽  
Juan-Bibiano Morales-Malacara ◽  
Iván Santamaría-Holek

2014 ◽  
Vol 20 (1) ◽  
pp. 55-76 ◽  
Author(s):  
Tom Froese ◽  
Nathaniel Virgo ◽  
Takashi Ikegami

Due to recent advances in synthetic biology and artificial life, the origin of life is currently a hot topic of research. We review the literature and argue that the two traditionally competing replicator-first and metabolism-first approaches are merging into one integrated theory of individuation and evolution. We contribute to the maturation of this more inclusive approach by highlighting some problematic assumptions that still lead to an ximpoverished conception of the phenomenon of life. In particular, we argue that the new consensus has so far failed to consider the relevance of intermediate time scales. We propose that an adequate theory of life must account for the fact that all living beings are situated in at least four distinct time scales, which are typically associated with metabolism, motility, development, and evolution. In this view, self-movement, adaptive behavior, and morphological changes could have already been present at the origin of life. In order to illustrate this possibility, we analyze a minimal model of lifelike phenomena, namely, of precarious, individuated, dissipative structures that can be found in simple reaction-diffusion systems. Based on our analysis, we suggest that processes on intermediate time scales could have already been operative in prebiotic systems. They may have facilitated and constrained changes occurring in the faster- and slower-paced time scales of chemical self-individuation and evolution by natural selection, respectively.


2018 ◽  
Vol 313 ◽  
pp. 306-315 ◽  
Author(s):  
Swati Tyagi ◽  
Subit K Jain ◽  
Syed Abbas ◽  
Shahlar Meherrem ◽  
Rajendra K Ray

Author(s):  
B. Kostet ◽  
M. Tlidi ◽  
F. Tabbert ◽  
T. Frohoff-Hülsmann ◽  
S. V. Gurevich ◽  
...  

The Brusselator reaction–diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.


Author(s):  
Theodore Kolokolnikov ◽  
Michael Ward ◽  
Justin Tzou ◽  
Juncheng Wei

For a large class of reaction–diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a ‘spotted-stripe’ solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.


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