Reactive and reactive-diffusive time scales in stiff reaction-diffusion systems

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.

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Validation and Calibration of Models for Reaction–Diffusion Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127498000929 ◽

1998 ◽

Vol 08 (06) ◽

pp. 1163-1182 ◽

Cited By ~ 15

Author(s):

Rui Dilão ◽

Joaquim Sainhas

Keyword(s):

Time Scales ◽

Numerical Solutions ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Reaction Diffusion Systems ◽

Space And Time ◽

Diffusion Systems ◽

Random Deviation ◽

Fixed Constant

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps (Δx and Δt) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction–diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite Δx and Δt, if the parameter γN=DΔt/(Δx)2 assumes a fixed constant value, where N is an odd positive integer parametrizing the algorithm. The error between the solutions of the discrete and the continuous equations goes to zero as (Δx)2(N+2) and the values of γN are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction–diffusion equations. Comparison between numerical and analytical solutions of reaction–diffusion equations give global discretization errors of the order of 10-6 in the sup norm. Circular patterns of traveling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of 10-3.

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