Reaction–diffusion systems for algorithmic composition

1996 ◽  
Vol 1 (3) ◽  
pp. 195-201 ◽  
Author(s):  
ANDREW MARTIN
Keyword(s):  
Computer Animation ◽  
Reaction Diffusion ◽  
Sound Synthesis ◽  
Reaction Diffusion Equations ◽  
Algorithmic Composition ◽  
Reaction Diffusion Systems ◽  
Fingerprint Enhancement ◽  
Alan Turing ◽  
Diffusion Systems ◽  
Biological Morphogenesis

Reaction–diffusion systems were first proposed by mathematician and computing forerunner Alan Turing in 1952. Originally intended as an explanation of plant phyllotaxis (the structure and arrangement of leaves in plants), reaction–diffusion now forms the basis of an area in biology which is as important as DNA research in the field of biological morphogenesis (Kauffman 1993). Reaction–diffusion systems were successfully utilised within the fields of computer animation and computer graphics to generate visually naturalistic patterning and textures such as animal furs (Turk 1991). More recently, reaction–diffusion systems have been applied to methods of half-tone printing, fingerprint enhancement, and have been proposed for use in sound synthesis (Sherstinsky 1994). The recent publication The Algorithmic Beauty of Seashells (Meinhardt 1995) uses various reaction–diffusion equations to explain patterned pigmentation markings on seashells. This article details an example of the application of reaction–diffusion systems to algorithmic composition within the field of computer music. The patterned data produced by reaction–diffusion systems is used to create a naturalistic soundscape in the piece cicada.

Bounds on the critical times for the Fisher-KPP equation

2021 ◽  
Vol 63 ◽  
pp. 448-468
Author(s):  
Marianito Rodrigo
Keyword(s):  
Upper And Lower Solutions ◽  
Critical Time ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Kpp Equation ◽  
Diffusive Process ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Comparison Functions

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

scholarly journals Validation and Calibration of Models for Reaction–Diffusion Systems

1998 ◽  
Vol 08 (06) ◽  
pp. 1163-1182 ◽  
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.

scholarly journals Insights from chemical systems into Turing-type morphogenesis

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
C. Konow ◽  
M. Dolnik ◽  
I. R. Epstein
Keyword(s):  
Reaction Diffusion ◽  
Behavioural Science ◽  
Future Directions ◽  
Reaction Diffusion Systems ◽  
Chemical Systems ◽  
Alan Turing ◽  
Diffusion Systems ◽  
Chemical Studies ◽  
Historical Role ◽  
Turing Mechanism

In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction–diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237 , 37–72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction–diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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.

Quantitative maximum principles and strongly coupled gradient-like reaction-diffusion systems

1983 ◽  
Vol 94 (3-4) ◽  
pp. 265-286 ◽  
Author(s):  
Nicholas D. Alikakos
Keyword(s):  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Diffusion Matrix ◽  
Strongly Coupled ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
All Solutions

SynopsisIn §§1 and 2, we consider mainly a system of reaction-diffusion equations with general diffusion matrix and we establish the stabilization of all solutions at t →∞. The interest of this problem derives from two separate facts. First, the sets that are useful for localizing the asymptotics cease to be invariant as soon as the diffusion matrix is not a multiple of the identity. Second, the set of equilibria is connected. In §3, we establish uniform L§ bounds for the solutions of a class of parabolic systems. The unifying feature in the problems considered is the lack of any conventional maximum principles.

Weakly invariant regions for reaction—diffusion systems and applications

2000 ◽  
Vol 130 (5) ◽  
pp. 1165-1180 ◽  
Author(s):  
Si Ning Zheng
Keyword(s):  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Chemical Processes ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Global Existence Of Solutions ◽  
Diffusion Systems ◽  
Invariant Regions ◽  
Semilinear Parabolic

The important theory of invariant regions in reaction-diffusion equations has only restricted applications because of its strict requirements on both the reaction terms and the regions. The concept of weakly invariant regions was introduced by us to admit wider reaction-diffusion systems. In this paper we first extend the L∞ estimate technique of semilinear parabolic equations of Rothe to the more general case with convection terms, and then propose more precise criteria for the bounded weakly invariant regions. We illustrate, by three model examples, that they are very convenient for establishing the global existence of solutions for reaction-diffusion systems, especially those from ecology and chemical processes.

Influence of mixed boundary conditions in some reaction–diffusion systems

1997 ◽  
Vol 127 (5) ◽  
pp. 1053-1066 ◽  
Author(s):  
Robert H. Martin ◽  
Michel Pierre
Keyword(s):  
Global Existence ◽  
Polynomial Growth ◽  
Existence Of Solutions ◽  
Mixed Boundary ◽  
Reaction Diffusion ◽  
Mixed Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Global Existence Of Solutions ◽  
Diffusion Systems

SynopsisWe analyse global existence of solutions to a system of two reaction–diffusion equations for whicha ‘balance’ law holds. The main aim is to make clear the influence of different combinations ofboundary conditions on global existence under the assumption that the nonlinearities satisfy polynomial growth estimates.

BOUNDS ON THE CRITICAL TIMES FOR THE GENERAL FISHER–KPP EQUATION

2021 ◽  
pp. 1-21
Author(s):  
MARIANITO R. RODRIGO
Keyword(s):  
Upper And Lower Solutions ◽  
Critical Time ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Kpp Equation ◽  
Diffusive Process ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Comparison Functions

Abstract The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

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