scholarly journals Gradient structures and geodesic convexity for reaction–diffusion systems

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120346 ◽  
Author(s):  
Matthias Liero ◽  
Alexander Mielke
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion System ◽  
Diffusion Equations ◽  
Mass Transportation ◽  
Drift Diffusion ◽  
Geodesic Convexity ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Entropy Functional

We consider systems of reaction–diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic λ -convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift–diffusion system, provide a survey on the applicability of the theory.

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

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.

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.

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.

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.

ABRAHAMS–TSUNETO REACTION–DIFFUSION SYSTEM IN SUPERCONDUCTIVITY AND SYMBOLIC COMPUTATION

2001 ◽  
Vol 12 (10) ◽  
pp. 1417-1423 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN ◽  
GUANG-MEI WEI
Keyword(s):  
Symbolic Computation ◽  
Reaction Diffusion ◽  
Analytic Solutions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Ginzburg Landau ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Exact Analytic Solutions ◽  
Self Consistent

The reaction–diffusion systems represent various problems in the real world. For the Abrahams–Tsuneto reaction–diffusion system arising in superconductivity, we perform computerized symbolic computation and find its new exact analytic solutions, which are solitonic. We see the possibility that by way of the shock waves, the self–consistent superconducting interaction drives the Ginzburg–Landau order parameter, which might be observable.

Replicating Spots in Reaction-Diffusion Systems

1997 ◽  
Vol 07 (05) ◽  
pp. 1149-1158 ◽  
Author(s):  
Kyoung J. Lee ◽  
Harry L. Swinney
Keyword(s):  
Cell Division ◽  
Numerical Simulations ◽  
Continuous Process ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Diffusion System ◽  
One Dimension ◽  
Reaction Diffusion Systems ◽  
Gel Layer ◽  
Diffusion Systems

We review the phenomenon of replicating spots in reaction-diffusion systems and discuss the mechanism of replication. This phenomenon was discovered in recent experiments on a ferrocyanide-iodate-sulfite reaction-diffusion system. Patterns form in a thin gel layer that is in contact with a continuously fed stirred reservoir. Patterns of spots are observed to undergo a continuous process of growth and multiplication through cell division and death through overcrowding. A similar phenomenon is also found in numerical simulations in one dimension on a four-species model of the ferrocyanide-iodate-sulfite reaction and in simulations in two dimensions of simpler two-species reaction-diffusion models: Gray–Scott model by J. Pearson and FitzHugh–Nagumo model by A. Hagberg and E. Meron.

scholarly journals A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

2015 ◽  
Vol 61 (1) ◽  
pp. 59-78 ◽  
Author(s):  
Monica-Dana Burlică ◽  
Daniela Roşu
Keyword(s):  
Fixed Point ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Nonlocal Initial Conditions ◽  
Diffusion Systems ◽  
Compactness Arguments ◽  
Fixed Point Techniques

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

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