On the minimal wave speed of wave fronts for reaction-diffusion equations

1992 ◽  
Vol 8 (3) ◽  
pp. 252-258 ◽  
Author(s):  
Mingxin Wang ◽  
Qixiao Ye
Keyword(s):  
Wave Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Wave Fronts ◽  
Minimal Wave Speed

SPREAD RATES UNDER TEMPORAL VARIABILITY: CALCULATION AND APPLICATION TO BIOLOGICAL INVASIONS

2011 ◽  
Vol 21 (12) ◽  
pp. 2469-2489 ◽  
Author(s):  
GUNOG SEO ◽  
FRITHJOF LUTSCHER
Keyword(s):  
Biological Invasions ◽  
Temporal Variability ◽  
Life Histories ◽  
Wave Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Coefficient Function ◽  
Compartment System ◽  
Minimal Wave Speed

In this paper, we introduce a technique to study the minimal wave speed in reaction-diffusion equations with temporal variability and apply it to two particular models for biological invasions. We use the exponential transform to avoid solving partial differential equations explicitly or finding inverse transforms. In a single reaction-diffusion equation with time-periodic coefficients, the minimal wave speed depends only on time-averages of each coefficient function. In a two-compartment system with mobile and stationary individuals, the invasion speed depends on the precise form of the coefficient functions and their temporal correlations; in some cases, a lower bound can be obtained. Our technique can be extended to more complex life histories of invading organisms.

EXTERNAL NOISE AND FRONT PROPAGATION IN REACTION-TRANSPORT SYSTEMS WITH INERTIA: THE MEAN SPEED OF FISHER WAVES

2002 ◽  
Vol 02 (04) ◽  
pp. R109-R124 ◽  
Author(s):  
WERNER HORSTHEMKE
Keyword(s):  
Wave Speed ◽  
Reaction Diffusion ◽  
Front Propagation ◽  
External Noise ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Transport Systems ◽  
The Mean ◽  
Reaction Transport ◽  
Fisher Waves

We review the effect of spatiotemporal noise, white in time and colored in space, on front propagation in systems of reacting and dispersing particles, where the particle motion displays inertia or persistence. We discuss the three main approaches that have been developed to describe transport with inertia, namely hyperbolic reaction-diffusion equations, reaction-Cattaneo systems or reaction-telegraph equations, and reaction random walks. We focus on the mean speed of Fisher waves in these systems and study in particular reaction random walks, which are the most natural generalization of reaction-diffusion equations. Hyperbolic reaction-diffusion equations account for inertia in the transport process in an ad hoc way, whereas the other reaction-transport systems have a proper macroscopic or microscopic foundation. For the former, external noise affects neither the mean wave speed nor the region in parameter space for which Fisher waves exist. For the latter, external noise increases the mean wave speed of Fisher waves and decreases the upper limit for the characteristic time of the transport process, below which propagating fronts exist.

scholarly journals Minimum wave speeds in monostable reaction–diffusion equations: sharp bounds by polynomial optimization

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200450
Author(s):  
Jason J. Bramburger ◽  
David Goluskin
Keyword(s):  
Lower Bounds ◽  
Wave Speed ◽  
Polynomial Optimization ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Upper And Lower Bounds ◽  
Wave Speeds ◽  
Minimum Wave Speed

Many monostable reaction–diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction–diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction–diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction–diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

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