Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach

Abstract

The use of reaction-diffusion models rests on the key assumption that the underlying diffusive process is Gaussian. However, a growing number of studies have pointed out the prevalence of anomalous diffusion, and there is a need to understand the dynamics of reactive systems in the presence of this type of non-Gaussian diffusion. Here we present a study of front dynamics in reaction-diffusion systems where anomalous diffusion is due to the presence of asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator, whose fundamental solutions are Levy $\alpha$-stable distributions. Numerical simulation of the fractional Fisher-Kolmogorov equation, and analytical arguments show that anomalous diffusion leads to the exponential acceleration of fronts and a universal power law decay, $x^{-\alpha }$, of the tail, where $\alpha$, the index of the Levy distribution, is the order of the fractional derivative.

arXiv commentNew version. Accepted for publication in Physical Review Letters

References in corpus (1)

• Superfast front propagation in reactive systems with anomalous diffusion

Cited by in corpus (44)

• A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations
• Time-space fabric underlying anomalous diffusion
• Matrix approach to discrete fractional calculus II: partial fractional differential equations
• Levy Flight Superdiffusion: An Introduction
• Non-diffusive transport in plasma turbulence: a fractional diffusion approach
• The influence of fractional diffusion in Fisher-KPP equations
• On reaction-subdiffusion equations
• The acceleration of evolutionary spread by long-range dispersal
• Fractional reaction-diffusion equations
• Non-Markovian Random Walks and Non-Linear Reactions: Subdiffusion and Propagating Fronts
• Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
• Truncation effects in superdiffusive front propagation with Lévy flights
• Quantum chaos dynamics in long-range power law interaction systems
• Solution of generalized fractional reaction-diffusion equations
• A foundational approach to the Lie theory for fractional order partial differential equations
• A Series of High Order Quasi-Compact Schemes for Space Fractional Diffusion Equations Based on the Superconvergent Approximations for Fractional Derivatives
• Two-dimensional SIR epidemics with long range infection
• Mesoscopic description of reactions under anomalous diffusion: A case study
• Non-Markovian Levy diffusion in nonhomogeneous media
• SIR epidemics with long range infection in one dimension
• Oscillatory instability in super-diffusive reaction -- diffusion systems: fractional amplitude and phase diffusion equations
• Spatial clustering of interacting bugs: Levy flights versus Gaussian jumps
• Space-time fractional reaction-diffusion equations associated with a generalized Riemann-Liouville fractional derivative
• Bayesian inference for a wavefront model of the Neolithisation of Europe
• Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable Processes
• Lévy flights in the infinite potential well as the hypersingular Fredholm problem
• Front propagation in reaction-diffusion systems with anomalous diffusion
• An efficient spectral-Galerkin method for fractional reaction-diffusion equations in unbounded domains
• On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation
• Hydrodynamic theory of scrambling in chaotic long-range interacting systems
• Spatial patterns of competing random walkers
• On the Melnikov method for fractional-order systems
• Expansion of fractional derivatives in terms of an integer derivative series: physical and numerical applications
• Long-range dispersal, stochasticity and the broken accelerating wave of advance
• Front Propagation Dynamics with Exponentially-Distributed Hopping
• Asymmetric Levy flights in nonhomogeneous environments
• A universal solution scheme for fractional and classical PDEs
• Anomalous beam diffusion near beam-beam synchro-betatron resonances
• Mesoscopic Fractional Quantum in Soft Matter
• Reaction-diffusion on random spatial networks with scale-free jumping rates via effective medium theory
• Sharp Space-Time Regularity of the Solution to Stochastic Heat Equation Driven by Fractional-Colored Noise
• Weakly non-linear dynamics in reaction -- diffusion systems with Lévy flights
• Dynamics of a 1-D model for the emergence of the plasma edge shear flow layer with momentum conserving Reynolds stress
• Pattern selection in the Schnakenberg equations: From normal to anomalous diffusion