Microscopic selection principle for a diffusion-reaction equation

Bramson, Maury; Calderoni, P.; Masi, Anna De; Ferrari, Pablo A.; Lebowitz, Joel L.; Schonmann, Roberto H.

doi:10.1007/bf01020581

Cited by 56 publications

(104 citation statements)

References 9 publications

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“…Here, we are interested especially in classes of problems in which the FKPP equation is obtained either in the large-scale limit (N → ∞) of many-particle systems or in the mean-field limit of physical problems that are discrete at a microscopic level [6,7,11,12,22,23].…”

Section: Introductionmentioning

confidence: 99%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

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Section: Introductionmentioning

confidence: 99%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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“…Equations of type (1) are obtained either as the large scale limit [5,8,13,14,15,16] or as the mean field limit [17] of physical situations which are discrete at the microscopic level (particles, lattice models, etc.) As the number of particles is an integer, the concentration h(x, t) could be thought as being larger than some ε, which would correspond to the value of h(x, t) when a single particle is present.…”

Section: Introductionmentioning

confidence: 99%

“…Equations describing the propagation of a front between a stable and an unstable state appear [1,2,3,4,5,6,7] in a large variety of situations in physics, chemistry and biology. One of the simplest equations of this kind is the Fisher-Kolmogorov [1,2] equation…”

Section: Introductionmentioning

confidence: 99%

Shift in the velocity of a front due to a cutoff

1997

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We consider the effect of a small cut-off ε on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off ε is to select a single velocity which converges when ε → 0 to the one predicted by the marginal stability argument. For small ε, the shift in velocity has the form K(log ε) −2 and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.

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“…The case q = 0 means no adhesion and reduces to the model of Refs. [5,6,7]. For nonzero q, it is much harder to a cell to diffuse if it has many neighbors.…”

Section: A Discrete Modelmentioning

confidence: 99%

“…We can ask for is the continuum analog of this model? It was shown [5,6,7] that for small proliferation rates the propagating fronts in this discrete system can be described by the Fisher-Kolmogorov equation [8] (FK):…”

Section: Introductionmentioning

confidence: 99%

Generalized Cahn-Hilliard equation for biological applications

2008

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Recently we considered a stochastic discrete model which describes fronts of cells invading a wound [1]. In the model cells can move, proliferate, and experience cell-cell adhesion. In this work we focus on a continuum description of this phenomenon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term. As in the discrete model, there are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similarly to those of Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in a good agreement with a numerical solution of the GCH equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak. To analyze this regime, we investigated relaxation dynamics for the Cahn-Hilliard equation without proliferation. We found that the relaxation process exhibits self-similar behavior. The results of continuum and discrete models are in a good agreement with each other for the different regimes we analyzed.

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Microscopic selection principle for a diffusion-reaction equation

Cited by 56 publications

References 9 publications

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Shift in the velocity of a front due to a cutoff

Generalized Cahn-Hilliard equation for biological applications

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