Grégoire Nadin

The non-local Fisher-KPP equation: travelling waves and steady states

We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel (x) and investigate the possible differences with the standard Fisher–KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transform is positive or if the length σ of the non-local interaction is short enough, then the only steady states are u ≡ 0 and u ≡ 1. Next, we study existence of the travelling waves. We prove that this equation admits travelling wave solutions that connect u = 0 to an unknown positive steady state u∞(x), for all speeds c ≥ c*. The travelling wave connects to the standard state u∞(x) ≡ 1 under the aforementioned conditions: 0 SRC=http://ej.iop.org/images/0951-7715/22/12/002/non313053in002.gif/> or σ is sufficiently small. However, the wave is not monotonic for σ large.

Traveling waves for the Keller–Segel system with Fisher birth terms

We consider the traveling wave problem for the one dimensional Keller-Segel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound on the solution and Ḣ(R) estimates. Solutions in the KPP case are built as a limit of traveling waves for the truncated birth rates (similar to ignition temperature in combustion theory). We also discuss some general bounds and long time convergence for the solution of the Cauchy problem and in particular linear and nonlinear stability of the non-zero steady state. Key-words: Chemotaxis; Traveling waves; Keller-Segel system; Reaction diffusion systems; Nonlinear stability. AMS Class. No. 35J60, 35K57, 92C17 1 The main result The growth of bacterial colonies undergoes complex biomechanical processes which underly the variety of shapes exhibited by the colonies. Usually cells divide and undergo active motion resulting in fronts of bacteria that are propagating. These fronts may be unstable leading to various patterns that have been studied for a long time, such as, for instance, spiral waves [16], aggregates [18] and dentrites [1, 10]. At least three elementary biophysical processes play commonly a central role in these patterns, and have been used in all modeling: (i) cell division which induces the growth of the colony, (ii) random cell motion – for instance, bacteria can swim in a liquid medium thanks to flagella, and (iii) chemoattraction through different molecules that the cells may release in their environment and that diffuse, leading to some kind of (possibly long distance) communication. Our purpose here is to study the existence of traveling waves and the linear and nonlinear stability of the steady states for a simple model combining these three effects. The macroscopic model describes the density of bacteria, denoted by u(t, x) below, and the chemoattractant concentration v(t, x) in the medium. It is a variant of the Keller-Segel system that has been widely studied in various contexts, see [5, 12, 19, 20] and references therein. ∗Departement de Mathematiques et Applications, Ecole Normale Superieure, CNRS UMR8553 , 45 rue d’Ulm, F 75230 Paris cedex 05 †Universite Pierre et Marie Curie-Paris6, UMR 7598 LJLL, Paris, F-75005 France and Institut Universitaire de France; email: perthame@dma.ens.fr ‡Department of Mathematics, University of Chicago, Chicago, IL 60637, USA; email: ryzhik@math.uchicago.edu

The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator

This paper is concerned with the periodic principal eigenvalue

Spreading speeds for one-dimensional monostable reaction-diffusion equations

k_\lambda(\mu)

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

associated with the operator

Some dependence results between the spreading speed and the coefficients of the space–time periodic Fisher–KPP equation

-\frac{d^2}{dx^2}-2\lambda\frac{d}{dx}-\mu(x)-\lambda^2

Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed

, where

Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations.

\lambda\in\mathbb{R}

Hyperbolic traveling waves driven by growth

and

What Is the Optimal Shape of a Fin for One-Dimensional Heat Conduction?

\mu