Lenya Ryzhik

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.

Existence and Non-Existence of Fisher-KPP Transition Fronts

We consider Fisher-KPP-type reaction–diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global-in-time solutions while creating a global-in-time bump-like solution. This is the first example of a medium in which no reaction–diffusion transition front exists. A weaker localized inhomogeneity leads to the existence of transition fronts, but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.

STABILITY IN A NONLINEAR POPULATION MATURATION MODEL

We consider models for population structured by maturation/maturation speed proposed by Rotenberg. It is a variant of transport equations for age-structured populations which presents particularly interesting mathematical difficulties. It allows one to introduce more stochasticity in the birth process and in the aging phenomena. We present a new method for studying the time asymptotics which is also illustrated on the simpler McKendrick–Von Foerster model. The nonlinear variants of these models are shown to exhibit either nonlinear stability or periodic solutions depending on the datum.

A short proof of the logarithmic Bramson correction in Fisher-KPP equations

In this paper, we explain in simple PDE terms a famous result of Bramson about the loga- rithmic delay of the position of the solutions u(t, x) of Fisher-KPP reaction-diffusion equations in R, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of u to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions u along their level sets to the profile of the minimal travelling front.

Stability of Generalized Transition Fronts

We study the qualitative properties of the generalized transition fronts for the reaction–diffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are globally exponentially stable for the solutions of the Cauchy problem. The results hold for reaction rates that have arbitrary spatial variations provided that the rate is uniformly positive and bounded from above.

Biomixing by Chemotaxis and Enhancement of Biological Reactions

Many phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. However, to the best of our knowledge, the question of reaction enhancement by chemotaxis has not yet received extensive treatment either analytically or numerically. We consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We prove that in the framework of our model, chemotaxis plays a crucial role. There is a rigid limit to how much the fertilization efficiency can be enhanced if there is no chemotaxis but only advection and diffusion. On the other hand, when chemotaxis is present, the fertilization rate can be arbitrarily close to being complete provided that the chemotactic attraction is sufficiently strong. Moreover, an interesting feature of the estimates on fertilization rate and timescales in the chemotactic case is that they do not depend on the amplitude of the reaction term.

On the nonlocal Fisher–KPP equation: steady states, spreading speed and global bounds

We consider the Fisher–KPP (for Kolmogorov–Petrovsky–Piskunov) equation with a nonlocal interaction term. We establish a condition on the interaction that allows for existence of non-constant periodic solutions, and prove uniform upper bounds for the solutions of the Cauchy problem, as well as upper and lower bounds on the spreading rate of the solutions with compactly supported initial data.

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

Self-Averaging from Lateral Diversity in the Itô–Schrödinger Equation

We consider the random Schrodinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Ito–Schrodinger stochastic partial differential equation which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions tends to zero but is large compared to the correlation length.

The logarithmic delay of KPP fronts in a periodic medium

We consider solutions of the KPP-type equations with a periodically varying reaction rate, and compactly supported initial data. It has been shown by M. Bramson in the case of the constant reaction rate that the lag between the position of such solutions and that of the traveling waves grows as (3/2) log(t). We generalize this result to the periodic case