Christophe Besse & Gregory Faye

Institut de Mathematiques de Toulouse

CNRS, UMR 5219

Universite Paul Sabatier

Toulouse

Proposition de stage – M1

Titre : Phenomene de propagation pour les equations de reaction-diffusion sur des graphes

Resume. L’objectif du stage est de se familiariser avec certaines proprietes qualitatives d’une classe

d’equations aux derivees partielles de type parabolique dites de reaction-diffusion. Plus precisement,

on s’interessera a la dynamique en temps long de ces equations lorsque ces dernieres sont posees sur

des graphes. Pour se faire, la strategie consistera a discretiser les equations a l’aide de la methode

des differences finies, et d’etudier plusieurs cas d’ecole.

Prerequis : connaissances de MATLAB ou Python, notions d’EDOs et d’EDPs, methodes des

differences finies.

Contacts : christophe.besse@math.univ-toulouse.fr

Contact : http://www.math.univ-toulouse.fr/∼cbesse/

Contact : gregory.faye@math.univ-toulouse.fr

Contact : https://www.math.univ-toulouse.fr/∼gfaye/

NB : Une gratification de stage est possible.

1

Descriptif detaille du projet

Ce projet s’interesse a l’etude d’equations de reaction-diffusion de la forme

∂tu = ∂2xu + f(u), t > 0, x ∈ Ω, (1)

ou Ω est un graphe avec une structure speciale. On considerera uniquement le cas ou Ω est l’union de

demi-droites se rejoignant en un point (quelques exemples de graphes (orientes) qui seront traites

durant le stage sont donnes dans la Figure 1). Et donc pour etre bien posee, l’equation (1) est

completee avec des conditions de compatibilites en la jonction plus une donnee initiale.THE FISHER-KPP EQUATION OVER SIMPLE GRAPHS 3

!"!#

$%& $'&$(&

!"

!

!#)

#*

!")

!"*

!#

Figure 1. (a) River network with lower-branch RL and upper-branche RU . (b) River network

with lower-branch RL and upper-branches RU1 , RU2 . (c) River network with upper-branch RU

and lower-branches RL1, RL2

.

Our analysis in this paper shows that the above dichotomy phenomenon generally does not stand

even very simple changes of the topological structure of the river. More precisely, if the homogeneous

river in (1.1) is replaced by a simple river system with two or three homogeneous river branches, as

described in Figure 1, with each branch having its own (usually di↵erent) waterflow speed , then

the population may persist in a very di↵erent fashion to that exhibited by (1.1). Indeed, we will

show a trichotomy phenomenon, according to the waterflow speeds of the river branches, namely,

(i) washing out: if the waterflow speed in every branch is no less than c, then the population

will be washed out in every branch as in case (I) for (1.1),

(ii) persistence at carrying capacity: if the waterflow speed in every upper branch is smaller

than c, then the population will persist at its carrying capacity in each branch as in case (II) for

(1.1),

(iii) persistence below carrying capacity: in all the remaining cases, the population will

persist at a positive steady-state strictly below the carrying capacity in every branch.

It turns out that this trichotomy behavior is also di↵erent from the finite graph case considered

in [8, 13, 14, 15] (see Remark 1.9 below), where the unique persistence state does not seem easily

distinguishable when the parameters are varied.

We now describe our results more precisely.

1.1. Two river branches. First, we consider the case that the river has two branches, and the

waterflow in each branch has a di↵erent constant speed, but otherwise the environment is homoge-

neous to the concerned species. Let RL := (0, +1) represent the lower river and RU := (1, 0)

stand for the upper river. Let wL, wU denote the density of the species in RL and RU , respectively.

Then, the evolution of the species is governed by the following reaction-di↵usion system:

(1.4)

8>>>>>>>>><>>>>>>>>>:

@twL DL@xxwL + L@xwL = fL(wL), x 2 RL, t > 0,

@twU DU@xxwU + U@xwU = fU (wU ), x 2 RU , t > 0,

wL(t, 0) = wU (t, 0), t > 0,

DLaL@xwL(t, 0) = DUaU@xwU (t, 0), t > 0,

wL(0, x) = w0(x), x 2 RL,

wU (0, x) = w0(x), x 2 RU ,

where the parameters DL, DU ,L,U , aL, aU are positive constants. The constants DL, DU are the

random di↵usion coecients of the species, L,U are the advection coecients (waterflow speeds),

and aL, aU account for the cross-section area of the river branches RL and RU , respectively. The

Figure 1 – Exemples de graphes etudies le stage (repris de [1]).

L’etude portera sur deux types de reaction :

— bistable : f(u) = u(1− u)(u− a), avec a ∈ (0, 1/2) ;

— monostable : f(u) = u(1− u).

La finalite du stage consistera a determiner (numeriquement) la solution de (1) a partir d’une

condition initiale de type indicatrice dont le support se situera dans l’une des branches du graphe.

Typiquement, en fonction du graphe, on etudiera si il y a extinction (convergence vers 0) ou bien

propagation (convergence vers 1). Dans le cas d’une propagation, on proposera un algorithme

permettant de calculer la vitesse d’invasion. On pourra s’appuyer sur les references [1, 2] pour la

description des modeles et le chapitre 4 du livre [3] pour se familiariser avec les fronts d’invasion

dans les equations de reaction-diffusion. Il s’agira donc de discretiser l’equation (1) en tenant compte

des proprietes du graphe considere.

References

[1] Y. Du, B. Lou, R. Peng and M. Zhou. The Fisher-KPP equation over simple graphs : Varied

persistence states in river networks. Preprint arXiv preprint arXiv :1809.06961, (2018).

[2] S. Jimbo and Y. Morita. Entire solutions to reaction-diffusion equations in multiple half-lines

with a junction. J. Differential Equations, 267 (2019), 1247–1276.

[3] B. Perthame. Parabolic Equations in Biology : Growth, Reaction, Movement and diffusion.

Lecture Notes on Mathematical Modeling in the Life Sciences, Springer, 2015.

2