Existence, positivity and stability for a nonlinear

⁄model of cellular proliferation

y zMostafa Adimy and Fabien Crauste

Year 2004

Laboratoire de Math´ematiques Appliqu´ees, FRE 2570

Universit´e de Pau et des Pays de l’Adour,

Avenue de l’universit´e, 64000 Pau, France

Abstract

Inthispaper, weinvestigatea systemof twononlinearpartialdiﬀerentialequations, arisingfroma

model of cellular proliferation which describes the production of blood cells in the bone marrow. Due

to cellular replication, the two partial diﬀerential equations exhibit a retardation of the maturation

variable and a temporal delay depending on this maturity. We show that this model has a unique

solution which is global under a classical Lipschitz condition. We also obtain the positivity of the

solutions and the local and global stability of the trivial equilibrium.

Keywords: nonlinear partial diﬀerential equation, age-maturity structured model, blood production sys-

tem, delay depending on the maturity, positivity, local and global stability.

1 Introduction

We analyse, in this paper, a mathematical model arising from the blood production system. It is based

on a system proposed by Mackey and Rudnicki [19], in 1994, to describe the dynamics of hematopoietic

stem cells in the bone marrow. The origin of this system is a model of Burns and Tannock [7] (1970) in

which each cell can be either in a proliferating phase or in a resting phase (also called G -phase). The0

resulting model is a time-age-maturity structured system.

Proliferatingcellsareinthecellcycle,inthattheyarecommittedtodivideattheendofthemitosis,the

so-called point of cytokinesis. After division, they give birth to two daughter cells which enter immediatly

the resting phase. Proliferating cells can also die by apoptosis, a programmed cell death.

The resting phase is a quiescent stage in the cellular development. Cells in this phase can not divide:

they mature and, provided they do not die, they enter the proliferating phase and complete the cycle.

The model in [19] has been analysed by Mackey and Rey [17, 18] in 1995, Crabb et al. [8, 9] in 1996,

Dyson et al. [10] in 1996 and Adimy and Pujo-Menjouet [3, 4] in 2001 and 2003. In these studies, the

authors assumed that all cells divide exactly at the same age.

⁄This paper has been published in Nonlinear Analysis: Real World Applications, 6, 337-366, 2005.

yE-mail: mostafa.adimy@univ-pau.fr

z fabien.crauste@univ-pau.fr

1M. Adimy and F. Crauste A nonlinear model of cellular proliferation

However, inthemostgeneralsituationinacellularpopulation, itisbelievedthatthetimerequiredfor

a cell to divide is not identical between cells (see Bradford et al. [6]). For example, pluripotent stem cells

(whichare the less mature cells) divide fasterthan committedstem cells, whicharethe moremature stem

cells. In 1993, Mackey and Rey [16] considered a model in which the time required for a cell to divide

is distributed according to a density, but the authors only made a numerical analysis of their model.

Dyson et al. [11, 12], in 2000, considered a time-age-maturity structured equation in which all cells do

not divide at the same age. They presented the basic theory of existence, uniqueness and properties of

the solution operator. However, in their model, they considered only one phase (the proliferating one),

and the intermediary ﬂux between the two phases is not represented. In 2003, Adimy and Crauste [2]

considered a model in which the proliferating phase duration is distributed according to a density with

compact support. They obtained global stability results for their model.

In this work, we consider the situation when the age at cytokinesis depends on the maturity of the cell

at the point of commitment, that means when it enters the proliferating phase. We assume that each cell

entering the proliferating phase with a maturity m divides at age ¿ =¿(m), depending on this maturity.

This hypothesis can be found, for example, in Mitchison [21] (1971) and John [13] (1981). This yields

to the boundary condition (11). To our knowledge, nobody has studied this model, except Adimy and

Pujo-Menjouet in [5], where they considered only a linear case.

We obtain a system of ﬁrst order partial diﬀerential equations with a time delay depending on the

maturity and a retardation of the maturation variable. We investigate the basic theory of existence,

uniqueness, positivity and stability of the solutions of our model.

The paper is organised as follows. In Section 2, we present the time-age-maturity structured model.

By using the characteristics method, we reduce this model to a time-maturity structured system, which is

formed by two partial diﬀerential equations with a time delay depending on the maturity and a nonlocal

dependence in the maturity variable. In Section 3, we ﬁrst give an integrated formulation of our model by

using the classical variation of constant formula and then we prove local existence of solutions, by using

a ﬁxed-point theorem, and their global continuation. We deduce the global existence. In Section 4, we

obtain the positivity of these solutions by developping a method described by Webb [24]. In Section 5,

we concentrate on the stability of the trivial equilibrium of the system and, in the last section, we discuss

the model and the asymptotic behaviour.

2 Biological background and equations of the model

Each cell is caracterised, in the two phases, by its age and its maturity. The maturity describes the

developmentofthecell. Itistheconcentrationofwhatcomposesacell, suchasproteinsorotherelements

one can measure experimentally. The maturity is supposed to be a continuous variable and to range from

m=0 to m=1 in the two phases.

Cells enter the proliferating phase with age a = 0 and they are committed to undergo cell division a

time ¿ later, so the age variable ranges from a = 0 to a = ¿ in the proliferating phase. We suppose that

proliferating cells can be lost by apoptosis with a rate ?.

At the cytokinesis age, a cell divides and gives two daughter cells, which enter immediatly the resting

phase, with age a = 0. A cell can stay its entire life in the resting phase, so the age variable ranges from

a = 0 to a = +1. The resting phase is a quiescent stage in the cellular development. In this phase,

cells can either return to the proliferating phase at a rate ﬂ and complete the cycle or die at a rate –

before ending the cycle. According to a work of Sachs [22], we suppose that the maturation of a cell and

the density of resting cells at a given maturity level determine the capacity of this cell for entering the

proliferating phase.

Wedenotebyp(t;m;a)andn(t;m;a)respectivelythepopulationdensitiesintheproliferatingandthe

2M. Adimy and F. Crauste A nonlinear model of cellular proliferation

resting phases at time t, with age a and maturity m. The conservation equations are

@p @p @(V(m)p)

+ + =¡?(m)p; (1)

@t @a @m

‡ ·¡ ¢@n @n @(V(m)n)

+ + =¡ –(m)+ﬂ m;N(t;m) n; (2)

@t @a @m

whereV(m)isthematurationvelocityand N(t;m)isthedensityofrestingcellsattime twithamaturity

level m, deﬁned by

Z +1

N(t;m)= n(t;m;a)da:

0

We suppose that the function V is continuously diﬀerentiable on [0;1], positive on (0;1] and satisﬁes

V(0)=0 and Z m ds

=+1; for m2(0;1]: (3)

V(s)0

Rm2 dsSince , with m <m , is the time required for a cell with maturity m to reach the maturity m ,1 2 1 2m V(s)1

then Condition (3) means that a cell with very small maturity needs a long time to become mature.

For example, if

pV(m) » ﬁm ; with ﬁ>0 and p‚1;

m!0

then Condition (3) is satisﬁed.

We suppose, throughout this paper, that ? and – are continuous and non-negative on [0;1]. The function

ﬂ is supposed to be positive and continuous.

Equations (1) and (2) are completed by boundary conditions which represent the cellular ﬂux between

the two phases. The ﬁrst condition,

Z +1

p(t;m;0)= ﬂ(m;N(t;m))n(t;m;a)da=ﬂ(m;N(t;m))N(t;m); (4)

0

describestheeﬄuxofcellsleavingtherestingphasetotheproliferatingone. Cellsenteringtheproliferating

phase with age 0 depend only on the population of the resting phase with a given maturity level.

The second boundary condition determines the transfer of cells from the point of cytokinesis to the

resting compartment.

We assume that a cell entering the proliferating phase with a maturity m2[0;1] divides at age ¿(m)>0,

and we require that ¿ is a continuously diﬀerentiable and positive function on [0;1] such that

10¿ (m)+ >0; for m2(0;1]: (5)

V(m)

Since V(0) = 0, this condition is always satisﬁed in a neighborhood of the origin. If we suppose that the

less mature cells divide faster than more mature cells, that is, if we assume, for example, that ¿ is an

increasing function, then Condition (5) is also satisﬁed.

If one consider a cell in the proliferating phase at time t, with maturity m 2 (0;1], age a and initial

maturity (that means at age a=0) m , then, naturally, we have0

Z m

ds

m •m and a= •¿(m ):0 0

V(s)m0

If m is the maturity of the cell at the cytokinesis point, then there exists a unique Θ(m) 2 (0;m) (the

maturity at the point of commitment) such that

Z m ds

=¿(Θ(m)); (6)

V(s)Θ(m)

3M. Adimy and F. Crauste A nonlinear model of cellular proliferation

because Condition (5) implies that the function

Z m

ds

me ! ¡¿(me)

V(s)m

is continuous and strictly decreasing from (0;m] into [¡¿(m);+1). Then, we can deﬁne a function

Θ:(0;1]!(0;1], where Θ(m) satisﬁes (6).

From a biological point of view, Θ(m) represents the initial maturity of proliferating cells that divide at

maturity m (at the point of cytokinesis). Then, from the deﬁnition, the age of a cell with maturity m at

the point of cytokinesis is ¿(Θ(m)).

Remark that Θ is continuously diﬀerentiable on (0;1] and satisﬁes

0<Θ(m)<m; for m2(0;1]:

This implies, in particular, that

Z m ds

lim Θ(m)=0 and lim =¿(0)<+1: (7)

m!0 m!0 V(s)Θ(m)

The property (7) means that cells with null maturity at the point of commitment keep a null maturity in

the proliferating phase.

The total number of proliferating cells at time t, with maturity m, is given by

Z ¿(Θ(m))

P(t;m)= p(t;m;a)da:

0

We consider the characteristic curves ´ : (¡1;0]£[0;1] ! [0;1], solutions of the ordinary diﬀerential

equation (

d´

(s;m) = V(´(s;m)); s•0 and m2[0;1];

ds

´(0;m) = m:

They represent the evolution of cells maturity to reach a maturity m at time 0 from a time s• 0. They

satisfy ´(s;0)=0 and ´(s;m)2(0;1] for s•0 and m2(0;1].

It is not diﬃcult to verify that, if m2[0;1], then Θ(m) is the unique solution of the equation

x=´(¡¿(x);m): (8)

Attheendoftheproliferatingphase,acellwithamaturitymdividesintotwodaughtercellswithmaturity

g(m). We assume that g : [0;1] ! [0;1] is a continuous and strictly increasing function, continuously

diﬀerentiable on [0;1) and such that g(m)• m for m2 [0;1]. We also assume, for technical reason and

without loss of generality, that

0lim g (m)=+1:

m!1

Then we can set

¡1g (m)=1; for m>g(1):

¡1This means that the function g :[0;1]![0;1] is continuously diﬀerentiable and satisﬁes

¡1 0(g )(m)=0; for m>g(1):

Note that the maturity m of the daughter cells just after division is smaller than g(1). Then, we must

have

n(t;m;0)=0; for m>g(1): (9)

4

eM. Adimy and F. Crauste A nonlinear model of cellular proliferation

If a daughter cell has a maturity m at birth, then the maturity of its mother at the point of cytokinesis

¡1 ¡1was g (m) and, at the point of commitment, it was Θ(g (m)). We set

¡1Δ(m)=Θ(g (m)); for m2[0;1]: (10)

Fromabiologicalpointofview, Δgivesthelinkbetweenthematurityofanewborncellandthematurity

of its mother at the point of commitment. Δ : [0;1]! [0;1] is continuous, continuously diﬀerentiable on

(0;1], withΔ(0)=0. Moreover, Δisstrictlyincreasingon(0;g(1))withΘ(m)•Δ(m)andΔ(m)=Θ(1)

for m2[g(1);1].

Then, we can give the second boundary condition,

¡ ¢¡1 0 ¡1

n(t;m;0)=2(g )(m)p t;g (m);¿(Δ(m)) ; for t‚0 and m2[0;1]: (11)

One can note that Expression (11) includes also Condition (9).

To complete the description of the model, we specify initial conditions,

p(0;m;a)=Γ(m;a); for (m;a)2[0;1]£[0;¿ ]; (12)max

and

n(0;m;a)=„(m;a); for (m;a)2[0;1]£[0;+1); (13)

where ¿ :=max ¿(m)>0. Γ and „ are assumed to be continuous, and the functionmax m2[0;1]

Z +1

„:m7! „(m;a)da (14)

0

is supposed to be continuous on [0;1].

We put

‰ Z ?t‡ ·¡ ¢ ¡ ¢

0»(t;m):=exp ¡ ? ´(¡s;m) +V ´(¡s;m) ds ;

0

for t‚0 and m2[0;1], and we deﬁne the sets

n o

Ω := (m;t)2[0;1]£[0;+1) ; 0•t•¿(Δ(m)) ;Δ

and n o

Ω := (m;t)2[0;1]£[0;+1) ; 0•t•¿(Θ(m)) :Θ

Proposition 2.1. Assume that the initial conditions „ and Γ satisfy, for m2[0;1],

¡ ¢

Γ(m;0)=ﬂ m;„(m) „(m): (15)

Then, the total populations of proliferating and resting cells, P(t;m) and N(t;m), satisfy, for m2 [0;1]

and t‚0,

¡ ¢@ @

P(t;m)+ (V(m)P(t;m))=¡?(m)P(t;m)+ﬂ m;N(t;m) N(t;m)

@t @m

8 ‡ ·¡ ¢

>…(m)»(t;m)Γ ´ ¡t;m ;¿(Θ(m))¡t ; if (m;t)2Ω ;Θ>> (16)< ‡ ·¡ ¢ ¡ ¢

¡ …(m)» ¿(Θ(m));m ﬂ Θ(m);N t¡¿(Θ(m));Θ(m) £>>> ¡ ¢:

N t¡¿(Θ(m));Θ(m) ; if (m;t)2= Ω ;Θ

5M. Adimy and F. Crauste A nonlinear model of cellular proliferation

‡ ·¡ ¢@ @

N(t;m)+ (V(m)N(t;m))=¡ –(m)+ﬂ m;N(t;m) N(t;m)

@t @m

8 ‡ ·¡ ¢

¡1 0 ¡1 ¡1> (17)2(g )(m)»(t;g (m))Γ ´ ¡t;g (m) ;¿(Δ(m))¡t ; if (m;t)2Ω ;> Δ<

+ ‡ ·> ¡ ¢ ¡ ¢: ‡(m)ﬂ Δ(m);N t¡¿(Δ(m));Δ(m) N t¡¿(Δ(m));Δ(m) ; if (m;t)2= Ω ;Δ

and

Z ¿(Θ(m))

P(0;m) = Γ(m):= Γ(m;a)da; (18)

0

N(0;m) = „(m); (19)

with

1

…(m)= ;

01+V(Θ(m))¿ (Θ(m))

and ¡ ¢

¡1 0 ¡1‡(m)=2(g )(m)» ¿(Δ(m));g (m) : (20)

Proof. Using(12), (13)andthedeﬁnitionsof P andN, weobtainimmediatlytheequations(18)and(19).

System (1)-(2) can be solved by using the method of characteristics. First, we obtain the following

representation of solutions of Equation (1),

‰

»(t;m)p(0;´(¡t;m);a¡t); for 0•t<a;

p(t;m;a)=

»(a;m)p(t¡a;´(¡a;m);0); for a•t:

The initial condition (12) and the boundary condition (4) give

p(t;m;a)=8 ¡ ¢

»(t;m)Γ ´(¡t;m);a¡t ; for 0•t<a;><

(21)

‡ ·¡ ¢ ¡ ¢>: »(a;m)ﬂ ´(¡a;m);N t¡a;´(¡a;m) N t¡a;´(¡a;m) ; for a•t:

Let m2[0;1] be given. By integrating Equation (1) with respect to the age, between 0 and ¿(Θ(m)), we

obtain

Z ¿(Θ(m)) ¡ ¢@ @

P(t;m)+ (V(m)p(t;m;a))da=¡?(m)P(t;m)+p(t;m;0)¡p t;m;¿(Θ(m)) :

@t @m0

One can note that

0 0¿ (Θ(m))Θ(m)V(m)¡1=¡…(m):

Since

Z ¿(Θ(m)) ¡ ¡ ¢@ @ 0 0(V(m)P(t;m))= V(m)p(t;m;a))da+¿ (Θ(m))Θ(m)V(m)p t;m;¿(Θ(m)) ;

@m @m0

and

p(t;m;¿(Θ(m)))=

8

> »(t;m)Γ(´(¡t;m);¿(Θ(m))¡t); if 0•t<¿(Θ(m));>>< ‡ ·¡ ¢

»(¿(Θ(m));m)ﬂ ´(¡¿(Θ(m));m);N t¡¿(Θ(m));´(¡¿(Θ(m));m) £

>>> ¡ ¢:

N t¡¿(Θ(m));´(¡¿(Θ(m));m) ; if ¿(Θ(m))•t;

6M. Adimy and F. Crauste A nonlinear model of cellular proliferation

then, using (4) and (8), we obtain Equation (16).

Thanks to (15) and by using the continuity of „, we show that lim n(t;m;a)=0.a!+1

So, by integrating Equation (2) with respect to the age, between 0 and +1, it follows that

‡ ·

@ @

N(t;m)+ (V(m)N(t;m))=¡ –(m)+ﬂ(m;N(t;m)) N(t;m)+n(t;m;0):

@t @m

From the equations (8) and (10), we deduce that

¡ ¢

¡1Δ(m)=´ ¡¿(Δ(m));g (m) :

Hence, from Equations (11) and (21), we obtain

8 ‡ ·¡ ¢

¡1 0 ¡1 ¡1> 2(g )(m)»(t;g (m))Γ ´ ¡t;g (m) ;¿(Δ(m))¡t ; if (m;t)2Ω ; Δ<

n(t;m;0)= ‡ ·> ¡ ¢ ¡ ¢: ‡(m)ﬂ Δ(m);N t¡¿(Δ(m));Δ(m) N t¡¿(Δ(m));Δ(m) ; if (m;t)2= Ω :Δ

Equation (17) follows immediatly.

Finally, we can remark that, if N is continuous, Condition (15) implies that the mappings (m;t) 7!‡ · ‡ ·¡ ¢ ¡ ¢

F t;m;N t¡¿(Δ(m));Δ(m) and(m;t)7!G t;m;N t¡¿(Θ(m));Θ(m) , withF :[0;+1)£[0;1]£

R!R and G:[0;+1)£[0;1]£R!R given by

8 ‡ ·¡ ¢

¡1 0 ¡1 ¡1>< 2(g )(m)»(t;g (m))Γ ´ ¡t;g (m) ;¿(Δ(m))¡t ; if (m;t)2Ω ;Δ

F(t;m;x)= (22)

> ¡ ¢:

‡(m)ﬂ Δ(m);x x; if (m;t)2= Ω ;Δ

and 8 ‡ ·¡ ¢

><…(m)»(t;m)Γ ´ ¡t;m ;¿(Θ(m))¡t ; if (m;t)2Ω ;Θ

G(t;m;x)= (23)

> ¡ ¢ ¡ ¢:

…(m)» ¿(Θ(m));m ﬂ Θ(m);x x; if (m;t)2= Ω ;Θ

are continuous.

This completes the proof.

One can remark that the solutions of Equations (17) and (19) do not depend on the proliferating popu-

lation. We extend N by setting

N(t;m)=„(m); for t2[¡¿ ;0] and m2[0;1]: (24)max

This extension does not inﬂuence our system. However, it will be useful in the following.

3 Local existence and global continuation

In this section, we are interested in proving the local existence of an integrated solution of Problem (16)-

(19). First, we consider an integrated formulation of Problem (16)-(19). We denote by C[0;1] the space

of continuous functions on [0;1], endowed with the supremum normjj:jj, deﬁned by

jjvjj= sup jv(m)j; for v2C[0;1]: (25)

m2[0;1]

Let us consider the unbounded closed linear operator A:D(A)‰C[0;1]!C[0;1] deﬁned by

' “

0 0D(A)= u2C[0;1] ;u diﬀerentiable on (0;1];u 2C(0;1]; lim V(x)u(x)=0

x!0

7M. Adimy and F. Crauste A nonlinear model of cellular proliferation

and ‰

0 0¡(–(x)+V (x))u(x)¡V(x)u(x); if x2(0;1];

Au(x)= 0¡(–(0)+V (0))u(0); if x=0:

Then, we have the following proposition, which characterise the operator (A;D(A)).

Proposition 3.1. The operator A is the inﬁnitesimal generator of the strongly continuous semigroup

(T(t)) deﬁned on C[0;1] byt‚0

(T(t)ˆ)(x)=K(t;x)ˆ(´(¡t;x)); for ˆ2C[0;1];t‚0 and x2[0;1];

where ‰ Z ?t‡ ·¡ ¢ ¡ ¢

0K(t;x)=exp ¡ – ´(¡s;x) +V ´(¡s;x) ds :

0

Proof. The proof is similar to the proof of Proposition 2.4 in [10].

We denote by C(Ω ) the space of continuous function on Ω , endowed with the normk:k , deﬁned byΘ Θ Ω£

kΥk := sup jΥ(m;a)j; for Υ2C(Ω ):Ω Θ£

(m;a)2Ω£

Now, we can consider an integrated formulation of Problem (16)-(19), given by the variation of constant

formula associated to the C -semigroup (T(t)) . That is the following deﬁnition.0 t‚0

Deﬁnition 3.1. Let Γ 2 C(Ω ) and „ be a function such that „ 2 C[0;1], with „ given by (14). AnΘ

integrated solution of Problem (16)-(19) is a continuous solution of the system

¡ ¢

N(t;m)=K(t;m)„ ´(¡t;m)

Z t ‡ ·¡ ¢ ¡ ¢

¡ K(t¡s;m)ﬂ ´(¡(t¡s);m);N s;´(¡(t¡s);m) N s;´(¡(t¡s);m) ds

(26)0

Z t ‡ ·¡ ¢

+ K(t¡s;m)F s;´(¡(t¡s);m);N s¡¿(Δ(´(¡(t¡s);m)));Δ(´(¡(t¡s);m)) ds;

0

and

¡ ¢

P(t;m)=»(t;m)Γ ´(¡t;m)

Z t ‡ ·¡ ¢ ¡ ¢

+ »(t¡s;m)ﬂ ´(¡(t¡s);m);N s;´(¡(t¡s);m) N s;´(¡(t¡s);m) ds

(27)0

Z t ‡ ·¡ ¢

¡ »(t¡s;m)G s;´(¡(t¡s);m);N s¡¿(Θ(´(¡(t¡s);m)));Θ(´(¡(t¡s);m)) ds;

0

for t‚0 and m2[0;1], where F and G are given by (22) and (23) and Γ is given by (18).

The extension given by (24) allows the second integrals, in the expressions (26) and (27), to be well

deﬁned.

In order to obtain a result of local existence for the solutions of System (26)-(27), we ﬁrst focus on

Equation (26). We show, in the next theorem, that Equation (26) has a unique local solution, which

depends continuously on the initial conditions.

8M. Adimy and F. Crauste A nonlinear model of cellular proliferation

Theorem 3.1. Assume that the mapping x7!xﬂ(m;x) is locally Lipschitz continuous for all m2[0;1],

that is, for all r >0, there exists L(r)‚0 such that

jxﬂ(m;x)¡yﬂ(m;y)j•L(r)jx¡yj; if jxj<r;jyj<r and m2[0;1]:

If Γ2 C(Ω ) and „ is a function such that „2 C[0;1], then, there exists T > 0 such that EquationΘ max

„;Γ(26) has a unique continuous solution N deﬁned on a maximal domain [0;T )£[0;1], and eithermax

„;ΓT =+1 or limsupkN (t;:)k=+1:max

¡t!Tmax

„;ΓFurthermore, N (t;:) is a continuous function of „ and Γ, in the sense that, if t2(0;T ), „ 2C[0;1]max 1

and Γ 2 C(Ω ), then there exist a continuous positive function C : [0;+1)!R and a constant " > 01 Θ

„ ;Γ22such that, for „ 2C[0;1] and Γ 2C(Ω ) such that N is deﬁned on [0;t]£[0;1] and2 Θ2

k„ ¡„ k<" and kΓ ¡Γ k <";1 2 Ω1 2 £

we get ‡ ·

„ ;Γ „ ;Γ1 1 2 2kN (s;:)¡N (s;:)k•C(t) k„ ¡„ k+kΓ ¡Γ k ; for s2[0;t]:1 2 Ω1 2 £

Proof. We put

r =k„k+1:

Let T >0 be ﬁxed. We consider the following set,

‰ ?

X(„)= N 2C([0;T]£[0;1]) ; N(0;:)=„ on [0;1] and sup jN(t;m)¡„(m)j•1 ;

(t;m)2[0;T]£[0;1]

where C([0;T]£[0;1]) is endowed with the uniform norm. X(„) is a non-empty closed convex subset of

C([0;T]£[0;1]).

We deﬁne the operator H :C([0;T]£[0;1])!C([0;T]£[0;1]) by

¡ ¢

H(N)(t;m)=K(t;m)„ ´(¡t;m)

Z t ‡ ·¡ ¢ ¡ ¢

¡ K(t¡s;m)ﬂ ´(¡(t¡s);m);N s;´(¡(t¡s);m) N s;´(¡(t¡s);m) ds

0

Z t ‡ ·¡ ¢

+ K(t¡s;m)F s;´(¡(t¡s);m);N s¡¿(Δ(´(¡(t¡s);m)));Δ(´(¡(t¡s);m)) ds:

0

H is continuous in C([0;T]£[0;1]). Our objective is to show that H is a contraction from X(„) into

itself.

Let N 2X(„). It is clear that H(N)(0;:)=„. On the other hand, we have, for (t;m)2[0;T]£[0;1],

ﬂ ﬂ ﬂ ﬂ¡ ¢

ﬂ ﬂ ﬂ ﬂH(N)(t;m)¡„(m) • K(t;m)„ ´(¡t;m) ¡„(m)

ﬂ ﬂZ t ‡ ·ﬂ ﬂ¡ ¢ ¡ ¢

ﬂ ﬂ+ K(t¡s;m)ﬂ ´(¡(t¡s);m);N s;´(¡(t¡s);m) N s;´(¡(t¡s);m) dsﬂ ﬂ

0

ﬂ ﬂZ t ‡ ·ﬂ ¡ ¢ ﬂ

ﬂ ﬂ+ K(t¡s;m)F s;´(¡(t¡s);m);N s¡¿(Δ(´(¡(t¡s);m)));Δ(´(¡(t¡s);m)) ds :ﬂ ﬂ

0

9M. Adimy and F. Crauste A nonlinear model of cellular proliferation

eSince K is continuous on [0;T]£[0;1], then there exists K‚0 such that

ejK(t;m)j•K; for (t;m)2[0;T]£[0;1]:

Since N 2X(„), then

jN(t;m)j•1+k„k=r:

fThis implies that there exists M := maxfM;k‡krL(r)g ‚ 0 such that, for (t;m) 2 [0;T]£ [0;1] and

s2[0;t], ﬂ ﬂ‡ ·¡ ¢ﬂ ﬂ

ﬂF s;´(¡(t¡s);m);N s¡¿(Δ(´(¡(t¡s);m)));Δ(´(¡(t¡s);m)) ﬂ•M;

where ﬂ ‡ ·ﬂ¡ ¢ﬂ ﬂ¡1 0 ¡1 ¡1fM := sup ﬂ2(g )(m)»(t;g (m))Γ ´ ¡t;g (m) ;¿(Δ(m))¡t ﬂ;

(m;t)2Ω¢

and ‡ is given by (20). Hence, we obtain that

ﬂ ﬂ ﬂ ¡ ¢ ﬂ

ﬂ ﬂ ﬂ ﬂ eH(N)(t;m)¡„(m) • K(t;m)„ ´(¡t;m) ¡„(m) +K(rL(r)+M)t:

¡ ¢

Let us recall that K(0;m) = 1, ´(0;m) = m and (t;m)7! K(t;m)„ ´(¡t;m) is continuous. Then, we

can choose T >0 such that

‰ ?

ﬂ ﬂ¡ ¢

ﬂ ﬂ esup K(t;m)„ ´(¡t;m) ¡„(m) +K(rL(r)+M)t <1: (28)

(t;m)2[0;T]£[0;1]

Consequently, ﬂ ﬂ

ﬂ ﬂH(N)(t;m)¡„(m) •1; for (t;m)2[0;T]£[0;1];

and H(X(„))‰X(„).

Now, we show that H is a contraction on X(„).

Let N 2X(„) and N 2X(„). Then,1 2

jH(N )(t;m)¡H(N )(t;m)j1 2

ﬂZ •t ‡ ·ﬂ ¡ ¢ ¡ ¢

ﬂ• K(t¡s;m) ﬂ ´(¡(t¡s);m);N s;´(¡(t¡s);m) N s;´(¡(t¡s);m)1 1ﬂ

0

ﬂ‚‡ · ﬂ¡ ¢ ¡ ¢

ﬂ¡ﬂ ´(¡(t¡s);m);N s;´(¡(t¡s);m) N s;´(¡(t¡s);m) ds2 2 ﬂ

ﬂ •Z t ‡ ·ﬂ ¡ ¢

ﬂ+ K(t¡s;m) F s;´(¡(t¡s);m);N s¡¿(Δ(´(¡(t¡s);m)));Δ(´(¡(t¡s);m))1ﬂ

0

‚ ﬂ‡ ·¡ ¢ ﬂ

ﬂ¡F s;´(¡(t¡s);m);N s¡¿(Δ(´(¡(t¡s);m)));Δ(´(¡(t¡s);m)) ds ;2 ﬂ

e•K(1+k‡k)L(r)T sup jN (t;m)¡N (t;m)j:1 2

(t;m)2[0;T]£[0;1]

Since r‚1 andk‡krL(r)•M, then Condition (28) implies that

e e eK(1+k‡k)L(r)T •K(1+k‡k)rL(r)T •K(rL(r)+M)T <1:

10