Existence positivity and stability for a nonlinear model of cellular proliferation

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Existence, positivity and stability for a nonlinear model of cellular proliferation? Mostafa Adimy† and Fabien Crauste‡ Year 2004 Laboratoire de Mathematiques Appliquees, FRE 2570 Universite de Pau et des Pays de l'Adour, Avenue de l'universite, 64000 Pau, France Abstract In this paper, we investigate a system of two nonlinear partial differential equations, arising from a 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 G0-phase).

  • univ-pau

  • cells just after

  • cellular proliferation

  • after division

  • mature stem cells

  • partial differential equations

  • nonlinear analysis

  • universite de pau et des pays de l'adour


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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 (

(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