JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

215, 499]513 Ž1997.

AY975654

Necessary and Sufficient Conditions for Asynchronous Exponential Growth in Age Structured Cell Populations with Quiescence O. Arino Laboratoire de Mathematiques Appliquees, ´ ´ I.P.R.A. Uni¨ ersite´ de Pau, 64000, Pau. France

E. Sanchez ´ Dpto. Matematicas, E.T.S.I. Industriales, U.P.M., c r Jose´ Gutierrez ´ ´ Abascal 2, 28006, Madrid, Spain

and G. F. Webb Department of Mathematics, 1326 Ste¨ enson Center, Vanderbilt Uni¨ ersity, Nash¨ ille, Tennessee 37240 Submitted by William F. Ames Received March 4, 1997

A linear model on age structured cell population is analyzed. The population is divided into proliferating and quiescent compartments. Necessary and sufficient conditions are established for the population to exhibit the asymptotic behavior of asynchronous exponential growth. The model is analyzed as a semigroup of linear operators which is shown to be eventually compact and irreducible. Q 1997 Academic Press

1. INTRODUCTION In the investigation of cell population dynamics it is important to consider the structure of the population with respect to individual properties such as age, size, or other physical characteristics. In structured cell 499 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

500

ARINO, SANCHEZ , AND WEBB ´

population dynamics the property of asynchronous Žor balanced. exponential growth is frequently observed. Asynchronous exponential growth occurs when a proliferating cell population converges Žafter multiplication by an exponential factor in time. to a characteristic distribution of structure that depends on the initial distribution of structure through a one-dimensional strictly positive projection. This behavior means that the ultimate distribution of the structure of cells will be strictly positive through all possible structure values no matter how the structure is initially distributed. The property of asynchronous exponential growth has been investigated in models of age structured cell populations by Webb w11x, Clement et al. w1x, and Iannelli w8x. The property of asynchronous exponential growth has been investigated in models of size structured cell populations by Diekmann et al. w2x and Greiner and Nagel w3x. In many cell populations not all cells are progressing to mitosis, but some are in a quiescent or resting state for an extended period of time. The property of asynchronous exponential growth has been investigated in models of size structured cell populations with proliferating and quiescent compartments in w5]7, 10x. In this paper we develop a model of an age structured cell population with proliferating and quiescent subpopulations. We establish necessary and sufficient conditions on the functions controlling transition between the two compartments to assure that the population has the property of asynchronous exponential growth. These conditions have the following interpretation: Asynchronous exponential growth occurs if and only if the youngest proliferating cells have the possibility to transit to the quiescent compartment and the oldest quiescent cells have the possibility to transit to the proliferating compartment. The techniques we use to prove this result are drawn from the theory of semigroups of positive linear operators in Banach lattices.

2. THE MODEL In this paper we analyze a linear model of cell population dynamics structured by age with two interacting compartments: proliferating cells and quiescent cells. Proliferating cells grow, divide, and transit to the quiescent compartment, whereas quiescent cells do not grow and can only transit back and forth to proliferation. We assume that an individual is fully characterized by its age and the state Žeither proliferating or quiescent . it is in. This means that all quantities that determine the development of an individual, such as growth and death rates and transition rates from one state to the other, depend only on age and state.

501

CELL POPULATIONS WITH QUIESCENCE

We assume that division is the only cause of cell loss and all daughter cells are born in the proliferating state. An individual in the quiescent compartment cannot divide as long as it stays in this state. We let t denote time, a age, and we denote the densities of cells in the proliferating and the quiescent state by pŽ a, t . and q Ž a, t ., respectively. Thus, for instance, Haa12 pŽ a, t . da is the number of proliferating cells which at time t have age between a1 and a2 . We can now write the balance equations for the two compartments:

­p ­t ­q ­t

q q

­p ­a ­q ­a

s ym Ž a . p y s Ž a . p q t Ž a . q, s s Ž a . p y t Ž a . q, a1

p Ž 0, t . s 2

H0

0 - a - a1 , t ) 0

m Ž a . p Ž a, t . da,

q Ž 0, t . s 0,

0 - a - a1 , t ) 0

t)0

Ž PQ .

t)0

p Ž a, 0 . s w Ž a . ,

0 - a - a1

q Ž a, 0 . s c Ž a . ,

0 - a - a1 ,

where m is the division rate, s is the transition rate from proliferating stage to the quiescent stage, and t is the transition rate from quiescent stage to proliferating stage. We assume that there exists a maximal age of division a1 , that is, cells older than a1 do not contribute to the renewal of the population. So, we simply neglect them and consider only the population of proliferating and quiescent individuals of age less than or equal to a1. Throughout the paper we make the following assumptions:

m g L`q Ž0, a1 .. There exists e 0 , 0 - e 0 - a1 , such that

HYPOTHESIS ŽH1..

;eg x 0, e 0 w ,

a1

Ha yem Ž a. da ) 0. 1

HYPOTHESIS ŽH2.. identically.

s , t g L1q Ž0, a1 .. The functions s , t do not ¨ anish

The natural choice for the state space is X s L1 Ž0, a1 . = L1 Ž0, a1 .. The solutions of the model ŽPQ. form a strongly continuous semigroup of positive linear bounded operators UŽ t .4t G 0 in X, according to the formula UŽ t .

p Ž ., t . w s , c q Ž ., t .

ž/ ž

/

t G 0,

ARINO, SANCHEZ , AND WEBB ´

502

where pŽ., t ., q Ž., t . are the solutions of ŽPQ. corresponding to given initial age distributions p Ž a, 0 . s w Ž a . ;

q Ž a, 0 . s c Ž a . ,

a)0

w , c g L1 Ž0, a1 .. Using the general theory of positive operator semigroups in Banach lattices, we can obtain the asymptotic behavior of solutions of ŽPQ.. In fact, we prove here that these solutions have asynchronous exponential growth. This means that there exist a real constant l* and a strictly positive rank one projection P on X such that ;

w g X, c

lim eyl *t U Ž t .

ž/

tªq`

w w sP . c c

ž/ ž/

l* is the Maltusian parameter and P Ž w , c .T the exponential steady state. We refer the reader to w9, 1x for the general theory of C0-semigroups of positive operators in Banach lattices. Asynchronous exponential growth of the semigroup UŽ t .4t G 0 results from two important properties it possesses: compactness and irreducibility. The main result we obtain here is to characterize the irreducibility of the semigroup in terms of the support of rates s and t . It will be useful in the following sections to consider ŽPQ. as a perturbation of two uncoupled problems: ­p ­t

q

­p ­a

s ym Ž a . p y s Ž a . p,

p Ž 0, t . s 2

a1

H0

m Ž a . p Ž a, t . da,

p Ž a, 0 . s ˆ p Ž a. ,

­q ­t

q

­q ­a

0 - a - a1 , t ) 0

Ž P.

t)0

0 - a - a1

s yt Ž a . q,

q Ž 0, t . s 0,

0 - a - a1 , t ) 0

Ž Q.

t)0

q Ž a, 0 . s qˆŽ a . ,

0 - a - a1 .

Problem ŽP. can be reduced to an integral equation for B Ž t . s pŽ0, t .. In fact, we have

¡ˆp Ž a y t . exp yH

p Ž a, t . s~

ž ž

a

ayt

Ž m Ž s . q s Ž s . . ds ,

/

¢B Ž t y a. exp yH Ž m Ž s . q s Ž s . . ds / , a

0

a)t a - t,

503

CELL POPULATIONS WITH QUIESCENCE

where ˆ p g L1 Ž0, a1 . is the initial age distribution, and a1

BŽ t. s 2

H0

m Ž a . eyH 0 Ž m Ž s.q s Ž s.. d s B Ž t y a . da, a

a - t.

Let T Ž t .4t G 0 be the semigroup associated for problem ŽP. in L1 Ž0, a1 .. T Ž t .4t G 0 is positive, eventually compact Žcompact for t G a1 ., and irreducible w1, Sect. 10.2x. Also, we can write a

Ž T Ž t . ˆp . Ž a. s Ž T Ž t y a. ˆp . Ž 0 . exp yH Ž m Ž s . q s Ž s . . ds ,

ž

/

0

t ) a.

Ž 1. The solution of problem ŽQ. is

½

q Ž a, t . s

a t Ž s . ds . , qˆŽ a y t . exp Ž yHayt 0,

a)t a - t,

where qˆ g L1 Ž0, a1 . is the initial age distribution. We denote by  SŽ t .4t G 0 the associated semigroup. It is obvious that ; t ) a1 , SŽ t . s 0. Using a ¨ ariation of constants formula, we can express the semigroup UŽ t .4t G 0 in terms of the semigroups T Ž t .4t G 0 ,  SŽ t .4t G 0 , p Ž ., t . s T Ž t . ˆ pq q Ž ., t . s S Ž t . qˆ q

t

H0 T Ž t y s . Ž t Ž .. q Ž ., s . . ds t

H0 S Ž t y s . Ž s Ž .. p Ž ., s . . ds

Ž 2. Ž 3.

or, in vector form UŽ t .

T Ž t. ˆp s qˆ 0

ž/ ž

q

t

H0

ž

0 SŽ t .

ˆp qˆ

/ž /

T Ž t y s. 0

0 SŽ t y s.

/ž

0

s Ž ..

t Ž .. ˆp UŽ s. ds. qˆ 0

/ ž/

3. COMPACTNESS OF THE SEMIGROUP UŽ t . Let W Ž.. s Ž wi j Ž... i, js1, 2 , W Ž0. s Id, be the fundamental matrix of the linear differential system pX Ž a . q9 Ž a .

ž / ž s

ym Ž a . y s Ž a . s Ž a.

t Ž a. yt Ž a .

p . q

/ž /

ARINO, SANCHEZ , AND WEBB ´

504

We make a change of the unknown variables p, q into new variables ˜ p, q, ˜ defined by

˜p p s W Ž a. . q q˜

ž/

ž/ Then ŽPQ. is transformed to

­˜ p

q

­t ­ q˜

q

­t

­˜ p ­a ­ q˜ ­a

s 0,

0 - a - a1 , t ) 0

s 0,

0 - a - a1 , t ) 0 a1

˜p Ž 0, t . s 2H m Ž a. Ž w11 Ž a. ˜p Ž a, t . q w12 Ž a. q˜Ž a, t . . da, 0

q˜Ž 0, t . s 0,

t)0 &

t)0

Ž PQ .

˜p Ž a, 0 . s w Ž a. ,

0 - a - a1

q˜Ž a, 0 . s c Ž a . ,

0 - a - a1 .

This problem can be reduced to an integral equation in ˜ pŽ0, t ., since the solutions are

with

q a y t , 0. s c Ž a y t . , 0,

a)t a-t

˜p Ž a y t , 0 . s w Ž a y t . , ˜p Ž 0, t y a. ,

a)t a-t

q˜Ž a, t . s

½ ˜Ž

˜p Ž a, t . s

½

¡2H m Ž a. w t

0

q2H t ˜p Ž 0, t . s~

a1

11

p Ž 0, t y a . da Ž a. ˜

m Ž a . w 11 Ž a . w Ž a y t . q w 12 Ž a . c Ž a y t . da, t - a1

¢2H

a1

0

m Ž a . w 11 Ž a . ˜ p Ž 0, t y a . da,

t ) a1 ,

where

w Ž a. c Ž a.

ž / ž s

˜p Ž a, 0 . ˆp Ž a. s Wy1 Ž a . q˜Ž a, 0 . qˆŽ a .

are the initial age distributions.

/

ž /

505

CELL POPULATIONS WITH QUIESCENCE

We introduce the notations K Ž a. s

Gwc

¡ ¢0,

2H Ž t . s~ t

a1

½

m Ž a . w 11 Ž a . , 0,

a g w 0, a1 x a ) a1

m Ž a . w 11 Ž a . w Ž a y t . q w 12 Ž a . c Ž a y t . da,

t - a1 t ) a1

so, ˜ pŽ0, .. is the unique solution of t

˜p Ž 0, t . s Gwc Ž t . q 2H K Ž a. ˜p Ž 0, t y a. da, 0

t ) 0.

LEMMA 1. The operator H : X ª L1 Ž0, 2 a1 . defined by H

w s˜ p Ž 0, .. c

ž/

is linear and bounded. Proof. Hypotheses ŽH1., ŽH2. imply ;T ) 0, Gwc g C Ž w 0, T x . .

K g L` Ž Rq . l L1 Ž Rq . ;

Then ;T ) 0, ˜ pŽ0, .. g C Žw0, T x. and there exists R g L1l o c ŽRq . such that t

˜p Ž 0, t . s Gwc Ž t . y H R Ž t y s . Gwc Ž s . ds 0

Žsee w8, Appendix II, Theorem 1.1x. and the lemma follows. THEOREM 1. The semigroup UŽ t .4t G 0 is e¨ entually compact, that is, the operators UŽ t . are compact for t G 2 a1. Proof. It suffices to prove that U˜Ž2 a1 . is a compact& operator, where U˜Ž t .4t G 0 is the semigroup associated with the problem ŽPQ.. Consider the operators defined by S : L1 Ž a1 , 2 a1 . ª L1 Ž 0, a1 . ,

Ž Sc . Ž .. s c Ž 2 a1 y ..

T : L1 Ž 0, 2 a1 . ª L1 Ž a1 , 2 a1 . , a1

Ž Tw . Ž t . s 2H K Ž a. w Ž t y a. da. 0

S is linear, bounded, and T is compact Žit is a convolution in L1 , see w4, Sect. 2.2, Theorem 2.5x.. Therefore, H (T ( S is compact, which proves the

ARINO, SANCHEZ , AND WEBB ´

506 compactness of

U˜Ž 2 a1 .

T w s Ž H (T ( S . Ž w , c . . c 0

ž/ ž

/

4. IRREDUCIBILITY OF THE SEMIGROUP We devote this section to establishing the main result of this paper, namely the characterization of irreducibility of the semigroup associated with ŽPQ. in terms of the support of the rates s , t . DEFINITION. A C0-linear semigroup T Ž t .4t G 0 in the Banach space X is irreducible iff ; x g Xq, x / 0, and x* g X *q, x* / 0, there is t ) 0 such that ² x*, T Ž t . x : ) 0. THEOREM 2. Under Hypotheses ŽH1., ŽH2., the semigroup UŽ t .4t G 0 is irreducible iff there are e 1 ) 0, e 2 ) 0 such that both the following conditions hold a1

;eg x 0, e 1 w ,

Ha yet Ž a. da ) 0,

Ž H3.

1

;eg x 0, e 2 w ,

e

H0 s Ž a. da ) 0.

Ž H4.

Proof. Sufficiency. We first claim that if the initial age distribution qˆ / 0, then there exists t 0 ) 0 such that pŽ., t 0 . / 0. If this is not true, we should have ; t G 0, pŽ., t . s 0. Then 0s

­p ­t

q

­p ­a

s t Ž a . q.

Therefore

­q ­t

q

­q ­a

s s Ž a. p y t Ž a. q s 0

from which q Ž a, t . s

q a y t. , 0,

½ ˆŽ

a)t a - t.

507

CELL POPULATIONS WITH QUIESCENCE

This is a contradiction to ŽH3., since this implies 0s

a1ya 0

Ha ya yet Ž a 1

0

q t . qˆŽ a0 . dt s

0

a1

Ha yeqˆŽ a . t Ž a. da ) 0 0

1

for some a0 g w0, a1 x such that qˆŽ a0 . / 0 and e gx0, e 1 w. Thus, without loss of generality we can suppose that initial age distribution ˆ p / 0. Let Ž w , c .T g L`q Ž0, a1 . = L`q Ž0, a1 ., where w , c are not both zero and let Ž ˆ p, qˆ.T g X with ˆ p / 0. Case 1. w / 0. Since T Ž t .4t G 0 is irreducible, there is t 0 G 0 such that ² w , T Ž t 0 . ˆ p : ) 0, where ²., .: means the usual duality product. Then,

¦Ž w , c .

T

, U Ž t0 . Ž ˆ p, qˆ.

T

;s¦w , p Ž ., t .;q¦c , q Ž ., t .; 0

0

G² w , p Ž ., t 0 .: G² w , T Ž t 0 . ˆ p: ) 0 and the irreducibility of the semigroup UŽ t .4t G 0 is proved. Case 2. c / 0. Denote J Ž t . s ²Ž w , c .T , UŽ t .Ž ˆ p, qˆ.T :. Then, J Ž t . s² w , p Ž ., t .: q² c , q Ž ., t .: G² c , q Ž ., t .:

¦

G c, s

a1

H0

t

;

H0 S Ž t y s . Ž s Ž .. p Ž ., s . . ds

c Ž a.

t

žH

tya

s Ž a y t q s . p Ž a y t q s, s . a

Haytqst Ž w . dw

=exp y

ž

/ /

ds da.

From Ž1., Ž2., Ž3., we obtain for a - t - a q s, p Ž a y t q s, s . G Ž T Ž s . ˆ p. Ž a y t q s. s Ž T Ž t y a. ˆ p . Ž 0 . exp y

ž

s ŽT Ž t. ˆ p . Ž a . exp

žH

aytqs

H0

a

aytqs

Ž m Ž w . q s Ž w . . dw

Ž m Ž w . q s Ž w . . dw

/

/

ARINO, SANCHEZ , AND WEBB ´

508 and then JŽ t. G

a1

H0 ?

ž

c Ž a. Ž T Ž t . ˆ p . Ž a. t

a

Htyas Ž a y t q s . exp Haytqs Ž m Ž w . q s Ž w .

ž

yt Ž w . . dw ds da.

/ /

Notice that, ;a g w0, a1 x, t

a

Htyas Ž a y t q s . exp Haytqs Ž m Ž w . q s Ž w . y t Ž w . . dw

ž

a

H0 t Ž w . dw

G exp y

ž

/

?

/

ds

a

H0 s Ž a . d a .

If we denote this last term by C Ž a., using ŽH4. we conclude that ;a g w 0, a1 x ,

C Ž a . ) 0.

Therefore JŽ t. G

a1

H0

C Ž a. c Ž a. Ž T Ž t . ˆ p . Ž a . da,

t ) a1 .

Since T Ž t .4t G 0 is irreducible, there is t 0 ) a1 such that J Ž t 0 . ) 0. Q.E.D. Necessity. Ža. Suppose that ŽH4. does not hold. Then, for some e ) 0, we have s Ž a. s 0, for a.e. a g w0, e x. We look for the solution q Ž a, t . of problem ŽPQ. in w0, e x = Rq, associated with the initial age distribution qˆŽ a. s 0, a g w0, e x, and qˆŽ a. / 0, a g Ž e , a1 x. It is straightforward to obtain from the equations of ŽPQ. that ; Ž a, t . g w 0, e x = Rq ,

q Ž a, t . s 0,

so, UŽ t .4t G 0 is not irreducible. Žb. Suppose that ŽH3. does not hold. Then, for some e ) 0, we have t Ž a. s 0, a g w a1 y e , a1 x. The initial age distributions ˆ p s 0, and qˆŽ a . s 0, a g w 0, a1 y e x ;

qˆŽ a . / 0, a g w a1 y e , a1 x

509

CELL POPULATIONS WITH QUIESCENCE

have the solutions of ŽPQ., p Ž a, t . s 0, q Ž a, t . s

½

0 - a - a1 , t G 0

0, a qˆŽ a y t . exp Ž yHay t t Ž s . ds . ,

a-t a)t

Žsince t Ž a. q Ž a, t . s 0 for 0 - a² a² a1 , t :0.. Therefore, the semigroup UŽ t .4t G 0 is not irreducible. The theorem is proved. The above characterization of irreducibility has the following biological interpretation: In order for the population to have a dispersion of any initial age distribution in p and q to an ultimate age distribution through all ages between 0 and a1 for both p and q, it is necessary and sufficient for ŽH3. and ŽH4. to hold. Condition ŽH3. prohibits the quiescent population from going extinct if qˆŽ a. s 0 for a g w0, a1 y e x. Condition ŽH4. prohibits the quiescent population from staying at 0 for Ž a, t . g w0, e x = Rq if qˆŽ a. s 0 for a g w0, e x.

5. ASYNCHRONOUS EXPONENTIAL GROWTH OF THE SOLUTIONS The asymptotic behavior of the semigroup UŽ t .4t G 0 follows immediately from Theorems 1 and 2 Žsee w1, Sect. 9.3x.. THEOREM 3. Under Hypotheses ŽH1., ŽH2., ŽH3., ŽH4., the semigroup UŽ t .4t G 0 has asynchronous exponential growth: There exists a real constant l* and a rank one projection P on X such that ;

w g X, c

lim eyl *t U Ž t .

ž/

tªq`

w w sP . c c

ž/ ž/

Moreo¨ er, l* s v 0 Ž A. Ž the growth bound of A, where A is the infinitesimal generator of the semigroup., and there exists Ž F, C .T g L1q Ž0, a1 . = L1q Ž0, a1 . and a strictly positi¨ e functional Ž F*, C*.T g L`q Ž0, a1 . = L`q Ž0, a1 . such that ;

w g X, c

ž/

P

w F s w ² F*, w : q ² C*, c : x . . c C

ž/

ž /

The general analysis of asymptotic behavior of solutions in the nonirreducible case is very complicated. We complete this section with an example showing what can happen when ŽH3., ŽH4. are not satisfied.

ARINO, SANCHEZ , AND WEBB ´

510

We make the following hypothesis HYPOTHESIS ŽH5.. There exists a0 gx0, a1w such that ;a g w 0, a0 x , m Ž a . s 0;

;a g w a0 , a1 x , t Ž a . s 0.

We will obtain the asymptotic behavior of the nonirreducible semigroup UŽ t .4t G 0 associated with this problem, from the analysis of its infinitesimal generator. The infinitesimal generator is the operator defined by A

w w9 mys sy q s c c9

t ? w yt c

ž/ ž / ž

/ ž /

with domain T

½

T

D Ž A . s Ž w , c . g X ; Ž w 9, c 9 . g X , w Ž 0 . a1

s2

H0

5

m Ž a . w Ž a . da, c Ž 0 . s 0 .

First of all, we look for the fundamental matrix of the differential problem

w9 ym y s s s c9

ž / ž

t ? w . yt c

/ ž /

Ž1. On w0, a0 x, we have m Ž a. s 0, and then w q c is constant. It is easy to obtain

w Ž a. c Ž a.

ž /

s W Ž a.

w Ž 0. , c Ž 0.

ž /

where a

W Ž a . sE Ž a .

y1



H0 t Ž b . E Ž b . db

1q

E Ž a . y1y

a

H0 t Ž b . E Ž b . db

a

H0 t Ž b . E Ž b . db E Ž a. y

and E Ž x . s exp

x

žH Ž 0

s Ž s . q t Ž s . . ds .

/

a

H0 t Ž b . E Ž b . db

0

511

CELL POPULATIONS WITH QUIESCENCE

Ž2. On w a0 , a1 x we have t Ž a. s 0 and then we obtain

w Ž a. c Ž a.

ž /

s V Ž a.

w Ž a0 .

ž / c Ž a0 .

,

where

V Ž a. s



ž

a

a

Ha Ž m Ž s . q s Ž s . . ds

exp y

0

s

/

0

Ha s Ž s . exp yHa Ž m Ž w . q s Ž w . . dw

ž

0

0

/

ds

1

0

.

This implies that the fundamental matrix H Ž H Ž0. s Id. is H Ž a. s

ž

h11 Ž a .

h12 Ž a .

h 21 Ž a .

h 22 Ž a .

/ ½ s

W Ž a. ,

a g w 0, a0 x

V Ž a . W Ž a0 . ,

a g w a0 , a1 x .

Consider the eigenvalue problem

Ž A y lI .

w 0 s . c 0

ž/ ž/

The general solution is

w Ž a. c Ž a.

ž /

s eyl a H Ž a .

w Ž 0. . c Ž 0.

ž /

The condition Ž w , c .T g DŽ A. provides a characteristic equation for the determination of the eigenvalues l: a1

w Ž 0. s 2

H0

eyl a h11 Ž a . m Ž a . w Ž 0 . da,

c Ž 0 . s 0.

That is, a0

1 s 2 exp y

ž

H0

= 1q

ž

=

ž

a1

Ha

0

a0

H0

Ž s Ž s . q t Ž s . . ds

t Ž b . exp

b

žH Ž 0

eyl am Ž a . exp y

ž

/

s Ž s . q t Ž s . . ds db

/ /

a

Ha Ž m Ž s . q s Ž s . . ds 0

/ /

da .

Ž 4.

ARINO, SANCHEZ , AND WEBB ´

512

˜. LEMMA 2. The characteristic equation Ž4. has a unique real root l Proof. Denote by F Ž l. the right hand side of Eq. Ž4.. Notice that F is a decreasing function, lim F Ž l . s q`;

lim F Ž l . s 0.

lªy`

l ªq`

˜ - 0. Then, the characteristic equation F Ž l. s 1 has a unique real root l THEOREM 4. Under Hypotheses ŽH1., ŽH2., ŽH5., the semigroup UŽ t .4t G 0 has the asymptotic beha¨ ior ˜

˜, lim eyl t U Ž t . s P

tªq`

˜ is a one dimensional projection on X Ž but not strictly positi¨ e .. where P Proof. From the general theory w11, Chap. 4x we can conclude that ˜ / 0 on X such that there exists a finite rank projection P ˜

˜ . F Me e Ž ly e .t , U Ž t . Ž Id y P

tG0

for some constants e ) 0, Me ) 0. From the solution of the eigenvalue problem we obtain that the geomet˜ Ži.e., the dimension of associate eigenspace. is one. ric multiplicity of l It suffices to prove that 2

˜ Id. s ker Ž A y l˜ Id. ker Ž A y l ˜ is also one. since then, the algebraic multiplicity of l 2 ˜ Ž . To obtain ker A y l Id , we start solving

Ž A y l˜ Id. ¨u s 00 ,

ž ¨u / g D Ž A.

ž / ž /

Ž 5.

and then we consider

w

w g D Ž A. . c

Ž A y l˜ Id. c s ¨u ,

ž/

ž/

ž /

The solution of Ž5. is

ž ¨u / s e

yl˜ a

H Ž a.

k , 0

ž/

where k is an arbitrary constant. The solution of Ž6. is then

w Ž a. c Ž a.

ž /

˜

s eyl a H Ž a .

ž

w Ž 0 . q ak . 0

/

Ž 6.

CELL POPULATIONS WITH QUIESCENCE

513

We impose Ž w , c .T g DŽ A.,

w Ž 0. s 2 w Ž 0.

a1

H0

˜

eyl a h11 Ž a . m Ž a . da q 2 k

a1

H0

˜

aey l a h11 Ž a . m Ž a . da

which, in view of Ž4., immediately implies k s 0. Then, the algebraic ˜ is also one and the theorem is proved. The claim that P multiplicity of l is not strictly positive follows from part Žb. of the necessity proof of Theorem 2.

REFERENCES 1. P. Clement, H. J. A. M. Heijmans, S. Angement, C. J. van Dujin, and B. de Pagter, ‘‘One-Parameter Semigroups,’’ North-Holland, Amsterdam, 1987. 2. O. Diekmann, H. J. A. M. Heijmans, and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 Ž1984., 227]248. 3. G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, in ‘‘Mathematics Applied to Science,’’ pp. 79]105, Academic Press, New York, 1987. 4. G. Gripenberg, S. O. London, and O. Staffans, ‘‘Volterra Integrals and Functional Equations,’’ Cambridge Univ. Press., Cambridge, UK, 1990. 5. M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Math. Biosci. 86 Ž1987., 67]95. 6. M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol. 28 Ž1990., 671]694. 7. M. Gyllenberg and G. F. Webb, Quiescence in structured population dynamics}Applications to tumor growth, in ‘‘Mathematical Population Dynamics’’ ŽArino et al., Eds.., Dekker, New York, 1991. 8. M. Iannelli, ‘‘Mathematical Theory of Age-Structured Population Dynamics,’’ Appl. Math. Monographs ŽGiardini, Ed.., Pisa, 1994. 9. R. Nagel, ŽEd.., One-parameter semigroups of positive operators, in ‘‘Lecture Notes in Math.,’’ Vol. 1184, Springer-Verlag, New YorkrBerlin, 1986. 10. B. Rossa, Asynchronous exponential growth in a size structured cell population with quiescent compartment, in ‘‘Mathematical Population Dynamics: Analysis of Homogeneity’’ ŽO. Arino et al., Eds.., Vol. 2, Wuerz Pub., Canada, 1995. 11. G. F. Webb, ‘‘Theory of Nonlinear Age-Dependent Population Dynamics,’’ Dekker, New York, 1985.