JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 25, 49953 997 ARTICLE NO. AY975654 Necessry nd Sufficien Condiions for Asynchronous Exponenil Growh in Age Srucured Cell Populions wih Quiescence O. Arino Lboroire de Mhemiques Appliquees, I.P.R.A. Uniersie de Pu, 64, Pu. Frnce E. Snchez Dpo. Memics, E.T.S.I. Indusriles, U.P.M., c Jose Guierrez Abscl 2, 286, Mdrid, Spin nd G. F. Webb Deprmen of Mhemics, 326 Seenson Cener, Vnderbil Uniersiy, Nshille, Tennessee 3724 Submied by Willim F. Ames Received Mrch 4, 997 A liner model on ge srucured cell populion is nlyzed. The populion is divided ino prolifering nd quiescen comprmens. Necessry nd sufficien condiions re esblished for he populion o exhibi he sympoic behvior of synchronous exponenil growh. The model is nlyzed s semigroup of liner operors which is shown o be evenully compc nd irreducible. 997 Acdemic Press. INTRODUCTION In he invesigion of cell populion dynmics i is imporn o consider he srucure of he populion wih respec o individul properies such s ge, size, or oher physicl chrcerisics. In srucured cell 499 22-247X97 $25. Copyrigh 997 by Acdemic Press All righs of reproducion in ny form reserved.
5 ARINO, SANCEZ, AND WEBB populion dynmics he propery of synchronous Ž or blnced. exponenil growh is frequenly observed. Asynchronous exponenil growh occurs when prolifering cell populion converges Žfer muliplicion by n exponenil fcor in ime. o chrcerisic disribuion of srucure h depends on he iniil disribuion of srucure hrough one-dimensionl sricly posiive projecion. This behvior mens h he ulime disribuion of he srucure of cells will be sricly posiive hrough ll possible srucure vlues no mer how he srucure is iniilly disribued. The propery of synchronous exponenil growh hs been invesiged in models of ge srucured cell populions by Webb, Clemen e l., nd Innelli 8. The propery of synchronous exponenil growh hs been invesiged in models of size srucured cell populions by Diekmnn e l. 2 nd Greiner nd Ngel 3. In mny cell populions no ll cells re progressing o miosis, bu some re in quiescen or resing se for n exended period of ime. The propery of synchronous exponenil growh hs been invesiged in models of size srucured cell populions wih prolifering nd quiescen comprmens in 57,. In his pper we develop model of n ge srucured cell populion wih prolifering nd quiescen subpopulions. We esblish necessry nd sufficien condiions on he funcions conrolling rnsiion beween he wo comprmens o ssure h he populion hs he propery of synchronous exponenil growh. These condiions hve he following inerpreion: Asynchronous exponenil growh occurs if nd only if he younges prolifering cells hve he possibiliy o rnsi o he quiescen comprmen nd he oldes quiescen cells hve he possibiliy o rnsi o he prolifering comprmen. The echniques we use o prove his resul re drwn from he heory of semigroups of posiive liner operors in Bnch lices. 2. TE MODEL In his pper we nlyze liner model of cell populion dynmics srucured by ge wih wo inercing comprmens: prolifering cells nd quiescen cells. Prolifering cells grow, divide, nd rnsi o he quiescen comprmen, wheres quiescen cells do no grow nd cn only rnsi bck nd forh o proliferion. We ssume h n individul is fully chrcerized by is ge nd he se Ž eiher prolifering or quiescen. i is in. This mens h ll quniies h deermine he developmen of n individul, such s growh nd deh res nd rnsiion res from one se o he oher, depend only on ge nd se.
CELL POPULATIONS WIT QUIESCENCE 5 We ssume h division is he only cuse of cell loss nd ll dugher cells re born in he prolifering se. An individul in he quiescen comprmen cnno divide s long s i sys in his se. We le denoe ime, ge, nd we denoe he densiies of cells in he prolifering nd he quiescen se by p, nd q, Ž,. respecively. Thus, for insnce, 2 p, d is he number of prolifering cells which ime hve ge beween nd 2. We cn now wrie he blnce equions for he wo comprmens: p p p p q,, q q p q,, pž,. 2 pž,. d, qž,., pž,. Ž., qž,. Ž.,, Ž PQ. where is he division re, is he rnsiion re from prolifering sge o he quiescen sge, nd is he rnsiion re from quiescen sge o prolifering sge. We ssume h here exiss mximl ge of division, h is, cells older hn do no conribue o he renewl of he populion. So, we simply neglec hem nd consider only he populion of prolifering nd quiescen individuls of ge less hn or equl o. Throughou he pper we mke he following ssumpions: YPOTESIS. L,. There exiss,,such h,, d. YPOTESIS 2., L Ž,.. The funcions, do no nish ideniclly. The nurl choice for he se spce is X L Ž,. L Ž,.. The soluions of he model Ž PQ. form srongly coninuous semigroup of posiive liner bounded operors U Ž.4 in X, ccording o he formul p,. U,, q,. ž / ž /
52 ARINO, SANCEZ, AND WEBB where p,., q,. re he soluions of Ž PQ. corresponding o given iniil ge disribuions pž,. Ž.; qž,. Ž.,,L Ž,.. Using he generl heory of posiive operor semigroups in Bnch lices, we cn obin he sympoic behvior of soluions of Ž PQ.. In fc, we prove here h hese soluions hve synchronous exponenil growh. This mens h here exis rel consn * nd sricly posiive rnk one projecion on X such h ž / ž / ž / * X, lim e U. * is he Mlusin prmeer nd Ž,. T he exponenil sedy se. We refer he reder o 9, for he generl heory of C-semigroups of posiive operors in Bnch lices. Asynchronous exponenil growh of he semigroup U Ž.4resuls from wo imporn properies i possesses: compcness nd irreducibiliy. The min resul we obin here is o chrcerize he irreducibiliy of he semigroup in erms of he suppor of res nd. I will be useful in he following secions o consider Ž PQ. s perurbion of wo uncoupled problems: p p p p,, pž,. 2 pž,. d, pž,. p ˆŽ., q q q,, qž,., qž,. qˆž.,. Ž P. Ž Q. Problem Ž P. cn be reduced o n inegrl equion for B p, Ž..In fc, we hve ˆpŽ. exp Ž Ž s. Ž s.. ds, pž,. ž / ž / B exp Ž s. Ž s. ds,,
ˆ CELL POPULATIONS WIT QUIESCENCE 53 where p L Ž,. is he iniil ge disribuion, nd ŽŽs.Žs.. ds B 2 e B d,. Le TŽ.4 be he semigroup ssocied for problem Ž P. in L Ž,.. TŽ.4 is posiive, evenully compc Ž compc for., nd irre- ducible, Sec..2. Also, we cn wrie ž / Ž T ˆp. Ž T ˆp. exp Ž Ž s. Ž s.. ds,. The soluion of problem Q is qˆ exp Ž s. ds, qž,. ½,, ˆ where q L Ž,. is he iniil ge disribuion. We denoe by S Ž.4 he ssocied semigroup. I is obvious h, S Ž.. Using riion of consns formul, we cn express he semigroup U Ž.4 in erms of he semigroups TŽ.4, S Ž.4, p., T ˆp TŽ s.. q, s. ds Ž 2. q,. S q ˆ SŽ s.. p, s. ds Ž 3. or, in.ž vecor form ˆp / T ž ˆp / U Ž qˆ S qˆ ž / TŽ s.. ˆp ž SŽ s. /ž. / ž qˆ/ U s ds. Ž. 3. COMPACTNESS OF TE SEMIGROUP U Le W Ž.. Žw Ž..., W ij i, j, 2 Id, be he fundmenl mrix of he liner differenil sysem p p. q q ž / ž / ž /
54 ARINO, SANCEZ, AND WEBB We mke chnge of he unknown vribles p, q ino new vribles p, q, defined by p p WŽ.. q Then PQ is rnsformed o ž / q ž / p p,, q q,, 2 p Ž,. 2 w pž,. w q Ž,. d, q,, PQ pž,. Ž., q Ž,. Ž.,. This problem cn be reduced o n inegrl equion in p Ž,., since he soluions re q Ž,. Ž., q Ž,. ½, wih Ž p,. Ž., pž,. ½ Ž p,., 2 w pž,. d 2 2 Ž p,. w w d, 2 w p, d,, where pž,. W ˆp ž / ž / Ž. q Ž,. ž qˆ / re he iniil ge disribuions.
CELL POPULATIONS WIT QUIESCENCE 55 We inroduce he noions wž.,, K ½, 2 w w 2 d, G, so, p Ž,.. is he unique soluion of p Ž,. G 2 K pž,. d,. LEMMA. The operor : X L, 2 defined by ž / Ž p,.. is liner nd bounded. Proof. ypoheses Ž., Ž 2. imply KL Ž R. L Ž R.; T, G CŽ,T.. Then T, p,. C, T nd here exiss R L Ž R. loc such h p Ž,. G RŽ s. G Ž s. ds Ž. see 8, Appendix II, Theorem. nd he lemm follows. TEOREM. The semigroup UŽ.4 is eenully compc, h is, he operors UŽ. re compc for 2. Proof. I suffices o prove h U Ž 2. is compc operor, where U Ž.4 is he semigroup ssocied wih he problem Ž PQ.. Consider he operors defined by S : L Ž,2. L Ž,., Ž S.Ž.. Ž 2.. T : L Ž,2. L Ž,2., Ž T. 2 K d. Ž S is liner, bounded, nd T is compc i is convoluion in L, see 4, Sec. 2.2, Theorem 2.5.. Therefore, T S is compc, which proves he
56 compcness of ARINO, SANCEZ, AND WEBB Ž T T S.Ž, / ž. / ŨŽ 2.. ž 4. IRREDUCIBILITY OF TE SEMIGROUP We devoe his secion o esblishing he min resul of his pper, nmely he chrcerizion of irreducibiliy of he semigroup ssocied wih Ž PQ. in erms of he suppor of he res,. DEFINITION. A C -liner semigroup TŽ.4 in he Bnch spce X is irreducible iff x X, x, nd x* X*, x*, here is such h ² x*, T x:. TEOREM 2. Under ypoheses Ž., Ž 2., he semigroup UŽ.4 is irreducible iff here re, 2 such h boh he following condiions hold,, d, Ž 3.,, d. Ž 4. 2 Proof. Sufficiency. We firs clim h if he iniil ge disribuion q, ˆ hen here exiss such h p,.. If his is no rue, we should hve, p,.. Then Therefore from which p p q. q q p q qˆž., qž,. ½,.
CELL POPULATIONS WIT QUIESCENCE 57 This is conrdicion o 3, since his implies Ž. qˆž. d qˆž. d for some, such h q nd, ˆ. Thus, wihou loss of generliy we cn suppose h iniil ge disribuion ˆp. Le Ž,. T L Ž,. L Ž,., where, re no boh zero nd le Ž ˆˆ p, q. T X wih ˆp. Cse.. Since TŽ.4 is irreducible, here is such h ², TŽ. p: ˆ, where ².,.: mens he usul duliy produc. Then, T T Ž,., U Ž ˆˆ p, q. ;, p,. ;, q,. ; ² :, p., ² ˆ:,T nd he irreducibiliy of he semigroup U Ž.4 Cse 2.. Denoe J ²Ž,. T,U Ž.Ž ˆˆ p,q. T :. Then, J ², p,. : ², q,. : ², q,. : Ž ;., S s. p., s ds ž p is proved. Ž s. pž s, s. From Ž.Ž.Ž., 2, 3, we obin for s, pž s, s. Ž TŽ s. ˆp. Ž s. ž / / s exp Ž w. dw ds d. s ž / Ž T ˆp. exp Ž Ž w. Ž w.. dw ž / s Ž T ˆp. exp Ž Ž w. Ž w.. dw
58 ARINO, SANCEZ, AND WEBB nd hen J Ž T ˆp. ž ž Ž s. exp Ž Ž w. Ž w. Noice h,,, s Ž w.. dw/ ds/ d. ž / s ž / Ž s. exp Ž w. Ž w. Ž w. dw ds exp w dw d. If we denoe his ls erm by C, using 4 we conclude h Therefore,, C. Ž ˆ. J C T p d,. Since TŽ.4 is irreducible, here is such h J. Q.E.D. Necessiy. Ž. Suppose h Ž 4. does no hold. Then, for some, we hve, for.e.,. We look for he soluion q, of problem Ž PQ. in, R, ssocied wih he iniil ge disribuion q, ˆ,, nd q, ˆ Ž,. I is srighforwrd o obin from he equions of Ž PQ. h Ž,., R, qž,., so, U Ž.4 is no irreducible. Ž b. Suppose h Ž 3. does no hold. Then, for some, we hve,,. The iniil ge disribuions ˆp, nd qˆ,, ; qˆ,,
CELL POPULATIONS WIT QUIESCENCE 59 hve he soluions of Ž PQ., pž,.,,, qž,. ½ qˆž. expž Ž s. ds., Žsince Ž q,. for ² ²,:.. Therefore, he semigroup U Ž.4 is no irreducible. The heorem is proved. The bove chrcerizion of irreducibiliy hs he following biologicl inerpreion: In order for he populion o hve dispersion of ny iniil ge disribuion in p nd q o n ulime ge disribuion hrough ll ges beween nd for boh p nd q, i is necessry nd sufficien for Ž 3. nd Ž 4. o hold. Condiion Ž 3. prohibis he quiescen populion from going exinc if q ˆ for,. Condiion Ž 4. prohibis he quiescen populion from sying for Ž,., R if q ˆ for,. 5. ASYNCRONOUS EXPONENTIAL GROWT OF TE SOLUTIONS The sympoic behvior of he semigroup U Ž.4follows immediely from Theorems nd 2 Žsee, Sec. 9.3.. TEOREM 3. Under ypoheses Ž., Ž 2., Ž 3., Ž 4., he semigroup UŽ.4 hs synchronous exponenil growh: There exiss rel consn * nd rnk one projecion on X such h ž / ž / ž / * X, lim e U. Moreoer, * Ž A. Ž he growh bound of A, where A is he infiniesiml generor of he semigroup., nd here exiss Ž,. T L Ž,. L Ž,. nd sricly posiie funcionl Ž *, *. T L Ž,. L Ž,. such h ž / ž / ž / ² : ² : X, *, *,.. The generl nlysis of sympoic behvior of soluions in he nonirreducible cse is very compliced. We complee his secion wih n exmple showing wh cn hppen when Ž 3., Ž 4. re no sisfied.
5 ARINO, SANCEZ, AND WEBB We mke he following hypohesis YPOTESIS Ž 5.. There exiss, such h,, ;,,. We will obin he sympoic behvior of he nonirreducible semigroup U Ž.4ssocied wih his problem, from he nlysis of is infiniesiml generor. The infiniesiml generor is he operor defined by wih domin ž / ž / ž / ž / A ½ T T 5 D A, X;, X, 2 d,. Firs of ll, we look for he fundmenl mrix of he differenil problem ž /. ž / ž / Ž. On,, we hve, nd hen is consn. I is esy o obin where ž / ž / WŽ., Ž b. EŽ b. db Ž b. EŽ b. db E Ž b. EŽ b. db E Ž b. EŽ b. db W E nd x ž / EŽ x. exp Ž s. Ž s. ds.
CELL POPULATIONS WIT QUIESCENCE 5 where Ž. 2 On, we hve nd hen we obin V ž / ž / VŽ., Ž. ž / Ž. s ž Ž. / exp s s ds s exp w w dw ds This implies h he fundmenl mrix Ž Id. is h h2 WŽ.,, ž h / ½ 2 h22 V WŽ.,,. Consider he eigenvlue problem The generl soluion is ž / Ž A I. ž/. ž / ž / e Ž.. T The condiion, D A provides chrcerisic equion for he deerminion of he eigenvlues :. 2 e h d,. Th is, ž Ž. / b ž ž Ž. / / ž ž / / 2 exp s s ds b exp s s ds db e exp Ž s. Ž s. ds d. Ž 4.
52 ARINO, SANCEZ, AND WEBB LEMMA 2. The chrcerisic equion Ž. 4 hs unique rel roo. Proof. Denoe by F he righ hnd side of Eq. Ž 4.. Noice h F is decresing funcion, lim F ; lim F. Then, he chrcerisic equion F hs unique rel roo. TEOREM 4. Under ypoheses Ž., Ž 2., Ž 5., he semigroup UŽ.4 hs he sympoic behior lim e U, where is one dimensionl projecion on X Ž bu no sricly posiie.. Proof. From he generl heory, Chp. 4 we cn conclude h here exiss finie rnk projecion on Xsuch h Ž. UŽ Id. Me, for some consns, M. From he soluion of he eigenvlue problem we obin h he geome Ž i.e., he dimension of ssocie eigenspce. is one. ric mulipliciy of I suffices o prove h 2 kerž A Id. kerž A Id. since hen, he lgebric mulipliciy of is lso one. Ž. 2 To obin ker A Id, we sr solving ž/ u AId, u DŽ A. Ž 5. nd hen we consider Ž. The soluion of 5 is ž / ž / ž / ž / u AId, DŽ A.. Ž 6. ž / ž / ž/ u k e Ž., where k is n rbirry consn. The soluion of Ž. 6 is hen Ž. k e Ž. ž /.ž /
CELL POPULATIONS WIT QUIESCENCE 53 T We impose, D A, 2 e h d2k e h d which, in view of Ž. 4, immediely implies k. Then, he lgebric mulipliciy of is lso one nd he heorem is proved. The clim h is no sricly posiive follows from pr Ž b. of he necessiy proof of Theorem 2. REFERENCES. P. Clemen,. J. A. M. eijmns, S. Angemen, C. J. vn Dujin, nd B. de Pger, One-Prmeer Semigroups, Norh-ollnd, Amserdm, 987. 2. O. Diekmnn,. J. A. M. eijmns, nd. R. Thieme, On he sbiliy of he cell size disribuion, J. Mh. Biol. 9 Ž 984., 227248. 3. G. Greiner nd R. Ngel, Growh of cell populions vi one-prmeer semigroups of posiive operors, in Mhemics Applied o Science, pp. 795, Acdemic Press, New York, 987. 4. G. Gripenberg, S. O. London, nd O. Sffns, Volerr Inegrls nd Funcionl Equions, Cmbridge Univ. Press., Cmbridge, UK, 99. 5. M. Gyllenberg nd G. F. Webb, Age-size srucure in populions wih quiescence, Mh. Biosci. 86 Ž 987., 6795. 6. M. Gyllenberg nd G. F. Webb, A nonliner srucured populion model of umor growh wih quiescence, J. Mh. Biol. 28 Ž 99., 67694. 7. M. Gyllenberg nd G. F. Webb, Quiescence in srucured populion dynmicsapplicions o umor growh, in Mhemicl Populion Dynmics Ž Arino e l., Eds.., Dekker, New York, 99. 8. M. Innelli, Mhemicl Theory of Age-Srucured Populion Dynmics, Appl. Mh. Monogrphs Ž Girdini, Ed.., Pis, 994. 9. R. Ngel, Ž Ed.., One-prmeer semigroups of posiive operors, in Lecure Noes in Mh., Vol. 84, Springer-Verlg, New YorkBerlin, 986.. B. Ross, Asynchronous exponenil growh in size srucured cell populion wih quiescen comprmen, in Mhemicl Populion Dynmics: Anlysis of omogeneiy Ž O. Arino e l., Eds.., Vol. 2, Wuerz Pub., Cnd, 995.. G. F. Webb, Theory of Nonliner Age-Dependen Populion Dynmics, Dekker, New York, 985.