Universite Pierre et Marie Curie – Paris 6

Laboratoire Jacques-Louis Lions

Age-Structured Nonlinear Renewal

Equations

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presentee et soutenue publiquement le

pour l’obtention du grade du

Docteur de l’Universite Pierre et Marie Curie-Paris 6(specialite Mathematiques Appliquees)

par

Suman Kumar TUMULURI

Composition du jury

Rapporteurs : AdimurthiMostafa Adimy

Examinateurs : Benoıt Perthame (Directeur)Jean-Pierre FrancoiseAdimurthiJerome JaffreMostafa AdimyPhilippe Michel

Laboratoire Jacques-Louis Lions – UMR 7598

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iA knowledgementsIt gives me immense pleasure to express my gratitude to Prof. Benoît Perthame. Iwould not have done whatever little bit of mathemati s I ould do during my PhD timewithout his support and en ouragement. He is always with me when I am down, espe iallywhen I lose myself 'among' lumsy symbols and strange graphs. He is an inspirational andhighly motivative mathemati ian for several young math students like me. I �nd my selflu ky to get an opportunity to work with him. He has lot of patien e to bear me, who hasvery little memory.I would like to deeply thank Prof. Jér�me Ja�ré for his support. The hospitality he hadshown to me during my early days in Paris is unforgettable.I would like to thank Prof. Adimurthy for providing su h an ex ellent opportunity to workwith Prof. Perthame. He is one of the typi al professors in TIFR who has good intera -tion with students and en ourages us to ask questions. When we ask him any questions,he patiently listens and rapidly answers. He taught me two ourses and one of them isMeasure Theory, whi h was a nightmare for me before I joined in TIFR. We had severalfruitful dis ussions during his visit to Paris in summer 2007.I would like to thank Prof. Veerappa Gowda for his valuable suggestions he had given mein numeri al omputing. I express my sin ere thanks to my tea hers Prof. V. Kannan,Prof. Amaranath in Hyderabad Central University, who taught me basi ourses in RealAnalysis, Topology and PDE, �uid dynami s. I would like to thank Mr. M. Uma Shankar,one of my favorite maths tea hers. I am grateful to Mr. M. Nageswara Rao for providingme an opportunity to tea h mathemati s and physi s at high s hool level in his institute.I would like to sin erely thank CEFIPRA for providing me s holarship during my PhDtenure.I would like to thank the Se retaries of LJLL and system administrator Dadras for histe hni al support. It is a great pleasure to remember my friends in this o asion. I wouldlike to thank Bhanu, Chandra, Rama Rao, Srinu in Guntur, Kavitha, Prashant, Srikanth,Vijay who were my lassmates in HCU, Dr. Mallikarjuna Rao, Mousomi, Pradeep, Su-riya Prabha in TIFR, Prashant Kandalla, Deepak, J.V.S. Surya, Manohar, Praneel, RaviTeja, Senthil, Sri Ram, Dr. Suresh Kumar, Vani in Maison de l'Inde, and Ebde for theirsupport. I would like thank Dr. Vin ent Calvez and Dr. Nejla who taught me how to typein LATEXwith great patien e.Last but not the least, it gives me immense pleasure to express my thanks to my parents,Vijaya Lakshmi and Madhuri for their emotional support and onstant en ouragement.

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To all my tea hers

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Table des matièresIntrodu tion 11 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Population dynami s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The model and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1 Existen e, uniqueness and uniform bound . . . . . . . . . . . . . . 73.2 Asymptoti behavior (perturbation result) . . . . . . . . . . . . . . 73.3 Asymptoti behavior ( on ave birth term) . . . . . . . . . . . . . . 83.4 Asymptoti behavior (redu tion to ODEs) . . . . . . . . . . . . . . 84 Existen e, uniqueness of the steady states and the linear stability . . . . . 94.1 Existen e of nonzero steady states . . . . . . . . . . . . . . . . . . . 104.2 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Numeri al study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Age-stru tured Renewal Equations 171.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Examples of Age Stru tured Models . . . . . . . . . . . . . . . . . . . . . . 191.3 Assumptions and Eigenelements . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Existen e of Solutions to the Linear Problem . . . . . . . . . . . . . . . . . 251.5 Nonlinear Problem, uniform bounds . . . . . . . . . . . . . . . . . . . . . 261.6 A Case with asymptoti De oupling . . . . . . . . . . . . . . . . . . . . . . 291.6.1 A parti ular solution . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6.2 Long time de oupling . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7 A on ave nonlinearity on birth term . . . . . . . . . . . . . . . . . . . . . 311.8 Stability for nonlinearities redu ing to ODE systems . . . . . . . . . . . . . 341.8.1 Redu tion to a system of ODE (Example 1) . . . . . . . . . . . . . 341.8.2 Stability of the linearized system (Example 1) . . . . . . . . . . . . 361.8.3 Nonlinear stability (Example 1) . . . . . . . . . . . . . . . . . . . . 37v

vi Table des matières1.8.4 Nonlinear stability (Example 2) . . . . . . . . . . . . . . . . . . . . 391.8.5 Nonlinear stability (Example 3) . . . . . . . . . . . . . . . . . . . . 411.8.6 Nonlinear stability (Kerma k-M Kendri k model) . . . . . . . . . . 421.8.7 Some numeri al results . . . . . . . . . . . . . . . . . . . . . . . . . 432 Steady State Analysis 472.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 Existen e of nontrivial steady state . . . . . . . . . . . . . . . . . . . . . . 492.3 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.2 Examples of linear stability . . . . . . . . . . . . . . . . . . . . . . 542.3.3 Examples of instability . . . . . . . . . . . . . . . . . . . . . . . . . 562.4 A model with onservation of total population . . . . . . . . . . . . . . . . 582.4.1 Existen e and uniqueness of steady state . . . . . . . . . . . . . . . 592.4.2 Linear stability revisited . . . . . . . . . . . . . . . . . . . . . . . . 602.4.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.5 Numeri al results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Numeri al Study 713.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2 First order numeri al study . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.2 A �rst order numeri al s heme and a priori estimates . . . . . . . . 733.2.3 Convergen e of the s heme and a proof of the main result . . . . . . 803.3 A se ond order s heme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.4 Numeri al results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 834 Perspe tives 894.1 Non- onstant growth rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2 Size stru ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3 Nonlinear �uxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Bibliographie 91

Introdu tionRésuméLes équations stru turées apparaissent dans de nombreux domaines de la biologie despopulations. La limitation des ressour es, introduits par Verhulst, onduisent à des mo-dèles ave des non-linéaritées sous formes intégrales. Les équations stru turées en âgesemblent les plus simples pour ommen er.Le hapitre 1 présente de nombreux exemples issus de l'épidémiologie, l'é ologie, l'on- ologie...et Il donne également des résultats généraux de onvergen e vers l'état station-naire non-nul par des méthodes de perturbation, d'entropie ou de rédu tion à des systèmesplus simples. On ne s'attend toutefois pas à des omportement toujours si simples.Le hapitre 2 étudie la stabilité linéaire de l'état stationnaire ave des hypothèsespermettant d'établir qu'il est unique. Ce i onduit à un problème spe tral que l'on nepeut résoudre analytiquement ou lassi�er en général. Nous donnons diverses stru turesmontrant que l'état stationnaire peut être stable ou instable (même dans le as de termesde naissan e dé roissants). Dans e adre on retrouve numériquement des solutions pé-riodiques stables déjà mises en éviden e par divers auteurs.Le hapitre 3 s'applique à l'étude de onvergen e, dans un adre général, des s hémasnumériques utilisés auparavant. Les di� ultés i i proviennent du terme de naissan e aubord non-linéaire et de l'absen e de bornes BV dans la variable naturelle. Ce i oblige àpasser par des estimations BV en temps a�n d'en déduire de la ompa ité né essaire àpasser à la limite. Les tests numériques montrent qu'un s héma d'ordre deux est né essairepour apturer les os illations transitoires générées par la nonlinéarité.

1

2 Introdu tionAbstra tThe stru tured equations appear in many areas of population biology. The limitationof resour es, introdu ed by Verhulst, leading to models with non-linear integral forms.The age stru tured equations seem the easiest to begin with.Chapter 1 presents several examples from epidemiology, e ology, on ology, et . It alsogives general results of onvergen e to the non-zero steady state by perturbation methods,entropy or redu tion to a simpler system. We do not anti ipate this kind of simple beha-vior always.Chapter 2 examines the stability of steady state with su� ient hypothesis for its uni-queness. This leads to a spe tral problem that an not be solved analyti ally or rankin general. We give various stru tures showing that the state may be stable or unstable(even in terms of low birth). In this ontext we numeri ally �nd stable periodi solutions,already put in eviden e by various authors.Chapter 3 applies to the study of onvergen e in a general, the numeri al s hemesused previously. Here the di� ulty omes from the nonlinear birth term in the boundary ondition and the absen e of BV in the natural variable. This requires an alternate ap-proa h via BV estimates in time to dedu e the ompa tness ne essary to pass to the limit.The numeri al tests show that a se ond order s heme is ne essary to apture the transientos illations generated by the nonlinearities.

1. Introdu tion 3�How do we know that, if we made a theory whi h fo uses its attention on phenomenawe disregard and disregards some of the phenomena now ommanding our attention, thatwe ould not build another theory whi h has little ommon with the present one but whi h,nevertheless explains just as many phenomena as the present theory� � [110℄1 Introdu tionIn last two enturies several mathemati ians, biologists, biophysi ists and several others ientists translated almost every well understood phenomena in biology and medi ine un-der reasonable hypothesis into the language of s ien e i.e mathemati al modelling1. Thisranges from ell y le of mi ro organism (eg. phytoplankton) to e ologi al diversity andevolution. The modelling helps up to a great extent to understand the phenomena in abetter way and sheds light on the hidden fa tors whi h in�uen e the phenomena. Weshould not forget that not all on epts of mathemati s whi h were used for modellingwere well established at the time of modelling but developed later by several great mindsvery systemati ally. In modelling Ordinary di�erential equations (ODEs), Partial di�e-rential equations (PDEs) played a ru ial role and enhan ed the e� a y of models. Thereis un anny list of mathemati al models in biology and obviously here we do not give allof them. Nevertheless in this hapter we present some of the models whi h have playedvital role in understanding biologi al phenomena and studied by many s ientists extensi-vely sin e they were proposed. Both mathemati s and biology are bene�ting from thesemodels.2 Population dynami sPopulation dynami s has been a entral �xture in mathemati al biology for severalde ades starting with Malthus exponential growth population model. The goals of popula-tion biology are to understand and predi t the dynami s of the population. To understand omplex ommunities with numerous spe ies intera ting with ea h other and the environ-ment one requires to understand the simpler systems of one or two spe ies �rst. So, wefo us on population dynami s of a single spe ies.Stru ture Individuals in biologi al populations di�er with regard to their physi al andbehavioral hara teristi s and therefore in the way they intera t with environment. Stu-dying this point e�e tively requires the use of stru tured models. Structured populationsare populations in whi h individuals di�er a ording to variables that a�e t their ferti-lity and mortality rates. These values an be phenotype or behavioral traits whi h aretransferred from parents to o�springs in the absen e of mutations. The main fo us ofpopulation dynami s has been hara terization of alterations in sizes, numbers and agedistribution of individuals.1This introdu tion has bene�ted from several ourses and le tures whi h I attended in Bangalore,Paris and Bar elona by Prof. Vanninathan, Prof.Perthame and Prof. Diekmann

4 Introdu tionThe fa tors whi h determines or ontrols the ell division of prototype spe ies or uni el-lular organisms (like E. Coli, yeast et ) are the mass of ell, its DNA ontent, the level of ertain growth proteins (eg. y lin) et . In the ell division models of su h ases we have`size' stru ture [81℄. Here size refers to one (or an be more) of above stated quantities.Why numbers are important ? In population dynami s, the entral fo us is on num-bers of individuals as the variable of interest. It is be ause of the possibility that smallnumber of individuals may have e�e ts on population stability, out of proportion to theirnumbers. For example, a small population of predators may play major role in regula-ting a prey population with a large population size. Diseases ranging from AIDS to theplague may have extreme e�e ts on the dynami s of the host population (see [55℄). Severalmathemati al biologists have been studying this not only be ause of the beautiful mathe-mati al stru ture in orporated in the models but also for its appli ability in ommer ialendeavors. For example it plays a ru ial role in �sheries, rops, environmental purposeslike reforesting and medi inal ontexts like epidemiology, tumor growth.Why is `age' an important stru ture ? In pra ti e animal population is often mea-sured by size with age stru ture used as an approximation to size stru ture. The study ofage stru tured models is onsiderably simpler than the study of general size stru turedmodels. This is be ause age in reases linearly with time whereas the linkage of size withtime is less predi table. Also the result of ell division for instan e in terms of size of thedaughter ells undergoes a larger variety than with age.Most populations onsist in set of individuals born over a range of past times with beha-viors that depends on their ages. For many spe ies there exist two stages given by juvenilestage and adult stage. The �rst one is devoted to maturation, growth and the se ond oneis for reprodu tion (see [22℄). For several spe ies, in parti ular for mammals the vitaldemographi fun tions like birth rate and death rate de rease with age (see [96℄). Someother spe ies exhibit delay in their reprodu tion. For example On orhyn hus Gorbus ha(widely known as Pink salmon) breeds at about two years old age and dies usually in twoweeks (see [71℄). Another example is Magi i ada (known as i ada) also breeds at an ageof 13-17 years and dies in few weeks (see [22℄). A very good dis ussion of aging and human an er an be found in [79℄. Here the authors announ e that in iden e of many tumors(most of ar inomas and leukemia's) in reases with age for a ombination of several rea-sons. Some of the reasons are environmental fa tors, de reased DNA repaired fun tion.Moreover they have shown that the in iden e rates sharply in reases on e one is past theage of 50 years. When we ome to the ellular levels the on ept of age is quite di�erentfrom hronologi al age. The hronologi al age of an organism is not ne essarily of cellularimportan e. An alternative approa h is to introdu e on epts like ` ellular maturity'. One an in orporate several biologi al subtleties regarding `age' and get an intri ate model forpopulation, but it is important to realize that the model is not easily analyzed (see [111℄).As a onsequen e we simplify it and treat spe ial ases of our interest.Competition Another important aspe t in population dynami s is ` ompetition'. Givena set of restri ted or limited resour es in the environment ea h individual has its own apa ity to get or utilize the resour e depending its ability to ompete with others. This ompetitive ability highly depends on the physiologi al quantities like size, age, on en-tration of ertain proteins et .Regularity It is important to realize that ontinuous fun tions are not appropriate for

3. The model and assumptions 5des ribing ertain biologi al quantities. For example the total population P (t) of someba teria at time t an only take integer values. If the number of ba teria is measured inunits of one million then P (t) an take values that di�er by 0.000001. In this ase it looksreasonable to make an approximation that P (t) is a ontinuous fun tion, provided theexperimental error in ounting ba teria is greater than 1. Similarly one an argue and getdi�erentiability.Therefore age stru tured population dynami s is an important bran h of mathemati albiology whi h attra ted many resear hers for obvious reasons. So far we have dis ussed theimportant modeling ingredients like age, stru ture, ompetition of our model of interest,i.e. a nonlinear renewal equation. At this moment, we re all brief history of basi popula-tion models involving di�erential equations. The earliest of the population model is dueto Sharpe�Lotka (see [99℄). M Kendri k, a physi ian introdu ed a linear age-stru turedmodel (where mortality and fertility rates depend only on age) in 1920's in the study ofepidemiology in his pioneering work [80℄. Rigorous analysis of these linear models wasa omplished by Feller (see [47℄), Bellman et al (see [11℄) using integral equations andLapla e transforms. Another breakthrough in this �eld is the Kerma k-M Kendri k modelfor infe tious disease in [69℄. Later many people have extended to this model to nonlinearPDE level. This model has been widely studied by many mathemati ians as well as biolo-gists (see [57℄, [58℄, [107℄, [108℄, [109℄, [106℄). Re ent a ount of the subje t an be foundin the book [36℄. Now we announ e the stru ture of the thesis and state main results thatwe prove later in subsequent hapters.3 The model and assumptionsConsider an isolated population living in an invariant habitat and also assume thatthere is no sex di�eren e. Let u(t, x) be the population density of age x, at time t. Letd, B be the age spe i� death rate, birth rates respe tively depending on weighted totalpopulation. Let ψ1, ψ2 be the ompetition weights re�e ting the age dependen e of everyindividual in the ontribution and ompetition for survival. In this situation the populationdynami s are governed by the following nonlinear M Kendri k equations, that is

∂∂tu(t, x) + ∂

∂xu(t, x) + d(x, S1(t))u(t, x) = 0, t ≥ 0, x ≥ 0,

u(t, 0) =

∫ ∞

0

B(x, S2(t))u(t, x)dx,(1)with a oupling, whi h we take as simple as possible

Si(t) =

∫ ∞

0

ψi(x)u(t, x)dx for i = 1, 2. (2)To deal with this equation, there are several te hniques developed by mathemati ians.Among them one is to use the hara teristi method to redu e the system (1)�(2) to Vol-terra integral equation and apply semigroup methods to the linear and nonlinear operators(see [6℄, [27℄, [30℄, [60℄, [52℄, [53℄, [58℄, [102℄, [107℄). Another te hnique is to redu e thesystem to an equivalent delay equations (see [31℄, [32℄, [35℄, [36℄, [81℄). Many appli ations

6 Introdu tionof the theory have been developed in various areas. We present some of them in Chapter1. Besides them here we give few more referen es whi h are not dis ussed in that hapter.We refer [63℄ for the re ent a ount of appli ations in demography. For appli ations inepidemiology one an refer [14℄, [15℄, [21℄, [49℄, [50℄. Perthame et al introdu ed the te h-nique alled General Relative Entropy (GRE) method to study the long time behavior ofthe linear renewal equation (see [85℄, [86℄). P. Mi hel used the GRE method to explain thelong time dynami s of a nonlinear system (1)�(2) with a parti ular type of nonlinearityin B (see [73℄, [84℄). Another appli ation of the GRE method an be found in [29℄. Inthat arti le the authors model the growth and size distribution of metastati tumors inthe earlier stages of the disease. Therefore the large time behavior of the size distributionshould be understood. In this ontext the authors prove the GRE inequality and followthe the pro edure presented in [85℄, [86℄.Assumptions Coming to the present s enario, in order to state our main results weneed some basi assumptions on d, B, ψ. These assumptions are so vital that most ofthem frequently appear throughout the thesis. To begin with we assume that the fun -tions d, B, ψ are ontinuous in S, nonnegative, lo ally bounded. Then, we also need theassumptionsB(., 0) ∈ L∞(R+) ∩ L1(R+), (3)

1 <

∫ ∞

0

B(x, 0)e−R x

0d(y,0)dydx <∞, (4)

0 <

∫ ∞

0

B(x,∞)e−R x

0d(y,∞)dydx < 1, (5)

1 < lima→∞

∫ ∞

0

B(x,∞)eax−R x

0d(y,∞)dydx = β∞ <∞, (6)

∂d(., .)

∂S> 0, (7)

d(.,∞) ∈ L∞(R+), d(., S) /∈ L1(R+), ∀ S ≥ 0, (8)∂B(., .)

∂S< 0. (9)There exists L > 0 su h that for all x, S1, S2 ≥ 0 we have

|B(x, S1) − B(x, S2)| ≤ L|S1 − S2|, |d(x, S1) − d(x, S2)| ≤ L|S1 − S2|. (10)Our last four assumptions use notation that are introdu ed later on. However we give themnow in order to gather all the assumptions. The eigenelements λ0, λ∞, N0, N∞, φ0, φ∞ arede�ned later in Chapter 1.There exist two maps D0, D∞ : R+ → R de�ned byD0(S) = inf

x

{(d(x, 0) − d(x, S) +

φ0(0)

φ0(x)

(B(x, S) − B(x, 0)

)}, (11)

D∞(S) = supx

{(d(x,∞) − d(x, S) +

φ∞(0)

φ∞(x)

(B(x, S) − B(x,∞)

)}. (12)

3. The model and assumptions 7Finally for the ompetition weight ψ(·), we assume that there are two positive onstantsC0

min and C0max su h that

C0minφ0(x) ≤ ψ(x) ≤ C0

maxφ0(x). (13)Also there are two positive onstants C∞min and C∞

max su h thatC∞

minφ∞(x) ≤ ψ(x) ≤ C∞maxφ∞(x). (14)3.1 Existen e, uniqueness and uniform boundFirst we want to prove the existen e, uniqueness of the solution to the system (1)�(2).In order to do that �rst prove a ru ial result that the unknown S(t) in (1)�(2) is a prioriuniformly bounded from above and below. This is given inProposition 1. Under assumptions (3)�(9), (11)�(14) and ∫∞

0φ0(x)u0(x) > 0 thereexists m,M > 0 su h that any weak solution in C

(R+;L1(R)

) of (1)�(2) satis�es m ≤S(t) ≤M ∀t > 0.Having done this we prove existen e of unique solution to the system (1)�(2) with strongassumption on d, B. This is given inTheorem 1. Let us assume (10), there exists M > 0 su h that B(., .), d(., .),ψ(·) < M and u0 ∈ L∞(0,∞) ∩ L1(0,∞), then there exists a unique weak solution u ∈C(R+;L1(R+)

) to (1)�(2).Sin e we have a priori bounds for S(t) we an relax the boundedness hypothesis on d, Band prove a general existen e and uniqueness result. This is given by the following maintheorem.Theorem 2. Assume (3)�(14), u0 ∈ L∞(0,∞)∩L1(0,∞), also assume ∫∞

0φ0(x)u0(x)dx >

0, then there exists a unique weak or distributional solution u ∈ C(R+;L1(R+)) to (1)�(2).Moreover there exists two positive onstants m, M su h thatm ≤ S(t) ≤M, ∀ t > 0.3.2 Asymptoti behavior (perturbation result)Next we turn our attention towards the long time behavior of the solution to the system(1)�(2). As we know that the dynami s of the solution of fully nonlinear system for largetimes is far from triviality, we try to understand the dynami s in several parti ular aseswhere d, B, ψ are in spe i� forms. Here we onsider three ases. In the �rst ase weassume

d(x, S) = d1(x) + d2(S), d1(x) ≥ 0, (15)B(x, S) ≡ B(x) ≥ 0, (16)

∫ ∞

0

B(x)e−D(x)dx = 1, D(x) =

∫ x

0

d1(x)dx. (17)

8 Introdu tionAt this point we introdu e a parti ular solution (variable separable solution) m(t, x) =n(t)N(x). Then N(x) solves

ddxN(x) + d1(x)N(x) = 0, x > 0,

N(0) =

∫ ∞

0

B(x)N(x)dx = 1.(18)In other words N = e−D(x). As usual we introdu e the adjoint problem for (1.34)

− ddxφ(x) + d1(x)φ(x) = φ(0)B(x), x > 0,

∫ ∞

0

φ(x)N(x)dx = 1.(19)With this information we an announ e the onvergen e theorem as follows.Theorem 3. Under the assumptions (15)�(17) and if there exists µ ≥ 0, δ > 0 su h that

B(x) ≥ µφ(x), µ + d2(S(t)) ≥ δ, then the solution of age stru tured equation (1)�(2)satis�es∫ ∞

0

|u(t, x) − n(t)N(x)|φ(x)dx ≤ e−δt∫ ∞

0

|u0(x) − n(0)N(x)|φ(x)dxwith n(t) =∫∞

0u(t, x)φ(x)dx.3.3 Asymptoti behavior ( on ave birth term)We study the ase where the boundary term has a nonlinearity of type

u(t, 0) = g(∫ ∞

0

B(x)u(t, x)dx), (20)where g : R+ → R+ is a fun tion satisfying ertain onditions (as on avity). A detaileddis ussion an be found in Chapter 1. In this ase also we an prove the onvergen e tothe steady states with appropriate weights following [68℄, [84℄.We numeri ally show the existen e of periodi solutions and o urren e of Hopf Bifur a-tion (see Chapter 2). We an do that when g′(n♯) is small enough and n♯ is the steadystate.3.4 Asymptoti behavior (redu tion to ODEs)Next we fo us on the ases where we an redu e the system (1)�(2) to a 2×2 system ofODE and study the dynami s of the system at ODE level. We make sure that the unknown

S(t) is arried through from PDE level to ODE level. In this mission we presented severalforms of B, d where we an su essfully prove the nonlinear stability of the steady stateby onstru ting a Lyapunov fun tion. Detailed dis ussion regarding this an be found in

4. Existen e, uniqueness of the steady states and the linear stability 9Chapter 1. Nevertheless here we illustrate one su h ase.Assume for some α > 0,B(x, S) = b1(S)e−αx + b2(S), bi > 0,

dbidS

(·) < 0 for i = 1, 2. (21)Also we assume the death rate is independent of aged(x, S) = d(S),

d

dSd(·) > 0. (22)Then we de�ne two quantities

S(t) =

∫ ∞

0

u(t, x)dx, Q(t) =

∫ ∞

0

e−αxu(t, x)dx.This means that the ompetition weight is assumed to be onstant ψ ≡ 1. It is obviousto �nd the ODE equivalent to (1)�(2) as

dS

dt(t) = [b2(S(t)) − d(S(t))]S(t) + b1(S(t))Q(t),

dQ

dt(t) = b2(S(t))S(t) + [b1(S(t)) − d(S(t)) − α]Q(t),

0 < Q(0) < S(0).

(23)For this system the existen e, uniqueness and global stability of the steady state are giventhe followingTheorem 4. Assume (4)�(5) then there exists a unique positive steady state (S, Q).Moreover assumeb2(S)

d(S)>

b1(S)

d(S) + α,then the steady state (S, Q) is globally and exponentially attra tive for the system (23).4 Existen e, uniqueness of the steady states and thelinear stabilityIn this se tion we brie�y present the on epts and main results that we prove inChapter 2. In Se tion 0.3, we have given examples where the steady state is globallyattra tive for B(x, S) (resp. d(x, S)) de reasing (resp. in reasing) with S. One naturalquestion is wether this is a generi situation. Another is to know in whi h ir umstan esother behaviors are possible. In the quest to sear h the answers to these questions, the�rst set of questions that we should ask ourself after ex luding the extreme ases (likeblow-up, extin tion of total population) is the following. Whether there exists any thenonzero steady state ? Is there uniqueness of the nonzero steady states ? We begin hereand ome up with a�rmative answer to this question. Throughout this se tion we assume(3)�(4), (7)�(9) from the set of assumptions that we stated in Se tion 0.3.

10 Introdu tion4.1 Existen e of nonzero steady statesIn this subse tion onsider more general ase in whi h the ompetition that in�uen esdeath rate and birth rates depends upon two di�erent weights ψ1, ψ2. In other words we onsiderd ≡ d(x, S1), B ≡ B(x, S2), Si =

∫ ∞

0

ψi(x)u(t, x)dx, for i = 1, 2.We set D(x, S) =∫ x0d(y, S)dy. Now we are ready to state our main result of this subse -tion. In this result we give some su� ient onditions for the existen e and uniqueness.Theorem 5. Assume (3)�(4), (7)�(9) then there exists at least one steady state to (1)�(2).Moreover assume that there exists α > 0 su h that ψ1 ≥ αψ2 and

(S1 − αS2

)DS(x, S1) < 1 +

α

∫ ∞

0

B(x, S2)DS(x, S1)e−D(x,S1)dx

∫ ∞

0

|BS(x, S2)|e−D(x,S1)dx

, (24)for every x > 0 and steady state S1, S2, then there exists unique positive steady state tothe system (1)�(2).4.2 Linear stabilityIn this subse tion we ome ba k to the original state with single ompetition weight,i.e., ψ := ψ1 ≡ ψ2. Now we linearize the system around the steady state and obtainthe hara teristi equation Γ(λ) = 0. This equation is vital in understanding the longtime behavior of the linearized system under the light of the prin iple of linear stability.A ording to this prin iple, stability of the linearized system depends upon the real part ofthe roots of the equation Γ(λ) = 0. Again, the main di� ulty is that the expression Γ(λ)is very intri ate. Now we present two results orresponding to stability and instabilityrespe tively.Lemma 1. (Stability) Assume (3)�(4), and that the mortality solely depends on age, i.e.,d(x, S) ≡ d(x) ∈ L∞(R) for all x, S ≥ 0. Then the steady state is linearly asymptoti allystable, in the following ases.(i) There is a µ > 0 su h that

−1 ≤ S

∫ ∞

0

e−D(x)BS(x, S)dx < 0,

ψ(x) = µB(x, S), ∀ x > 0,(ii)Re

∫ ∞

0

ψ(x)e−D(x)−λxdx ≥ 0, ∀ Re(λ) > 0,for instan e when there is a µ > 0 and r ≥ ‖d‖∞ su h thatψ(x)e−D(x) ≡ µe−rx ∀ x > 0.

4. Existen e, uniqueness of the steady states and the linear stability 11At this stage we announ e that we have onstru ted an example where the birth rateB(x, S) is de reasing with S but still we have the instability of the steady state. Details an be found in Chapter 2. Now we state a general result for linear instability in thefollowingProposition 2. (Instability) Assume BS(x, S) > 0, dS(x, S) < 0,

0 <

∫ ∞

0

B(x, 0)e−R x

0d(y,0)dydx < 1,

1 <

∫ ∞

0

B(x,∞)e−R x

0d(y,∞)dydx <∞,and dS(x, S) is bounded then the steady state exists and is linearly unstable.After that we turn our attention towards the ase where the total population is onstant at all times. This kind of models are naturally seen in neural networks (see[91℄). Conservation is be ause the mortality rate is equal to the fertility rate, i.e.,

d ≡ B. (25)We normalize the total population by∫ ∞

0

u(t, x)dx =

∫ ∞

0

u0(x)dx = 1, ∀t ≥ 0.In this ase also we prove the existen e and uniqueness of the steady state. As usual welinearize around the steady state. But obtaining the hara teristi equation is not straight-forward like non onservative ase. We have to be areful while hoosing the onstraintsas there is redundan y of the onstraints that ome into pi ture in this business. As aresult we get the hara teristi equation and the zero ondition (see Chapter 2). Now wepresent a ase where we an expli itly ompute the roots of the hara teristi equation.This is when the rate d is independent of age and in reases with S, i.e.,d ≡ d(S), d′(S) > 0. (26)Assume the ompetition weight is given by

ψ(x) = e−mx for some m > 0. (27)Proposition 3. Assume (26)�(27) andd′(S) <

(d(S) +m)2

m,for all steady states S ≥ 0 then the steady state is unique and is linearly stable. The steadystate S is linearly unstable if

d′(S) >(d(S) +m)2

m.

12 Introdu tionWe on lude this subse tion with some omments on periodi solutions. There are very fewresults regarding the existen e of nontrivial periodi solutions in age stru tured models(see [12℄, [28℄, [97℄). Usually they are build using bifur ation theory, parti ularly by Hopfbifur ation (see [25℄, [61℄, [62℄, [112℄). But unfortunately there is no Hopf bifur ationtheorem available in general ase ( f. [77℄). Problems with boundary term of the type (20)are studied using enter manifold theory by [37℄, [38℄, [77℄. In [84℄ the GRE method isused to study its long time behavior. We give numeri al eviden e of su h periodi solutionin a parti ular ase.5 Numeri al studyIn this se tion we outline the results we prove in the Chapter 3. In this hapter, �rstwe present a semi-impli it �rst order upwind s heme. These are motivated by the modelswhere the �rst order term indu es x dependen y as in [42℄. More sophisti ated s hemesare ne essary to deal with the models with growth rate depending on time. Usually thedis retization of the equation relays on the hara teristi method. For instan e Runge-Kutta methods are used along the hara teristi s to integrate the system (1)�(2) in [4℄.Method of hara teristi s is used to integrate SIR, SIS models in epidemiology by Iannelliet al. in [59℄. We prove the onvergen e of this numeri al s heme. Se ondly we design ase ond order s heme and ompare the latter with the �rst order s heme on spe i� omplexdynami s. In order to do that we begin with the main assumptions in this se tion. Assume(10), assume that there exists d, d, B, ψ > 0 su h thatd < d(., .) < d, B(., .) < B, 0 ≤ ψ1(.), ψ2(.) < ψ. (28)To state our main results �rst we introdu e the following notation. We dis retize thepositive quarter plane by hoosing an uniform mesh width △x > 0 and an uniform timestep △t > 0. A typi al mesh point (tn, xj) := (n△t, j△x) for n, j = 0, 1, . . . .Assume the CFL ondition

λ :=△t

△x≤ 1. (29)For the sake of ompletion we introdu e the following standard notation, we denote thenodal points xj+ 1

2

:= xj + △x2

for j ≥ 0, and the ell averageuj(t) :=

1

△x

∫ xj+ 1

2

xj− 1

2

u(t, y)dy.Furthermore for n ≥ 0 we de�neu0j =

1

△x

∫ xj+ 1

2

xj− 1

2

u0(x)dx, Sni = △x∞∑

j=1

ψi(xj)unj , for i = 1, 2,

dj(S) =1

△x

∫ xj+ 1

2

xj− 1

2

d(y, S)dy, Bj(S) =1

△x

∫ xj+1

2

xj− 1

2

B(y, S)dy,

dnj = dj(Sn1 ), Bn

j = Bj(Sn2 ).

5. Numeri al study 13As the hara teristi s move with positive speed, one an simply use the following semi�impli it upwind s heme. For n ≥ 0, we set

un+1j = unj − λ(unj − unj−1) −△tdnj u

n+1j for 1 ≤ j ≤M, n ≥ 0,

un0 = △x

M∑

j=1

Bnj u

nj ,

Sni = △xM∑j=1

ψi(xj)unj , for i = 1, 2.

(30)By solving this di�eren e equation we get dis rete solution whi h we use to onstru tpie ewise onstant solutions at ontinuous level. With the help of these notation at dis retelevel, we introdu e pie ewise onstant fun tions at ontinuous level. This is given by

u0△x(x) :=

∞∑

j=1

u0j1I{xj−1<x<xj},

Si,△t(t) :=

∞∑

n=1

Sni 1I{tn≤t<tn+1}, for i = 1, 2,

B△x(x, t) :=∞∑

j=1

Bnj 1I{xj−1<x<xj}, for tn ≤ t < tn+1,

d△x(x, t) :=

∞∑

j=1

dnj 1I{xj−1<x<xj}, fortn ≤ t < tn+1,

u△x(t, x) :=∞∑

j=1

unj 1I{xj−1<x<xj}, for tn ≤ t < tn+1,

β△x(t) :=

∫ ∞

0

B△x(x, S2,△t(t))u△x(t, x)dx, for tn ≤ t < tn+1.With this set of notation we state our main theorem as follows.Theorem 6. We assume (10), (28)�(29), u0(x) ∈ BV (R) and assume△xk → 0, △tk → 0as k → ∞. Then the full sequen e uk := u△xkof solutions to (30) satis�es

ukk→∞−→ u, in C([0, T ];L1(R+)

)∩ L∞

([0, T ];L∞(R+) ∩ L1(R+)

)and a.e.,for some fun tions u, b, S1, S2 satisfying

u ≥ 0, u ∈ L∞([0, T ];L1(R+) ∩ L∞(R+)

),

b(t) =

∫ ∞

0

B(x, S2(t))u(t, x)dx ∈W 1,∞([0, T ]

),

Si(t) =

∫ ∞

0

ψi(x)u(t, x)dx ∈W 1,∞([0, T ]

) for i = 1, 2and u is a distributional solution to equation (1)�(2).

14 Introdu tionIn order to prove the main theorem �rst we prove the a priori BV estimates and thenwe prove the onvergen e of the dis rete solution at subsequen e level. Finally we useuniqueness to get the full sequen e onvergen e to the weak solution. This is demonstratedin the following two resultsProposition 4. Under assumptions of Theorem 6 the fun tion u△x(t, x) satis�es for allt ∈ [0, T ] the estimates,(i)(a)

∫ ∞

0

|u△x(t, x)|dx ≤

∫ ∞

0

|u0△x|e

(B−d)n△tdx,

(i)(b) ‖u△x(t)‖∞ ≤ max(‖u0

△x‖∞, Bmaxs≤t

∫ ∞

0

|u△x(s, x)|dx),

(ii) ‖∂

∂tu△x(t, x)‖M1(R+) ≤ λe(A−d)T

[‖∂

∂xu0△x(x)‖M1(R+) + d

∫ ∞

0

|u0△x(x)|dx

]

(iii) ‖ ∂∂xu△x(t, x)‖M1(R+) ≤ e(A−d)T

[‖ ∂∂xu0△x(x)‖M1(R+) + d

∫ ∞

0

|u0△x(x)|dx

]

+de(B−d)(n+1)△t

∫ ∞

0

|u0△x(x)|dx.Moreover if we assume xu0 ∈ L1(R+) then

(iv)

∫ ∞

△x

(x−△x

2)|u△x(t, x)|dx ≤

∫ ∞

△x

(x−△x

2)|u0

△x(x)|dx+

∫ t

0

∫ ∞

0

|u△x(τ, x)|dxdτ.Having these BV estimates for u△x, we are able to prove the following onvergen e resultfor subsequen es uniform of (u△x).Proposition 5. Under the hypothesis of Theorem 6, for △x, △t → 0 we have b△t ∈C([0, T ]), Si,△t ∈ C[0, T ], for i = 1, 2, ∀ T > 0. There exists u ∈ L∞

([0, T ]; L1(R+) ∩

L∞(R+)), b ∈ C([0, T ]) and Si,△t ∈ C([0, T ]) for i=1,2 su h that for a subsequen e

u△x∗⇀ u weak- *, in L∞

([0, T ];L1(R+) ∩ L∞(R+)

), a.e.,

Si,△t(t)−→Si(t) in C([0, T ]) for i=1,2,β△x(t) −→ b(t) in C([0, T ]),

d△x(x, t)−→d(x, S1(t)) in C([0, T ];Lplo (R+)), for 1 ≤ p <∞,

B△x(x, t)−→B(x, S2(t)) in C([0, T ];Lplo (R+)

) for 1 ≤ p <∞.We on lude this se tion by presenting a se ond order s heme. A omprehensive review ofhigh order ENO s hemes an be found in [101℄. Here we use slope re onstru tion te hniqueto in rease the order of the s heme. The onstru tion is as follows. At ea h time level tnfor n ≥ 0 using (30) we re onstru t a pie ewise linear approximation of the solution of(1)�(2) on the intervals on (xj− 1

2

, xj+ 1

2

) for j ≥ 1 and is given byv(tn, x) = Wj(t

n, x) := vj(tn) + (x− xj)σ

nj , xj− 1

2

< x < xj+ 1

2

, (31)where σj is the numeri al derivative of v in the interval (xj− 1

2

, xj+ 1

2

) at time tn. We de�neapproximation near the boundary without re onstru tion usingv(tn, x) = v0(t

n), for 0 ≤ x ≤△x

2. (32)

5. Numeri al study 15Therefore we annot expe t the se ond order a ura y at the boundary. Now we turn ourattention towards the slope σnj whi h is very ru ial in the re onstru tion. We de�ne theso- alled min-mod re onstru tion asσnj = minmod { vj(tn) − vj−1(t

n)

△x,vj+1(t

n) − vj(tn)

△x

}, (33)with minmod(a, b) =

{0, ab ≤ 0sgn(a) min(|a|, |b|), ab > 0.With this re onstru tion the standard se ond order s heme in spa e reads as, for 1 ≤ j ≤

M, n ≥ 0,

vn+1j = vnj − λ(V n

j+ 1

2

− V nj− 1

2

) −△td(xj, Sn1 )vn+1

j ,

V n1

2

= △xM∑

j=1

B(xj , Sn2 )vnj ,

Sni = △xM∑j=1

ψi(xj)vnj , for i = 1, 2.

(34)Regarding numeri al �uxes at ell interfa es V n

j+ 1

2

we hoose upwind s heme. For 1 ≤ j ≤

M , we setV nj+ 1

2

= vnj +△x

2σnj . (35)Hen e (34)�(35) onstitute the se ond order s heme. Comparison of this s heme with the�rst order s heme an be found in Chapter 3. For the sake of ompletion here we presentone example where we have ompared both the s hemes. We begin with a simple ase inwhi h we have age independent mortality term given by

d(x, S1) =c1S1 + ε

1 + S1,where c1, ε > 0. Noti e that d is in reasing with S1 if and only if c1 > ε. To keep thesystem to be simple, the birth term has been taken by (to be onsistent with previousnotation)

B(x, S2) :=2

1 + S2

.Further we take the ompetition weights ψ1, ψ2 to be{ψ1 = x

311{0≤x≤1.5} + (2 − x) 11{1.5≤x≤2},

ψ2 = x 11{0≤x≤1} + (2 − x) 11{1≤x≤2}.We begin with a pie ewise onstant initial data given byu0(x) := 11{0≤x≤12.5} for x ≥ 0. (36)In this example we start with the parameters c1 = 0.1, ε = 0.01 and the initial pro�leis given by (36) and in this ase we observe the better onvergen e rate (see Figure 1(a),Figure 1(b)).

16 Introdu tion

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

(a) 0 5 10 15 20 250

2

4

6

8

10

12

(b)Fig. 1 � Solution at time t = 10 (dotted line), 15 (dashed line), 20 (dash-dotted line), 25( ontinuous line) with the initial data given by (36), c1 = 0.1, ε = 0.01 using, (a) : the�rst order s heme, (b) : the se ond order s heme.

Chapitre 1Age-stru tured Renewal EquationsPublised in Sele ted Topi s On Can er Modelling Genesis - Evolution - Immune Compe-tition - Therapy, (Editors : Bellomo, N., Chaplain, M., De Angelis, E.).Abstra tThe renewal equation plays a entral role in modeling population biology and appearsin various domains ranging from ell proliferation and tumor growth to epidemiology, ellmotion and e ology. Being simple and with a dire t interpretation, it an be onsideredas a �rst step towards more elaborate mathemati al des riptions.The linear renewal equation is well understood and there are several mathemati always to express the main behavior of its solutions : they exhibit exponential growth orde ay of the population with a rate and pro�le that an be entirely hara terized, hen epossible appli ations to an er therapy.However, the theory for nonlinear models is mu h more ompli ated. Several beha-viors are possible ( haoti , periodi , or stable steady states). In this Chapter we give anintrodu tion to this theory with a spe ial interest on ases where there is an exponentiallyattra tive steady state.1.1 Introdu tionIn this hapter we onsider an usual nonlinear age stru tured population model whi harises in many di�erent ontexts. One of them is the des ription of ell proliferation andthus tumor growth, another is metastatisis size distribution. It an be written as thePartial Di�erential Equation (PDE in short) on the unknown fun tion n(x, t) ≥ 0 whi hrepresents the population density of individuals of age x, at time t,

∂∂tn(t, x) + ∂

∂xn(t, x) + d(x, S(t))n(t, x) = 0, t ≥ 0, x ≥ 0,

n(t, 0) =∫∞

0B(x, S(t))n(t, x)dx,

n(0, x) = n0(x) ≥ 0.

(1.1)The ve tor valued fun tion S(t) =(S1(t), S2(t), · · · , Sk(t)

), represents the environmentalfa tors whi h depend on the solution n(x, t) itself, with a oupling, whi h we take as17

simple as possibleSi(t) =

∫ ∞

0

ψi(x)n(t, x)dx, 1 ≤ i ≤ k. (1.2)Also B ≥ 0, d ≥ 0 represent the birth and death rates respe tively. Throughout this hapter our interest is on ells, ell y le and related topi s, therefore we onsider a singlepopulation living isolated, in an invariant habitat, all of individuals being equal and thereis no sex di�eren e.The model (1.1)�(1.2) arises in many examples issued from population biology. Histo-ri ally, it is the �rst PDE introdu ed in biology and the linear equation (when d and b donot depend upon S) is usually known after the name of M Kendri k [80℄ who introdu edit for epidemiology, and Feller [48℄ made an extensive study through Markov pro esses.The linear equation is also known as the VonFoerster equation be ause he was �rst touse it for modeling ell ultures. It is well understood and there are several mathemati always to express the main behavior of its solutions : they exhibit exponential growth orde ay of the population with a rate and pro�le that an be entirely hara terized. Forexample, using the General Relative Entropy (GRE) inequalities ([85, 86, 93℄), one anprove that solutions to the linear model satisfy the long time asymptoti s∫φ(x)

∣∣n(x, t) e−λt − ρN(x)∣∣ dx −−−−→

t→∞ 0, (1.3)for some real number ρ > 0 and appropriate fun tions φ and N (see Se t. 1.3). Hen edepending on the sign of λ, we on lude that either the population will grow for ever orget extin t.As for nonlinear models, the most famous was proposed by Kerma k and M Kendri kfor epidemiology with ontinuous state (age in the disease), [69℄. Nowadays, these mo-dels are used in various domains ranging from epidemiology to e ology, medi ine and ell ultures. We give several examples and referen es in the Se t. 1.2. The �rst mathemati alstudy of su h nonlinear equations is due to Gurtin, Ma Camy, [53℄, and thus (1.1)�(1.2)is sometimes referred to as the Gurtin-Ma Camy model, in the ase ψ1 = ψ2 ≡ 1 atleast. Existen e, uniqueness, stability results of solutions of this model was dis ussed in[24, 51, 53℄. Afterwards it has been vastly studied by several mathemati ians using variouste hniques as semigroup theory, entropy GRE methods, Lapla e transforms. To deal withthis model, the basi te hnique whi h many people (in luding Gurtin and Ma Camy)used was to apply the method of hara teristi s to onvert this problem to system ofVolterra integral equations (see [53, 57, 58, 107, 17℄). The papers [33, 34℄ and the book[102℄ ontain a re ent a ount of the theory. Here we will try to avoid this artefa t anddeal with PDE methods, some of them an be extended to more elaborate models as, e.g.,size stru tured models [81, 93, 94, 85, 83℄.An important aspe t in the linear model leading to the behavior (1.3) is that it doesnot take resour es into a ount. This is the main drawba k of the linear model. In thesystem (1.1)�(1.2) we over ome this and onsidered the onsumption of resour es like nu-trients, by introdu ing nonlinearity in birth and death terms. In many of the examples we18

will onsider later, these are limitating the possible growth and indu e extra-growth whenthe solution gets to small. In other words, these models ontain the lassi al assumptionof Verhulst (for a mere ODE) as a basi ingredient. However due to the delay indu edby the boundary ondition in (1.1), this limitation an have dramati e�e ts as existen eof haoti solutions, periodi solutions, `os illating' solutions, see [41, 87, 84℄. Travelingwaves were also studied [40℄, periodi for ing are treated in [26℄. Of ourse stability ques-tions are also a major issue, studied by many authors, see also [16, 78℄. In this Chapterwe pay a spe ial interest on global a priori bounds and ases where we an prove thatthere is an exponentially attra tive non-zero steady state.We begin with several examples of nonlinearities taken from the literature (Se t.1.2). Then, we introdu e our main assumptions (Se t. 1.3). Existen e theory and uni-form bounds on the solutions require some work whi h we perform in Se ts. 1.4, 1.5.Our �rst original result on erns a ase where we an prove the asymptoti behaviour asn(x, t) ∼ N(x)n(t), see Se t. 1.6. Then, we re all in Se t. 1.7, the long time asymptoti sresult of Ph. Mi hel [84℄ on ' on ave type' nonlinearities on the birth term. After thesegeneral ases we on lude in Se t. 1.8 with spe ial nonlinearities for whi h we an redu ethe system to ODE systems and prove again their exponential stability.1.2 Examples of Age Stru tured ModelsMany variants of the system (1.1)�(1.2) have been proposed in the literature in va-rious area of biology, leading to di�erent hoi es of the model parameters d, B and ψ. Inthis se tion we present some spe i� examples. Of ourse this presentation is in ompletebut we hope it an give a general view of the broad use of su h models. We begin withexamples from epidemiology, e ology, ba terial ell ulture and on lude with the ell po-pulation models whi h arise in medi al appli ations. This se tion is far from exhaustiveand we refer for instan e to [89℄ ï¾1

2for an original modelling of the a tin- ytoskeleton insymmetri lamellipodial fragments, to [90℄ for an apli ation to neuros ien e, to [8℄ for a annibalism model (this is one of the most interesting phenomena in population studies,see [32℄ for an evolutionary point of view), here the authors arrive to equation (1.1) with

ψ = 1.Metastasis size distribution A linear version of equation (1.1) arises in modeling on- ology. In [66℄, the authors propose it as a dynami al model for the growth and sizedistribution of multiple metastati tumors. The density n(t, x) represents tumors of sizex and the birth term are metastoti tumors that grow (with a x depenent velo ity to takeinto a ount Gomperts law).Epidemiology As mentioned earlier, the �rst age stru tured model in epidemiology goesba k to Kerma k and M Kendri k, [69, 80℄. It has been widely studied by many mathema-ti ians as well as biologists (see [57, 58, 107℄). Here we brie�y introdu e the model whi hdes ribes the propagation of a virus in a population. Let Σ(t), n(t, x), R(t) denote thetotal sus eptible population, infe tive population and total re overed population at time19

t respe tively, re�e ting the e�e t of virus. Here the age stru ture is in orporated intothe density of infe tive population n and x represents the age in the disease. Let BΣ, dΣdenote birth rate, mortality rate of sus eptible population, BR(x), dn denote the rate of onversion of infe ted ells into re overed ells, mortality rate of infe ted ells. It is verynatural to assume that dn > dΣ. Another main assumption in this model is that indivi-duals are infe ted from en ounters between sus eptibles and infe ted individuals with agex in the disease with the rate ψ(x). Therefore the total infe tion rate is S(t) de�ned by

S(t) =

∫ ∞

0

ψ(x)n(t, x)dx,and thus the Kerma k-M kendri k model is de�ned, for t > 0, x > 0, byd

dtΣ(t) = BΣ − dΣΣ(t) − S(t)Σ(t),

{∂∂tn(t, x) + ∂

∂xn(t, x) +

[dn(x) +BR(x)

]n(t, x) = 0,

n(t, 0) = S(t)Σ(t),

d

dtR(t) =

∫ ∞

0

BR(x)n(t, x)dx.In the book [36℄ one an �nd a re ent a ount of the subje t. At this point we would liketo noti e that if we make the quasistati hypothesis on Σ(t), we arrive at0 = BΣ − dΣΣ(t) − S(t)Σ(t), Σ(t) =

BΣ

dΣ + S(t).This model falls in the lass studied by Ph. Mi hel [84℄ that we re all in Se t. 1.7.E ology Our next examples on ern models in e ology. To begin with, we refer thereader to the book [88℄ whi h ontains a full analysis of ro odiles population based onage stru tured equations. Bees et al (see [7℄) studied Dero eras reti ulatum populationdynami s (these spe ies of slugs ause the majority of the damage to agri ultural ropsand turn out to be a pest of global e onomi importan e. Another example of appli ationin e ology an be found in [46℄ where a similar model des ribes the density n(t, x) of aparti ular tree (Pinus Cembra) in a forest. Let S(t) des ribes the population of a ertainbird (Nu ifraga Caryo ata tes) that helps disseminating the seeds. They arrive to a variantwhere the equation on S(t) is

d

dtS(t) + µ(S(t))S(t) = S(t)

∫ t

−∞

k(t− x)P (s, t)dswhere P (s, t) is the total population of trees at time t whi h were born at time s, s ≤ t. Ifthe time s ale for birds reprodu tion is faster than that of Pinus Cembra (these trees don'tprodu e seeds in �rst forty years !) then we arrive to a model as (1.1)�(1.2), transformingthis delay integral into an age stru tured equation.20

Cell proliferation As mentioned earlier, modeling ell ultures by age stru tured equa-tions is an old subje t, see [106, 23, 5℄ for instan e. Gyllenberg proposed a nonlinear agestru tured model for ba terial ulture growth in a ontinuous fermentation pro ess (see[54℄). He studied the existen e, uniqueness, positivity, and boundedness of solutions, theexisten e of equilibrium solutions and the stability of equilibria of the model. In his model,the growth, death and �ssion rates of the ells are nonlinear fun tions of the substrate on entration in the rea tor tank.More general and realisti age-stru tured models were obtained by di�erent authors.We re all now some of them be ause they use PDEs in higher dimensions. The �rst one wasproposed by Rotenberg [98℄, still for ells, who introdu es a maturation velo ity variableµ ∈ [0, 1]. It is the ratio between biologi al age and physi al age. Let n(t, x, µ) be thedensity of population, then it satis�es

∂∂tn(t, x, µ) + ∂

∂xn(t, x, µ) + d(x, µ)n(t, x, µ) =

∫K(x, µ, µ′)n(t, x, µ′)dµ′,

n(t, 0, µ) =∫b(x′, µ, µ′)n(t, x′, µ′)dµ′dx′,

n((0, x, µ) = n0(x, µ).Here K(x, µ, µ′) is the probability of a hange of maturation velo ity from µ to µ′. Forstability and long time asymptoti s results via GRE method and existen e of periodi solutions see [87℄.Medi al s ien es, tumor growth Another model was proposed by Ma key and Rey[76℄ to study the produ tion of red blood ells (hematopoiesis) has been attra ted byseveral people. In this model the main biologi al assumption is the life period of any ell is divided into the quies ent and proliferating phases. The ells in quies ent phase an't divide, but they mature and if they don't die, then they will enter the proliferatingphase. There, when they don't die by apoptosis, they will divide and give birth to twodaughter ells whi h are in quies ent phase. Hen e this age stru tured model takes intoa ount maturity m and is the following oupled system of two nonlinear equations. Fort ≥ 0, x ≥ 0, m ≥ 0,

∂∂tp(t, x,m) + ∂

∂xp(t, x,m) + ∂

∂m

[V (m)p(t, x,m)

]+ d1(m)p(t, x,m) = 0,

p(t, 0, m) = b2(m,N(t,m)),

N(t,m) =∫∞

0n(t, x,m)dx,

{∂∂tn(t, x,m) + ∂

∂xn(t, x,m) + ∂

∂m

[V (m)n

]+[d2(m) + b2(m,N(t,m))

]n = 0,

n(t, 0, m) = 2∫∞

0b1(x,m)p(t, x, G−1(m))dx.Here p(t, x,m) and n(t, x,m) denotes the population density of proliferating ells andresting ells at time t, having age x, with maturity m, with mortality rates d1, d2 res-pe tively. Cell division rate or birth rate is b1. The main assumptions in this model is

V (0) = 0 and ∀m,∫ m0

dm′

V (m′)= ∞, V (·) is in reasing. Se ond assumption is G : R+ → R+su h that G ∈ C1(R+), G is in reasing, G(0) = 0 and G(m) < m. Moreover b2(., N) is21

de reasing in N. This model is related to an er modeling, it has nonvanishing steadystates and possesses periodi solutions (whi h might represent some lassi al blood di-sease alled Chroni Myeloid Leukemia). Adimy, Pujo-Menjouet simpli�ed this model byassuming that all ells in proliferating phase divide exa tly at age τ (see [3℄) and obtai-ned existen e, uniqueness results and global exponential stability. A more general asewas onsidered by Adimy and Crauste in whi h ells in proliferating phase divide at agedistributed between τ and τ with ontinuous density fun tion x 7→ k(x,m) supported in[τ , τ ] (see [2℄). An important observation made in [2, 3℄ is if the time of repli ation is largeenough, then the destru tion of population of stem ells a�e ts the total population ingreater extent and the total population goes extin t in �nite time.A further example in higher dimension is a oupled nonlinear model whi h takesthe ontent of y lin/ y lin dependent kinases(CDKs) into a ount has been studied byPerthame et al. in [9℄ for omparing healthy tissues and tumoral tissues. Any tissue omprises two ompartments namely proliferative and quies ent ompartments. The �rstone represents omplete ell y le with G1, S, G2, M phases. Cy lins/CDKs omplexesare the most ru ial ontrol mole ules in phase transitions. Ea h phase has its parti ular y lins/CDKs. The main idea behind the modeling is the proliferating ells grow anddivide whereas quies ent ells don't possess physiologi al evolution. Let p(t, x, c), q(t, x, c)be the densities of proliferating and quies ent ells at time t with age a and ontentc in y lin/CDKs. Let L(x, c), F (x, c) denote demobilisation rate from proliferation toquies en e and the rate of ell division. Let d1, d2 be the rates of apoptosis of proliferating ells and quies ent ells respe tively. Let Γ1(x, c) be the evolution speed of y lin/CDKsand Γ0 > 0 be a onstant. With this information the model reads as an age stru turedmodel

∂∂tp(t, x, c) + ∂

∂xp(t, x, c) + ∂

∂c

(Γ1p(t, x, c)

)

−[L(x, c) + F (x, c) + d1

]p(t, x, c) +G(S(t)) q(t, x, c),

p(t, 0, c) =∫b(x, c, c′)p(t, x, c′)dc′,

∂∂tq(t, x, c) = L(x, c) p(t, x, c) −

[G(S(t)) + d2

]q(t, x, c),Here G(S) is the rate at whi h quies ent ells reenter the proliferative phase. S(t) denotestotal weighted population given by

S(t) =

∫ ∞

0

∫ ∞

0

ψ1(x, c)p(t, x, c) + ψ2(x, c)q(t, x, c)dxdc.Typi al boundary data and appropriate initial data has been given to lose the model.Finally let us mention an original model des ribing the ovulatory pro ess. Clément etal. [42℄ have derived an age and maturity stru tured model, with either proliferative ordi�erentiated phases, depending on the ell maturity level and ells di�erentiated phasewill never re enter proliferative phase. In [42℄, nonlo al nonlinearities also arise throughhormones produ tion. For instan e Folli ule�Stimulating Hormone a ts on folli ular ellsand are des ribed by a bio hemi al dynami al model.22

1.3 Assumptions and EigenelementsIn this se tion, we present our basi assumptions and de�nitions whi h will be usedfrom now on. We always assume that the fun tions d, B, ψ are ontinuous in S, nonne-gative, lo ally bounded. Then, we also need the assumptionsB(., 0) ∈ L∞(0,∞) ∩ L1(0,∞), B(x, .) ∈ L∞

loc(0,∞) for all x ≥ 0, (1.4)1 <

∫ ∞

0

B(x, 0)e−R x

0d(y,0)dydx <∞, (1.5)

0 <

∫ ∞

0

B(x,∞)e−R x

0d(y,∞)dydx < 1, (1.6)

1 < lima→∞

∫ ∞

0

B(x,∞)eax−R x

0d(y,∞)dydx = β∞ <∞. (1.7)There exists L > 0 su h that for all x, S1, S2 ≥ 0 we have

|B(x, S1) − B(x, S2)| ≤ L|S1 − S2|, |d(x, S1) − d(x, S2)| ≤ L|S1 − S2|. (1.8)∂d(., .)

∂S> 0, (1.9)

d(.,∞) ∈ L∞(0,∞), d(x, .) ∈ L∞loc(0,∞) for all x ≥ 0, (1.10)

∂B(., .)

∂S< 0. (1.11)There exist two maps D0, D∞ : R+ → R de�ned by

D0(S) = infx

{(d(x, 0) − d(x, S) +

φ0(0)

φ0(x)

(B(x, S) −B(x, 0)

)}, (1.12)

D∞(S) = supx

{(d(x,∞) − d(x, S) +

φ∞(0)

φ∞(x)

(B(x, S) − B(x,∞)

)}. (1.13)Finally for the ompetition weight ψ(·), we assume that there are two positive onstants

C0min and C0

max su h thatC0

minφ0(x) ≤ ψ(x) ≤ C0maxφ0(x). (1.14)Also there are two positive onstants C∞

min and C∞max su h that

C∞minφ∞(x) ≤ ψ(x) ≤ C∞

maxφ∞(x). (1.15)Our last two assumptions use notations that are introdu ed later on. However we givethem now in order to gather all the assumptions. The eigenelements λ0, λ∞, N0, N∞, φ0, φ∞are de�ned as follows.Consider the eigenvalue problem orresponding to S = 0

ddxN0(x) + (d(x, 0) + λ0)N0(x) = 0, ∀ x ≥ 0,

N0(0) =∫∞

0B(y, 0)N0(y)dy = 1,

N0(·) ≥ 0, N0 ∈ L1(R+) ∩ L∞(R+).

(1.16)23

The orresponding adjoint equations for above equations are{

− ddxφ0(x) + (d(x, 0) + λ0)φ0(x) = φ0(0)B(x, 0), ∀ x ≥ 0,

∫∞

0φ0(x)N0(x)dx = 1, φ0(·) ≥ 0, φ0 ∈ L∞(R+).

(1.17)The eigenvalue problem orresponding to S = ∞ is also written as

ddxN∞(x) + (d(x,∞) + λ∞)N∞(x) = 0, ∀ x ≥ 0,

N∞(0) =∫∞

0B(y,∞)N∞(y)dy = 1,

N∞(·) ≥ 0, N∞ ∈ L∞loc(R

+).

(1.18)Similarly for this set of equations at S = ∞, we have the following adjoint equation{

− ddxφ∞(x) + (d(x,∞) + λ∞)φ∞(x) = φ∞(0)B(x,∞), ∀ x ≥ 0,

∫∞

0φ∞(x)N∞(x)dx = 1, φ∞(·) ≥ 0, φ∞ ∈ L∞(R+).

(1.19)We on lude this se tion with a result on erning existen e and uniqueness of solutionsto eigenvalue problems, orresponding adjoint problems and signs of eigenvaluesTheorem 7. Under assumptions (1.4)�(1.7), the problems (1.16)�(1.19) have unique so-lutions, and we have the inequalities λ0 > 0, λ∞ < 0. Moreover there exists C > 0, for allr > 0 we have

N0(x) ≤ e−λ0x, N∞(x) ≤ e−λ∞x, φ0(x) ≤ C, φ∞(x) ≤ β∞φ∞(0)e(λ∞−r)xfor all x ≥ 0.Démonstration. We �rst onsider the dire t problems on N0 and N∞. A dire t omputa-tion shows that N0(x) is given byN0(x) = e−

R x

0(λ0+d(y,0))dy.Consider the map λ 7−→

∫ ∞

0

B(x, 0)e−λxe−R x

0d(y,0)dydx and observe that it is integrablefor all λ > 0, ontinuous and de reasing. Moreover by (1.5)

limλ→0

∫ ∞

0

B(x, 0)e−λxe−R x

0d(y,0)dydx > 1,

limλ→∞

∫ ∞

0

B(x, 0)e−λxe−R x

0d(y,0)dydx = 0.Therefore there exists a unique λ0 satisfying (1.16) whi h is positive.A similar argument using (1.6) and (1.7) proves that λ∞ is negative. Next we turn tothe adjoint problem. One an also ompute φ0 expli itly to get

φ0(x) =φ0(0)

e−R x

0(λ0+d(y,0))dy

∫ ∞

x

B(y, 0)e−R y

0(λ0+d(x′,0))dx′dy, (1.20)

φ∞(x) =φ∞(0)

e−R x

0(λ∞+d(y,∞))dy

∫ ∞

x

B(y,∞)e−R y

0(λ∞+d(x′,∞))dx′dy. (1.21)24

Clearly φ0, φ∞ ≥ 0. Using (1.4) we have B(., 0) ∈ L1(0,∞), therefore φ0 ∈ L∞(0,∞).Moreover we an hoose φ0(0) in order to normalize as follows∫ ∞

0

N0(x)φ0(x)dx = φ0(0)

∫ ∞

0

xB(x, 0)e−R x

0(λ0+d(x′,0))dx′dx = 1.This normalization is possible be ause B(., 0) ∈ L∞(0,∞) and λ0 > 0. Now we prove asimilar result on φ∞. A ording to (11), we denote an upper bound by L′, d(x,∞) ≤ L′ <

∞. From the expli it formula (1.21) for φ∞(x), we getφ∞(x) ≤ φ∞(0)e(λ∞+L′)x

∫ ∞

x

B(y,∞)e−R y

0(λ∞+d(x′,∞))dx′dyand thus

φ∞(x) ≤ φ∞(0)e(λ∞−r)x

∫ ∞

x

e(−λ∞+L′+r)yB(y,∞)e−R y

0d(x′,∞)dx′dy.Now thanks to (1.7), we obtain φ∞(x) ≤ β∞φ∞(0)e(λ∞−r)x, for any r > 0, with λ∞ < 0.Again it an be normalized be ause N∞ has growth as e−λ0x.1.4 Existen e of Solutions to the Linear ProblemIn this se tion we onsider that S(t) is a given lo ally bounded fun tion. We re allqui kly basi fa ts, whi h are used later on, about the linear renewal equation

∂∂tn(t, x) + ∂

∂xn(t, x) + d(x, S(t))n(t, x) = 0, t ≥ 0, x ≥ 0,

n(t, 0) =∫∞

0B(x, S(t))n(t, x)dx,

n(0, x) = n0(x) ∈ L1loc(R

+).

(1.22)We de�ne the notion of weak (distributional) solutions as usualDe�nition 1. A fun tion n ∈ L1loc(R

+ × R+) satis�es the renewal equation (1.22) inweak sense if ∫∞

0B(t, x)|n(t, x)|dx ∈ L1

loc(R+) and for all T > 0, for all test fun tions

Ψ ∈ C1comp

([0, T ] × [0,∞)

) su h that Ψ(T, x) ≡ 0, we have−

∫ T

0

∫ ∞

0

n(t, x){ ∂∂t

Ψ(t, x) +∂

∂xΨ(t, x) − d(x, S(t))Ψ(t, x)

}dx dt

=

∫ ∞

0

n0(x)Ψ(0, x)dx+

∫ T

0

Ψ(t, 0)

∫ ∞

0

B(x, S(t))n(t, x)dx dt.General uniqueness results an be proved, whi h means that other notions of solutionsare all equivalent.Theorem 8. Let there exist M > 0 su h that B(., .), d(., .) < M , S(t) ≥ 0, S(t) ∈L∞

loc(R+),n0 ∈ L∞(0,∞)∩L1(0,∞) then there is a unique weak solution n ∈ C(R+;L1(R+))solving (1.22). Moreover n(t, x) ≥ 0 whenever n0 ≥ 0, and

∫ ∞

0

|n(t, x)|dx ≤ e||(B−d)+||∞t

∫ ∞

0

|n0(x)|dx. (1.23)25

We refer to [93℄ for proofs of su h results. In parti ular the GRE method allows formore pre ise results as the long time asymptoti s (1.3) when S(t) ≡ S is independent oftime.1.5 Nonlinear Problem, uniform boundsIn this se tion we prove existen e and uniqueness along with a priori bounds on S(t).First we state our main theorem asTheorem 9. Assume (1.4)�(1.15), n0 ∈ L∞(0,∞)∩L1(0,∞), also assume ∫∞

0φ0(x)n0(x)dx >

0, then there exists a unique weak or distributional solution n ∈ C(R+;L1(R+)) to (1.1)�(1.2). Moreover, be ause λ0 > 0, λ∞ < 0 (see Theorem 7) there exists two positive onstants m, M su h thatm ≤ S(t) ≤M, ∀ t > 0.This theorem is a onsequen e of Theorem 10 and Proposition 6 below. In the hypothe-sis, ∫∞

0φ0(x)n0(x)dx > 0 is a te hni al one, it tells us that the initial population densityshould be positive on a subset having nonzero measure, of [0,∞). Under this onditionthe estimate we present here tells us that the weighted population ontinue for ever andit neither blows up, nor goes extin t even at in�nite time. Hen e these bounds open thequestion of long time behavior whi h is treated later.Before proving the main theorem we give a statement with stronger hypothesis on

B, d.Theorem 10. Let us assume (1.8), there exists M > 0 su h that B(., .), d(., .),ψ(·) < M and n0 ∈ L∞(0,∞) ∩ L1(0,∞), then there exists a unique weak solution n ∈C(R+;L1(R+)) to (1.1)�(1.2).Démonstration. We prove this result by the Bana h �xed point theorem. First, we setX = C([0, T ]) with the sup norm, later we hoose T (very small). Let X+ be the set ofall nonnegative ontinuous fun tions on [0, T ] and de�ne Λ = ||(B − d)+||∞. For S(t) ∈

C([0, T ]), thanks to Theorem 8, we have n(t, x) ∈ C([0, T ];L1(R+)

) solving

∂∂tn(t, x) + ∂

∂xn(t, x) + d(x, S(t))n(t, x) = 0, t ∈ [0, T ], x ≥ 0,

n(t, 0) =

∫ ∞

0

B(x, S(t))n(t, x)dx,

n(0, x) = n0(x) ≥ 0.

(1.24)Now de�ne a map T : X+ → X+ byS(t) 7−→

∫ ∞

0

ψ(x)n(t, x)dx. (1.25)To prove the Theorem 10, it is enough to prove T is a ontra tion map.26

Let n1(t, x), n2(t, x) be the solutions of (1.24) orresponding to S1(t), S2(t), thenn := n1 − n2 satis�es

∂∂tn(t, x) + ∂

∂xn(t, x) + d(x, S1(t))n(t, x) +

[d(x, S1) − d(x, S2)

]n2 = 0, t ∈ [0, T ], x ≥ 0,

n(t, 0) =

∫ ∞

0

B(x, S1(t))n(t, x) +[B(x, S1) − B(x, S2)

]n2 dx,

n(0, x) ≡ 0.Now |n(t, x)| satis�es

∂∂t|n| + ∂

∂x|n| + d(x, S1(t))|n| ≤ |d(x, S1(t)) − d(x, S2(t))|n2, t ∈ [0, T ], x ≥ 0,

|n(t, 0)| ≤

∫ ∞

0

(B(x, S1(t))n(t, x) + |B(x, S1(t)) − B(x, S2(t))|n2

)dx,

|n(0, x)| ≡ 0. (1.26)Therefore by integration in age, we haved

dt

∫ ∞

0

|n(t, x)|dx =

∫ ∞

0

∂

∂t|n| dx

≤

∫ ∞

0

−∂

∂x|n| − d(x, S1(t))|n| + |d(x, S1(t)) − d(x, S2(t))|n2 dx

≤ |n(t, 0)| −

∫ ∞

0

d(x, S1(t))|n| dx+ L|S1(t) − S2(t)|

∫ ∞

0

n2(t, x) dx

≤

∫ ∞

0

(B(x, S1(t))n(t, x) + |B(x, S1(t)) − B(x, S2(t))|

)n2 dx

−

∫ ∞

0

d(x, S1(t))|n|dx+ L|S1(t) − S2(t)|

∫ ∞

0

n2(t, x) dx.

≤ Λ

∫ ∞

0

|n(t, x)| dx + 2L|S1(t) − S2(t)|

∫ ∞

0

n2(t, x)dx

≤ Λ

∫ ∞

0

|n(t, x)|dx + 2L(

sup0≤t≤T

|S1(t) − S2(t)|)|n0|L1eΛt.Gronwall's lemma gives

∫ ∞

0

|n(t, x)|dx ≤ 2Lt|n0|L1e2Λt(

sup0≤t≤T

|S1(t) − S2(t)|),and thus

sup0≤t≤T

∫ ∞

0

|n(t, x)|dx ≤ 2LT |n0|L1e2ΛT(

sup0≤t≤T

|S1(t) − S2(t)|). (1.27)From this we dedu e that

sup0≤t≤T

|TS1 − TS2| = sup0≤t≤T

|

∫ ∞

0

ψ(x)(n1(t, x) − n2(t, x))dx|

≤M sup0≤t≤T

∫ ∞

0

|n(t, x)|dx

≤ 2MLT |n0|L1e2ΛT(

sup0≤t≤T

|S1(t) − S2(t)|).27

Now we hoose T small enough su h that T be omes a ontra tion map.Hen e we proved existen e and uniqueness of solution of (1.1)�(1.2).Proposition 6. Under assumptions (1.4)�(1.7), (1.9)�(1.15) and ∫∞

0φ0(x)n0(x) > 0there exists m,M > 0 for S(t) whi h is an unknown in the oupled system (1.1)�(1.2)su h that m ≤ S(t) ≤M ∀t > 0.Démonstration. To prove this proposition we use a te hnique developed Carrillo et al. in[20℄ based on adjoint problem. First we treat the lower bound. We de�ne an auxiliaryfun tion

S0(t) =

∫ ∞

0

φ0(x)n(t, x)dx. (1.28)A duality omputation (related to the GRE method ) using (1.1), (1.17) leads tod

dtS0(t) = λ0S0(t) +

∫ ∞

0

(d(x, 0) − d(x, S(t)

)φ0(x)n(t, x) dx

+

∫ ∞

0

φ0(0)[B(x, S(t)) −B(x, 0)])n(t, x)dx.

≥ λ0S0(t) + D0(S(t))S0(t)

≥ λ0S0(t) + D0(C0maxS0(t))S0(t). (1.29)Last inequality holds be ause D0 is de reasing with D0(0) = 0. As long as C0

maxS0(t) issmaller than D−10 (−λ0), S0 is in reasing. Therefore we obtain

S0(x) ≥ min{S0(0),

D−10 (−λ0)

C0max

}, ∀t ≥ 0.Finally we exploit the assumption (1.14) to get

S(x) ≥ m := min{C0

minS0(0),C0

min

C0max

D−10 (−λ0)

}, ∀t ≥ 0.To prove the other inequality we use the same te hnique by de�ning another auxiliaryfun tion

S∞(t) =

∫ ∞

0

φ∞(x)n(t, x)dx. (1.30)We follow the same methodology to get an upper bound for S(·). A parti ular hoi e ofM, we get after repeating similar exer ise is given by

M = max{C∞

maxS∞(0),C∞

max

C∞min

D−1∞ (−λ∞)

}.

28

1.6 A Case with asymptoti De ouplingFor this se tion, we onsider parti ular variants of above birth and death rates. Inthis ase we redu e the renewal equation to a single O.D.E. where we an easily on ludeabout the onvergen e of the solution to the steady state. It is whend(x, S) = d1(x) + d2(S), d1(x) ≥ 0, (1.31)

B(x, S) ≡ B(x) ≥ 0, (1.32)∫ ∞

0

B(x)e−D(x)dx = 1, D(x) =

∫ x

0

d1(x)dx. (1.33)Here, the mortality rate d is the sum of the mortality rate due to inherent spe ies agingd1(x) ≥ 0 and �u tuations (extra birth or death) d2. In parti ular d2 has no sign but it isnatural to keep d1(x) + d2(S) nonnegative.Our method relies on the study of the following linearized state given by

ddxN(x) + d1(x)N(x) = 0, x > 0,

N(0) =

∫ ∞

0

B(x)N(x)dx = 1.(1.34)In other words N = e−D(x). As usual we introdu e the adjoint problem for (1.34)

− ddxφ(x) + d1(x)φ(x) = φ(0)B(x), x > 0,

∫ ∞

0

φ(x)N(x)dx = 1.(1.35)1.6.1 A parti ular solutionBefore going to the main onvergen e result whi h we prove in next subse tion weprove the followingProposition 7. With assumptions (1.32)�(1.33), the system (1.1)�(1.2) admits the par-ti ular solution n(t, x) = n(t)N(x) with n(t) given by the di�erential equation

d

dtn(t) + d2(kn(t))n(t) = 0, (1.36)withk =

∫ ∞

0

ψ(x)N(x)dx. (1.37)Therefore the nonlinear problem admits these parti ular family of solutions to (1.36)(parametrized by the initial value n(0)) whi h an be solved by standard O.D.E. methods.In parti ular from this example we an insight the long time asymptoti s of the solu-tion. The onditions λ0 > 0, λ∞ < 0 here means d2(0) < 0, d2(∞) > 0. Assuming alsod′2(·) > 0 as usual we see that n(t) −→ n > 0 as t −→ ∞ with n the unique solution tod2(kn) = 0. 29

1.6.2 Long time de ouplingNow let us turn our attention towards a long time asymptoti s result for the generalsolutions to the nonlinear system (1.1)�(1.2). We prove exponential onvergen e to thesolution obtained by variables separable method. We use L1 onvergen e with a properweight fun tion whi h will be introdu ed later.This an be done introdu ing n(t) =∫∞

0n(t, x)φ(x), whi h satis�es (we leave the al ulation to the reader)

d

dtn(t) + n(t)d2(S(t)) = 0. (1.38)In this subse tion we explain why these solutions built previously attra t all other traje -tories. Here we adapt a lassi al argument for paraboli systems, whi h an be found forinstan e in [68℄. Namely, we prove the long time asymptoti s result for this modelTheorem 11. Under the assumptions (1.31)�(1.33) and if there exists µ ≥ 0, δ > 0su h that B(x) ≥ µφ(x), µ + d2(S(t)) ≥ δ, then the solution of age stru tured equation(1.1)�(1.2) satis�es

∫ ∞

0

|n(t, x) − n(t)N(x)|φ(x)dx ≤ e−δt∫ ∞

0

|n0(x) − n(0)N(x)|φ(x)dxwith n(t) =∫∞

0n(t, x)φ(x)dx.Démonstration. We prove this onvergen e result with the help of a ombination of aperturbation argument and of the duality method. One an easily ompute to arrive at

∂

∂tn(t)N(x) +

∂

∂xn(t)N(x) +

(d1(x) + d2(S(t))

)n(t)N(x) = 0.We subtra t this expression from (1.1) to get

∂∂t

(n− nN) + ∂∂x

(n− nN) +(d1(x) + d2(S(t))

)(n− nN) = 0,

(n− nN)(t, 0) =

∫ ∞

0

B(x)(n− nN)dx.Let us denote h(t, x) = n(t, x) − n(t)N(x). With this notation we have by onstru tion∫ ∞

0

h(t, x)φ(x)dx = 0, (1.39)further

∂∂t|h(t, x)| + ∂

∂x|h(t, x)| +

(d1(x) + d2(S(t))

)|h(t, x)| = 0,

|h(t, 0)| = |

∫ ∞

0

B(x)(h(t, x))dx|.30

We integrate against φdx to and use (1.39) arrive at0 =

d

dt

∫ ∞

0

|h(t, x)|φdx− φ(0)∣∣∫ ∞

0

B(x)h(t, x)dx∣∣

+

∫ ∞

0

φ(0)B(x)|h(t, x)|dx+ d2(S(t))

∫ ∞

0

|h(t, x)|φdx,

=d

dt

∫ ∞

0

|h(t, x)|φdx− φ(0)∣∣∫ ∞

0

(B(x) − µφ(x)

)h(t, x)dx

∣∣

+

∫ ∞

0

φ(0)B(x)|h(t, x)|dx+ d2(S(t))

∫ ∞

0

|h(t, x)|φdx,

=d

dt

∫ ∞

0

|h(t, x)|φdx+(µ+ d(S(t))

)∫ ∞

0

|h(t, x)|φdxFrom hypothesis we have µ+ d(S(t)) ≥ δ, therefored

dt

∫ ∞

0

|n− nN |φdx+ δ

∫ ∞

0

|n− nN |φdx ≤ 0.From this the announ ed result follows.1.7 A on ave nonlinearity on birth termHere we re all the result of Ph. Mi hel [84℄ based on ideas from the GRE method. He onsiders a parti ular nonlinearity in the age stru tured equation but that still ontainsseveral interesting appli ations, and in parti ular the Kerma k-M Kendri k model of epi-demiology (see Se t. 1.2) in the quasisasti ase0 = BΣ − dΣΣ(t) − S(t)Σ(t).His result des ribes the long time asymptoti s of solutions : the nontrivial steady state isglobally attra tive.The problem is given by

∂∂tn(t, x) + ∂

∂xn(t, x) + d(x)n(t, x) = 0, t ≥ 0, x ≥ 0,

n(t, 0) = g(∫∞

0B(x)n(t, x)dx

),

n(0, x) = n0(x) ≥ 0.

(1.40)Here and as usual, d and B are nonnegative fun tions and g(·) is a ontinuous and nonli-near fun tion that satis�es assumptions whi h are des ribed later on. Now our goal is toobtain that the solution to (1.1)�(1.2) onverges to the nonzero steady state.Changing the notation for g(·) by g(r·) and B in rB if ne essary, we may assume∫ ∞

0

B(x)e−D(x)dx = 1, with D(x) =

∫ x

0

d(y)dy.31

Moreover let us make the following assumption on the nonlinearity g (that is more generalbut ontains in reasing and on ave fun tions)

∃ N0 > 0, g(N0) = x0,

x < g(x) < N0, for x < N0,

x0 < g(x) < x, for N0 < x.

(1.41)With this hypothesis it is lear that the orresponding steady state problem

ddxN(x) + d(x)N(x) = 0, x ≥ 0,

N(0) = g(∫∞

0B(x)N(x)dx

),

(1.42)admits the unique solution N(x) = N0e−D(x). As an immediate onsequen e we obtainthat

g(∫ ∞

0

B(x)N(x)dx)

= N0 =

∫ ∞

0

B(x)N(x)dx.On e interpreted in this way, the orresponding adjoint problem is given by{

− ddxφ(x) + d(x)φ(x) = φ(0)B(x), x ≥ 0,

∫∞

0φ(x)N(x)dx = 1, φ(·) ≥ 0.

(1.43)We are now able to state the nonlinear stability result of [84℄Theorem 12. Under the assumption (1.41), as t → ∞, any solution n(t, x) to (1.40),with nonzero initial data, onverges to the solution N(x) to (1.42), i.e.,∫ ∞

0

∣∣n(t, x) −N(x)∣∣φ(x)dx −→ 0 as t −→ ∞.Démonstration. As we did in previous se tions we try to get a ontra tion inequality andwrite

∂

∂t[n(t, x) −N(x)]+ +

∂

∂x[n(t, x) −N(x)]+ + d(x)[n(t, x) −N(x)]+ = 0,

[n(t, 0) −N(0)]+ =[g(∫ ∞

0

B(x)n(t, x)dx)− g(∫ ∞

0

B(x)N(x)dx)]

+.32

After integrating against φ(x)dx to obtaind

dt

∫ ∞

0

[ n(t, x) −N(x)]+φdx

= −

∫ ∞

0

( ∂∂x

[n(t, x) −N(x)]+φ(x) − d(x)[n(t, x) −N(x)]+φ)dx

= φ(0)[g(∫ ∞

0

B(x)n(t, x)dx)− g(∫ ∞

0

B(x)N(x)dx)]

+

+

∫ ∞

0

(d

dxφ(x) − d(x))[n(t, x) −N(x)]+dx

= φ(0)[g(∫ ∞

0

B(x)n(t, x)dx)− g(∫ ∞

0

B(x)N(x)dx)]

+

−

∫ ∞

0

φ(0)B(x)[n(t, x) −N(x)]+dx.Let us onsider the times t where ∫∞

0B(x)n(t, x)dx ≥

∫∞

0B(x)N(x)dx, then from (1.41)we noti e that

g

(∫ ∞

0

Bndx

)− g

(∫ ∞

0

BNdx

)≤

∫ ∞

0

Bndx−

∫ ∞

0

BNdx ≤

∫ ∞

0

B[n−N ]+dx.Therefore [g(

∫ ∞

0

Bndx) − g(

∫ ∞

0

BNdx)]

+≤

∫ ∞

0

B[n−N ]+dx.For the other values of t, i.e, ∫∞

0B(x)n(t, x)dx ≤

∫∞

0B(x)N(x)dx then the above inequa-lity is straightforward and hen e we have obtained

d

dt

∫ ∞

0

[n(t, x) −N(x)]+φdx ≤ 0.Using similar arguments we getd

dt

∫ ∞

0

[n(t, x) −N(x)]−φdx ≤ 0.Finally we getd

dt

∫ ∞

0

|n(t, x) −N(x)|φdx ≤ 0. (1.44)Using standard ompa tness argument (for instan e see [82℄, [87℄, [93℄) we obtain the onvergen e of the solution to (1.40) towards the steady state N(x).To on lude this se tion, we point out that the assumption (1.41) is nearly optimalfor the above stated stability result. In [84℄, the reader may �nd many extensions, morepre ise results and ounterexamples. 33

1.8 Stability for nonlinearities redu ing to ODE sys-temsIn general, the problem of long time asymptoti s for system (1.1)�(1.2) is not om-pletely solved. Any approa h takes advantage of inbuilt spe ial properties of the systemand pro eeds by giving su� ient onditions to on lude the long time behavior. One ofthe well known te hniques to deal with su h kind of problems is to redu e the originalequation or system to a system of ODE. For instan e Iannelli [58℄ redu ed both linear andnonlinear renewal equations having parti ular stru ture. In the nonlinear ase he studiedlo al behavior around nonzero steady state. In this se tion �rst we dis uss a ase in whi hwe an get nonlinear stability and the solution of (1.1)�(1.2) onverges to the non zerosteady state. Finally we give some examples in whi h we redu e (1.1)�(1.2) to a 2 × 2system of ODE. We pay attention to this problem and we would like to use this pathbe ause it serves as a simpli�ed ba kground before studying the more general problem.1.8.1 Redu tion to a system of ODE (Example 1)A lassi al ase that an be redu ed to a di�erential system has been studied byIannelli (see [53, 58℄). Here in Example 1, we slightly modify that to study its linear andnonlinear stability. It is when for some α > 0,B(x, S) = b1(S)e−αx + b2(S) (1.45)withbi > 0,

dbidS

(·) < 0 for i = 1, 2.Also we assume the death rate is independent of aged(x, S) = d(S),

d

dSd(·) > 0. (1.46)Then we de�ne two quantities

S(t) =

∫ ∞

0

n(t, x)dx, Q(t) =

∫ ∞

0

e−αxn(t, x)dx. (1.47)Whi h means that the ompetition weight is assumed to be onstantψ ≡ 1. (1.48)Now (1.1) be omes

∂∂tn(t, x) + ∂

∂xn(t, x) + d(S(t))n(t, x) = 0, t ≥ 0, x ≥ 0,

n(t, 0) = b1(S(t))Q(t) + b2(S(t))S(t),

n(0, x) = n0(x) ≥ 0.

(1.49)34

Integrating (1.49) against dx and e−αxdx, we arrive to the system on S(·) and Q(·)

dS

dt(t) = [b2(S(t)) − d(S(t))]S(t) + b1(S(t))Q(t),

dQ

dt(t) = b2(S(t))S(t) + [b1(S(t)) − d(S(t)) − α]Q(t),

0 < Q(0) < S(0).

(1.50)Remark 1. With this notation if α = 0, then (1.1)�(1.2) redu es to S = Q anddS

dt(t) = [b(S(t)) − d(S(t))]S(t), b = b1 + b2.If d(S)−b(S) is independent of total population i.e. S, then we get the lassi al Malthusianequation with parameter d − b. If d(S) − b(S) = k1 − k2S for onstants k1, k2 > 0, thenour model (1.1)�(1.2) turned out to be another lassi al model of Verhulst with intrinsi growth onstant k1 and arrying apa ity k1

k2(see Se t. 1.1). In general from (1.8.1),(1.46)one an see that S(t) −→ S as t −→ ∞, with S being unique solution of b(S) = d(S) if itexists, moreover n(t, x) −→ b(S)Se−d(S)x as t −→ ∞. Even though the system (1.1)�(1.2)is redu ed to a single equation we an noti e that it is still more elaborate than the exampleof Se t. 1.6.1.In order to prove that S(t) and Q(t) are omparable, we have theLemma 2. If S(t), Q(t) are solutions of (1.50) then there exists ρ > 0 depending on

(S(0), Q(0)), su h that ρS(t) < Q(t) < S(t).Démonstration. One an easily omputed

dt

(Q(t)

S(t)

)=(b2(S) + b1(S)

Q

S

)(1 −

Q

S) − α

Q

S. (1.51)As b2(0) > 0, for �xed initial data (S(0), Q(0)), from (1.51) it follows that the quantity

Q(t)

S(t) annot be arbitrary lose to zero. This implies there exists ρ su h that ρS(t) < Q(t).In (1.51) we observe that as soon as Q(t)

S(t)> 1, the quantity Q(t)

S(t)starts de reasing and sin einitial data satis�es 0 < Q(0) < S(0), therefore Q(t) < S(t). This proves our assertion.Now we study the long time behavior of non zero steady state of (1.50).Lemma 3. Assume, similar to the onditions λ0 > 0, λ∞ < 0 (see Theorem 7), that

b2(0)

d(0)+

b1(0)

d(0) + α> 1,

b2(∞)

d(∞)+

b1(∞)

d(∞) + α< 1, (1.52)then there exists a unique positive steady state (S, Q).35

Démonstration. It is enough to prove that there exists (S, Q) 6= (0, 0) whi h solves{

(b2(S) − d(S))S + b1(S)Q = 0,

b2(S)S + [b1(S) − d(S) − α]Q = 0,and thus

Q =d(S) − b2(S)

b1(S)S,

b1(S)b2(S) = (b2(S) − d(S))(b1(S) − d(S) − α).We �nally redu e these equations to

Q =d(S) − b2(S)

b1(S)S,

b2(S)

d(S)+

b1(S)

d(S) + α= 1.

(1.53)First we show that there exists a unique positive solution to the se ond equation in (1.53).Now onsider F : R+ → R+ de�ned byF (S) =

b2(S)

d(S)+

b1(S)

d(S) + α. (1.54)Observe that F is a de reasing fun tion. By (1.52) there exists a unique S > 0 satisfyingse ond equation of (1.53). Again from se ond equation of (1.53) we have b2(S) < d(S),this assures Q > 0.1.8.2 Stability of the linearized system (Example 1)By �rst order Taylor's expansion around (S, Q) we get the linearized system asso iatedwith (1.50). The Ja obian matrix for that system is given by

J =

(b2 − d+ b′2S − d′S + b′1Q b1

b2 + b′2S + b′1Q− d′Q b1 − d− α

)at S=S

(1.55)We omputeTr(J) = b1(S) + b2(S) − 2d(S) − α + (b′2(S) − d′(S))S + b′1(S)Q.From (1.53) we have

b2(S) < d(S), b1(S) < d(S) + α. (1.56)Combining this with (1.8.1),(1.46) we on lude thatTr(J) < 0. (1.57)36

On the other hand we an omputeDet(J) =

(b2(S) − d(S)

)(b1(S) − d(S) − α

)− b1(S)b2(S)

+(b′2(S)S − d′(S)S + b′1(S)Q

)(b1(S) − d(S) − α

)

− b1(S)(b′2(S)S + b′1(S)Q− d′(S)Q

).From (1.53) we an redu e this expression to

Det(J) = 0 +(b′2(S)S − d′(S)S + b′1(S)Q

)(b1(S) − d(S) − α

)

− b1(S)(b′2(S)S + b′1(S)Q− d′(S)Q

).By (1.8.1), (1.46) and (1.56) we have

Det(J) > 0. (1.58)Therefore both eigenvalues of linearized version of (1.50) have negative real part. Hen e(1.50) is linearly stable around the steady state (S, Q).1.8.3 Nonlinear stability (Example 1)We re all the de�nition of steady state in (1.53) and assumeθ :=

b2(S)

d(S)>

b1(S)

d(S) + α:= 1 − θ. (1.59)Theorem 13. Under the assumptions (1.8.1)�(1.47), (1.52), (1.59), the steady state

(S, Q) is globally and exponentially attra tive for the system (1.50).Our proof is based on a Lyapunov fun tion for the system whi h shows a ontra tioninequality. We do not have a global Lyapunov fun tion in the ase when the oppositeinequality in (1.59), however the numeri al results in Se t. 1.8.7 seem to indi ate thatglobal exponential stability an hold true in general.Démonstration. First let us �x initial point (S(0), Q(0)). As a �rst step, from (1.50) we omputed

dt(S − S) + d(S)(S − S) + (d(S) − d(S))S = b2(S)(S − S)

+ (b2(S) − b2(S))S + b1(S)(Q− Q) + (b1(S) − b1(S))Q,

d

dt(Q− Q) + (d(S) + α)(Q− Q) + (d(S) − d(S))Q = b2(S)(S − S)

+ (b2(S) − b2(S))S + b1(S)(Q− Q) + (b1(S) − b1(S))Q.This givesd

dt|S − S| + d(S)|S − S| + |d(S) − d(S)|S = b2(S)|S − S|

− |b2(S) − b2(S)|S + b1(S)(Q− Q)sgn(S − S) − |b1(S) − b1(S)|Q,37

d

dt|Q− Q| + (d(S) + α)|Q− Q| + (d(S) − d(S))Qsgn(Q− Q)

= b2(S)(S − S)sgn(Q− Q) + (b2(S) − b2(S))Ssgn(Q− Q)

+ b1(S)|Q− Q| + (b1(S) − b1(S))Qsgn(Q− Q).At this stage we have two possibilities, either sgn(S − S) = sgn(Q− Q) or sgn(S − S) 6=sgn(Q− Q). In both the ases by (1.53), (1.63) we have

d

dt

(θ|S − S| + (1 − θ)|Q− Q|

)< 0. (1.60)Therefore

(θ|S(t) − S| + (1 − θ)|Q(t) − Q|

)ց L ≥ 0 as t −→ ∞. (1.61)As a se ond step we noti e that for �xed initial data (S(0), Q(0)), the urve (S(t), Q(t)) annot ome arbitrary lose to zero. Indeed (1.60), together with Lemma 2, implies that

S(t), Q(t) in rease if they are lose enough to zero. Hen e we getS(t), Q(t) ≥ Σmin > 0.In third step we prove exponential onvergen e. We noti e that the solutions are boundedthanks to (1.61) and thus there is a γ depending on (S(0), Q(0)) su h that

|d(S) − d(S)| + |b2(S) − b2(S)| ≥ γ|S − S|. (1.62)Now we hoose θ su h that1 − θ < θ < θ, θ(1 − θ) <

(1 − θ +

γΣmin

d(S)

)θ. (1.63)Finally we ompute to obtain

d

dt

(θ|S(t) − S| + (1 − θ)|Q(t) − Q|

)< −c

(θ|S(t) − S| + (1 − θ)|Q(t) − Q|

).where

c = min{

(d(S) + α)(θ − θ),d(S)

θ

((1 − θ +

γΣmin

d(S)

)θ − θ(1 − θ)

), (1 − θ)d(S)

}.Finally we get the exponential onvergen e

(θ|S(t) − S| + (1 − θ)|Q(t) − Q|

)< −c

(θ|S(0) − S| + (1 − θ)|Q(0) − Q|

)e−ct (1.64)Hen e we proved the announ ed result. 38

1.8.4 Nonlinear stability (Example 2)With the notations (S,Q) used in last example, we now onsider another ase in whi hwe an redu e the equation (1.1) to a 2×2 system. This is whenB(x, t) = b1

(Q(t)

S(t)

)e−αx + b2

(Q(t)

S(t)

), α > 0, (1.65)

bi > 0,d

dPbi(P ) < 0 for i = 1, 2. (1.66)

d = d(S(t), Q(t)

), (1.67)with

d

dSd(S, .) > 0,

d

dQd(., Q) > 0. (1.68)Finally the ompetition weight is hosen as

ψ ≡ 1. (1.69)Now let us de�ne a new variable P (t) := Q(t)S(t)

. Noti e that P (·) < 1. With this notation(1.1) be omes with the help of system in previous example

d

dtS(t) = S(t)

(b2(P (t)) − d

(S(t), P (t)S(t)

)+ P (t)b1(P (t))

),

d

dtP (t) = (1 − P (t))

[P (t)b1(P (t)) + b2(P (t))

]− αP (t).

(1.70)Lemma 4. Assume (1.66)�(1.68),d(0, 0) ≤ b2(1) and d(∞,∞) ≥ b1(0) + b2(0),then the system (1.70) has a unique nonzero steady state (S, P ).Démonstration. First let us fo us on se ond equation of (1.70). We prove that there existsa unique positive root to the fun tion

F (P ) = (1 − P )(Pb1(P ) + b2(P )) − αP.We have F (0) = b2(0) > 0, F (1) = −α < 0, therefore F vanishes at least on e in (0,1).We an write F asF (P ) =

{b2(0) if P = 0,PG(P ) if P 6= 0,where G is given by

G(P ) = b1(P )(1 − P ) − b2(P ) +b2(P )

P− α.The zeros of F are pre isely those of G and G is de reasing. Observe that G(P ) is positivefor small values of P and G(1) = F (1) whi h is negative. This proves the existen e and39

uniqueness of a zero of G, onsequently the same for the fun tion F . As a on lusion we hara terize the steady state byb1(P )(1 − P ) − b2(P ) +

b2(P )

P= α. (1.71)Now we turn towards the �rst equation of (1.70). Observe d(S, PS) in reases as S in- reases. This fa t together with our hypothesis on d on�rms the existen e and uniquenessof steady state S, i.e.,

b2(P ) − d(S, P S

)+ P b1(P ) = 0. (1.72)This proves our laim.Now we introdu e the notations b1 = b1(P ) and b2 = b2(P ), in order to simplifyexpressions. Now we are ready to prove the stability result for (1.70).Theorem 14. Under the assumptions of Lemma 4, the steady state (S, P ) for (1.70) isglobally and exponentially attra tive.Démonstration. Again we prove a ontra tion inequality. First we �x initial values S(0) >

0, P (0) > 0. We begin in the same manner as we did for in Example 1, by onsideringthe Lyapunov fun tionald

dt|P − P | = −|b1 − b1|P (t) + b1|P − P | + |b1 − b1|(P (t))2 − b1(P + P )|P − P |

− |b2 − b2| + |b2 − b2|P (t) − b2|P − P | − α|P − P |

=[|b2 − b2| + |b1 − b1|P

](P − 1) + (b1(1 − P ) − b2 − α)|P − P |.By (1.71) we have

d

dt|P − P | < −

b2P|P − P |, (1.73)and thus

|P (t) − P | ≤ |P (0) − P | e−βt, (1.74)where β = b2P. This shows that P (t) −→ P as t −→ ∞. Next we treat the unknown S(t).One an easily al ulate, using (1.72),d

dtS(t) = S

(b2 − b2 + Pb1 − P b1 + d(S, PS) − d(S, PS)

)

+ S(d(S, P S)| − d(S, PS)

).For su� iently large t, the sign of d

dtS(t) is same as sign of d(S, P S)| − d(S, PS) be ause

P (t) −→ P as t −→ ∞. Sin e we have uniqueness of the steady state S and d(S, PS) isin reasing with S, therefore using (1.66), (1.68), (1.74) one an �nd C1, C2 > 0 su h thatd

dt|S − S| ≤ C1e

−βt − C2|S − S|, ∀t > 0.With this, we get the announ ed result. 40

1.8.5 Nonlinear stability (Example 3)In this example we onsider more general situation. Here the death term depends onlyon total population S, birth term depends on total population S and P , whi h is theratio between weighted total population Q, total population S. As usual we assume deathterm in reases with its argument. Regarding monotoni ity of birth term, our assumptionin this example is di�erent from the assumptions we had in previous examples. That is,B(x, S(t)) = b1(S(t), P (t))e−αx + b2(S(t), P (t)), (1.75)

b3(S(t), P (t)) := P (t)b1(S(t), P (t)) + b2(S(t), P (t)), (1.76)b3(S(t), P (t)) > 0,

d

dSb3(S, .) < 0,

d

dPb3(., P ) < 0, (1.77)

d = d(S(t)) > 0,d

dSd(S) > 0. (1.78)As usual we hoose the ompetition weight

ψ ≡ 1.For future use, we introdu e the notationsS := log S, P := − log(1 − P ). (1.79)Thanks to (1.70), we get

d

dtS(t) = b3

(S(t), P (t)

)− d(S(t)),

d

dtP (t) = b3

(S(t), P (t)

)− α P (t)

1−P (t).

(1.80)Lemma 5. Assume (1.75)�(1.78) and there exists Σ > 0 su h thatb3(Σ,

d(Σ)

d(Σ) + α

)< d(0) < b3(0,

d(0)

d(0) + α)then the system (1.80) has unique steady state (S, P ) > (0, 0).Démonstration. Let (S, P ) be a steady state. It is hara terised by the solution of

b3(S, P ) = d(S) = αP (t)

1 − P (t).From later equality we obtain P = d(S)

d(S)+α. Hen e d(S) = b3(S,

d(S)

d(S)+α) whi h has uniquesolution from the hypothesis. Positivity and uniqueness of steady state are straightforward.Hen e we proved our assertion.Theorem 15. Under the assumptions of Lemma 5 the steady state (S, P ) for the system(1.80) is globally exponentially attra tive. 41

Démonstration. We build a Lyapunov fun tional as follows. We set S := log S, P =− log(1 − P ). With this notations we immediately see that

sgn(S − S) = sgn(S − S), sgn(P − P ) = sgn(P − P ).After a small omputation one an getd

dt|S(t) − S| = −|b3(S, P ) − b3(S, P )| − |b3(S, P ) − b3(S, P )|sgn(S − S)sgn(P − P )

− |d(S) − d(S)|,

d

dt|P (t) − P | = −|b3(S, P ) − b3(S, P )|sgn(S − S)sgn(P − P ) − |b3(S, P ) − b3(S, P )|

−α|P − P |

(1 − P )(1 − P ).For given initial data (S(0), P (0)), by adding last two equations we obtain that S(t), P (t)are bounded. Therefore there exists δ > 0 depending on (S(0), P (0)) su h that δ ≤

d′(S(t)) <∞. Finally we have proved the ontra tion inequalityd

dt

(|S(t) − S| + |P (t) − P |

)≤ −|d(S) − d(S)| −

α|P − P |

(1 − P )(1 − P )

≤ −δ|S(t) − S| + α|P (t) − P |)

≤ −C(|S(t) − S| + |P (t) − P |

),where C = min{δ, α}. Our assertion easily follows from this.1.8.6 Nonlinear stability (Kerma k-M Kendri k model)We ome ba k to the Kerma k-M Kendri k model ([69, 80℄) already presented inSe t. 1.2. Re all that in Se t. 1.7, we proved a stability result for this system when Σ(t)is quasistati .Now we present a simple and parti ular ase in whi h this model an be redu ed to a

2×2 system. In this subse tion we prove, with a di�erent method, a global and exponentialstability result for the non-trivial steady state. This is a ase whi h is not motivated byepidemiologi apli ations, but by its simpli ity.dΣ(x) = dn(x) +BR ≡ d, B > d2, (1.81)

ψ(x) ≡ 1. (1.82)With these assumptions we may integrate the equation on n(t, x) of Kerma k-M Kendri kmodel, see Se t. 1.2. And it redu es to the system of O.D.E. with S(t), Σ(t) as unknowns,and is given by {ddt

Σ(t) = B − dΣ(t) − S(t)Σ(t),

ddtS(t) = −dS(t) + S(t)Σ(t).

(1.83)Let us denote the nonzero steady state of system (1.83) by (Σ, S). With this notation wehave (Σ, S) =(d, B

d− d) (this is where we need B > d2).42

Proposition 8. Under the assumptions (1.81)�(1.82) and with initial data satisfyingΣ(0), S(0) > 0, the 'nonzero' steady state (Σ, S) for the system (1.83) is globally exponen-tially stable.We re all that the hemostat problem leads to mu h more ompli ated systems, seefor instan e [93℄ and the referen es therein.Démonstration. We introdu e H(t) := Σ(t) + S(t) and H := Σ + S. With this notationwe have

d

dt

(H(t) − H

)= −d

(H(t) − H

),whi h implies ∣∣H(t) − H

∣∣ =∣∣H(0) − H

∣∣e−dt. (1.84)We use the fa t that Σ = d to obtaind

dt

(S(t) − S

)= S(t)

(Σ − Σ

)= S(t)

(H(t) − H − S(t) + S

).For simpli ity, we denote by h := |H(0) − H|. With this notation, by straightforward omputation we obtain

d

dt

(S(t) − S

)≤(S(t) − S

)(− S + he−dt

)+ hSe−dt.Gronwall's lemma gives exponential onvergen e of S(t) to the steady state S, i.e., thereexists two positive onstants c, C su h that

∣∣S(t) − S∣∣ < Ce−ct for all t > 0. (1.85)Announ ed result follows from (1.84) and (1.85).1.8.7 Some numeri al resultsIn this se tion we present some numeri al eviden es on erning the stability resultswe have dis ussed in the previous se tion and in parti ular the �rst example dis ussed insubse tion 1.8.1. As stated in subse tion 1.8.3, the proof of the global onvergen e resultrelies on the ondition (1.59). However we have proved in Se . 1.8.2 that the non-zerosteady state is always stable under more general onditions. Therefore the question leftopen is to know if ondition (1.59) is really ne essary for exponential onvergen e to thenon-zero steady state. Our numeri al results indi ate that this ondition might not bene essary and are based on two examples we des ribe below.As a �rst example, we hoose

b1(S) =1

1 + S, b2(S) =

k

1 + S, d(S) =

S

1 + S, (1.86)43

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Fig. 1.1 � Solutions of (1.50) with the hoi e of b1, b2, d given by (1.86) ; Left : S(t)( ontinuous line) and Q(t) (dashed line) with parameters k = 0.2, α = 0.58, Right :S(t) ( ontinuous line) and Q(t) (dashed line) with parameters k = 0.4, α = 0.48. Theseindi ate that ondition (1.59) might be too strong, and global onvergen e to the steadystate holds true more generally.with k > 0 a onstant. One an easily he k that a hoi e of α whi h violates the ondition(1.59) is given by0 < α < min

{1 − 2k

k,k − 2 +

√(k + 2)2 + 4

2

}. (1.87)We have omputed numeri ally the traje tories of solutions of (1.50), where b1, b2, d aregiven by (1.86) with various hoi es of parameters α, k satisfying (1.87). All the ases wehave omputed turned out to exhibit global onvergen e. Two ases are depi ted in theFig. 1.1.As a se ond example

b1(S) =1 + k

1 + S, b2(S) =

1

1 + S, d(S) =

S

1 + S. (1.88)A straightforward omputation shows that the ondition

α <2k

3(1.89)is su� ient to violate (1.59). To on lude this se tion we show two more ases withdi�erent hoi es of the parameters k, α satisfying the ondition (1.89), see Fig. 1.2.

44

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

1

2

3

4

5

6

7

Fig. 1.2 � Solutions of (1.50) with the hoi e of b1, b2, d given by (1.88) ; Left : S(t)( ontinuous line) and Q(t) (dashed line) with parameters k = 1, α = 0.1, Right : S(t)( ontinuous line) and Q(t) (dashed line) with parameters k = 10, α = 1.A knowledgement Suman Kumar TUMULURI has been supported by Indo Fren hCEFIPRA Proje t 3401-2.

45

46

Chapitre 2Steady State AnalysisAbstra tIn the analysis of a nonlinear renewal equation it is natural to anti ipate the existen e ofnonzero steady states and deal with the question of their stability. Su� ient onditions forexisten e and uniqueness for this steady states are given. Study of the linearized versionof the renewal equation around the steady state helps to the great extent to have insightof some ompli ated dynami s of full problem. At this stage the �rst eigenvalue of thesteady state plays a vital role. Chara teristi equation, a fun tional equation whose rootsare the eigenvalues is derived and studied in spe ial ases. Similar study is arried outon a nonlinear model motivated by neuros ien e in whi h the total population is onserved.AMS ode 35F15, 35L60, 35Q80, 92D25Key words Existen e of steady states, linear stability, nonlinear renewal equation, per-iodi solutions.2.1 Introdu tionAt the beginning population dynami s is on erned with hanges in `number of indivi-duals' due to mortality and fertility. When people realized that both mortality and fertilitymay depend on stage of development of organism and other environmental fa tors, po-pulation models started in orporating submodels. Consider an isolated population livingin an invariant habitat. We also assume that there is no sex di�eren e. Let u(t, x) be thepopulation density of age x, at time t. Let d, B be the age spe i� death rate, birth ratesrespe tively. If individuals are independent from ea h other then equations will be linearand the study of su h equations is relatively easy. When dependen e arises (for example' ompetition for food') we introdu e environmental intera tion variables (quantities likestru tured ompetition weights, weighted populations). Let ψ1, ψ2 be the ompetitionweights. In this situation the population dynami s are governed by the following nonli-47

near M Kendri k equations

∂∂tu(t, x) + ∂

∂xu(t, x) + d(x, S1(t))u(t, x) = 0, t ≥ 0, x ≥ 0,

u(t, 0) =

∫ ∞

0

B(x, S2(t))u(t, x)dx,

u(0, x) = u0(x) ≥ 0, u0(x) ∈ BV (R+),

(2.1)with a oupling, whi h we take as simple as possibleSi(t) =

∫ ∞

0

ψi(x)u(t, x)dx for i = 1, 2. (2.2)This kind of model has been extensively studied by many authors. For instan e answersfor lassi al questions of existen e and uniqueness an be found in [95℄.The genesis of stability analysis is with the existen e of steady states of (2.1)�(2.2). Thereare several approa hes to deal with this issue, Farkas et al used semi group theory andgave some su� ient onditions for existen e of stationary solution [45℄. Lapla e transformte hniques are used in this ontext by Iannelli, Webb (see [58℄, [107℄). More re ent abs-tra t methods and te hniques an be found in [31℄, [35℄, [77℄.In this hapter we use analyti al methods to obtain su� ient onditions for existen e anduniqueness of steady state of (2.1)�(2.2). We also study linearized stability and show howperiodi solution an appear. Here we mention two ases, a simple possible nonlinearityon birth and a onservative model motivated by neuros ien e. In order to do that webegin with some basi assumptions on vital rates and ompetition weights whi h we willuse throughout this hapter.Throughout this paper, we assume the mortality rate d, the fertility rate B, are posi-tive and ontinuously di�erentiable and the ompetition weights ψ1, ψ2 are positive andintegrable. For every S ≥ 0, we assume that the birth rate and the mortality rates satisfyB(., S) ∈ L∞(R+) ∩ L1(R+),

∂B(., .)

∂S< 0, (2.3)

d(.,∞) ∈ L∞(R+),∂d(., .)

∂S> 0, (2.4)

d(., S) ∈ L1loc(R

+),

∫ ∞

0

d(x, S)dx = ∞, (2.5)1 <

∫ ∞

0

B(x, 0)e−R x

0d(y,0)dydx <∞, (2.6)

0 <

∫ ∞

0

B(x,∞)e−R x

0d(y,∞)dydx < 1. (2.7)The assumption (2.6) implies that the eigenvalue of steady state problem at S1 = S2 = 0is positive and (2.7) implies that the eigenvalue of the same at S1 = S2 = ∞ is negative.As we know from [95℄, this implies that the solution to (2.1)�(2.2) remain bounded in L1.The long time behavior of su h solutions is however far from obvious. Our purpose hereis to study the linear stability of the steady states and possibly appearan e of periodi solutions. 48

This paper is organized as follows. We begin with the question of existen e and uniquenessof nonzero steady states for the system (2.1)�(2.2). We dis uss this and give some su�- ient onditions to obtain a�rmative answer to this question in Se tion 2.2. In Se tion 2.3we linearize the system (2.1)�(2.2) around the nonzero steady state and get the hara -teristi equation. We analyze this hara teristi equation to draw on lusions regardingthe linear stability. In Se tion 2.4 we present a model in whi h we have onservation oftotal population. We study the linear stability of this model via hara teristi equation. InSe tion 2.5 we present a parti ular nonlinear variant of (2.1)�(2.2) and study numeri ally.We show the existen e of periodi solutions and o urren e of Hopf bifur ation.2.2 Existen e of nontrivial steady stateTo understand the long time behavior (like blow-up, extin tion, onvergen e to asteady state) of solutions to the renewal equation (2.1)�(2.2), it is essential to have learideas about its steady states. This problem an be redu ed to a �xed point whi h as wewill see, leaves pla e for omplex behaviors. In this se tion we present some su� ient onditions on mortality, fertility rates and ompetition weights su h that we an haveexisten e and uniqueness of nonzero steady state. For ompletion, we de�ne the systemof steady state equations orresponding to (2.1)�(2.2) as

ddxU(x) + d(x, S1)U(x) = 0, x ≥ 0,

U(0) =

∫ ∞

0

B(x, S2)U(x)dx,

Si =

∫ ∞

0

ψi(x)U(x)dx for i = 1, 2.

(2.8)Observe that U ≡ 0 is always a steady state. The linearized version of the renewal equation(2.1)�(2.2) around this zero steady state reads as

∂∂tu(t, x) + ∂

∂xu(t, x) + d(x, 0)u(t, x) = 0, t ≥ 0, x ≥ 0,

u(t, 0) =

∫ ∞

0

B(x, 0)u(t, x)dx,

u(0, x) = u0(x).

(2.9)One an readily he k that the �rst eigenvalue of the steady state equation orrespondingto (2.9) is positive under the assumption (2.6). Therefore the zero steady state U ≡ 0 isglobally unstable and it an be proved that in parti ular ase the dynami s of (2.1)�(2.2)never ome lose to zero (see [8℄, [95℄).Now we turn our attention towards the existen e and uniqueness of solutions for (2.8).Noti e that any solution of (2.8) is given byU(x) = U0e

−D(x,S1) for D(x, S) =

∫ x

0

d(y, S)dy, (2.10)49

where U0, S1, S2 solves the oupled systemSi = U0

∫ ∞

0

ψi(x)e−D(x,S1)dx, for i = 1, 2, (2.11)

∫ ∞

0

B(x, S2)e−D(x,S1)dx = 1. (2.12)Now we give a set of su� ient onditions for the existen e and uniqueness of solution to(2.8).Proposition 9. Assume (2.3)�(2.7) then there exists a solution to (2.8). Moreover assumethat there exists α > 0 su h that ψ1 ≥ αψ2 and

(S1 − αS2

)DS(x, S1) < 1 +

α

∫ ∞

0

B(x, S2)DS(x, S1)e−D(x,S1)dx

∫ ∞

0

|BS(x, S2)|e−D(x,S1)dx

, (2.13)for every x > 0 and steady state S1, S2, then the equation (2.8) has a unique positivesolution.Remark 2. Assume ψ1 ≡ ψ2 then S1 = S2 and α = 1. Therefore the ondition (2.13) isful�lled. Hen e (2.3)�(2.7) are enough to have existen e and uniqueness.Démonstration. To prove this proposition it is enough to show that there exists a uniquetriplet (U0, S1, S2) satisfying (2.11)�(2.12).First we prove an existen e result. We solve the two equations in (2.11) to get U0, S2 interms of S1 and substitute in (2.12) to solve for S1. In order to do that �rst we rewriteU0 in terms of S1 using (2.11) to end up with

U0(S1) =S1∫∞

0ψ1(x)e−D(x,S1)dx

. (2.14)From (2.14) we an ompute S2 as a fun tion of S1 and it is given byS2(S1) = U0(S1)

∫ ∞

0

ψ2(x)e−D(x,S1)dx.We an redu e (2.12) to a relation f(S1) = 0 where

f(S1) :=

∫ ∞

0

B(x, U0(S1)

∫ ∞

0

ψ2(y)e−D(y,S1)dy

)e−D(x,S1)dx− 1. (2.15)We observe that U0(0) = 0 and U0(∞) = ∞. From (2.6) it follows that f(0) > 0 and from(2.7) we obtain f(∞) < 0. Therefore we have existen e of steady state.Se ond we prove uniqueness. To prove this we show that the fun tion f is de reasing. Astraightforward omputation gives

dU0

dS1

=

1 + U0(S1)

∫ ∞

0

DS(x, S1)ψ1(x)e−D(x,S1)dx

∫ ∞

0

ψ1(x)e−D(x,S1)dx

> 0. (2.16)50

One an ompute (with short hand notations)f ′(S1) =

(U ′

0

∫ ∞

0

ψ2e−Ddx− U0

∫ ∞

0

ψ2DSe−Ddx

) ∫ ∞

0

BSe−Ddx

−

∫ ∞

0

BDSe−Ddx

=[( ∫ ∞

0

ψ2e−Ddx+ U0

∫ ∞

0

ψ1DSe−Ddx

∫ ∞

0

ψ2e−Ddx

−U0

∫ ∞

0

ψ2DSe−Ddx

∫ ∞

0

ψ1e−Ddx

)∫ ∞

0

BSe−Ddx

−

∫ ∞

0

ψ1e−Ddx

∫ ∞

0

BDSe−Ddx

]/∫ ∞

0

ψ1e−Ddx. (2.17)We noti e that the sign of f ′ is the sign of the numerator in (2.17) and we denote thenumerator (2.17) by N and re all that ψ1 ≥ αψ2. Consequently we obtain from (2.13)

N <

∫ ∞

0

ψ2e−D[1 +DS(αS2(S1) − S1

) ∫ ∞

0

BSe−Ddy − α

∫ ∞

0

BDSe−Ddy

]dx

< 0.Therefore f is de reasing at every steady state. This proves the uniqueness of the solutionto (2.8).Following the same lines of the uniqueness proof in this proposition we prove anotherresult.Lemma 6. Assume (2.3), (2.4), d is solely age independent, i.e., d = d(S) supp(ψ1) ⊆[1,∞], supp(ψ2) ⊆ [0, 1] then the steady state is unique.Démonstration. To prove this result we re all the equation (2.17) in the uniqueness partof the Proposition 9 and show that the numerator of f ′, i.e., N is negative. It is lear thatN is the sum of two terms and the se ond term is negative from (2.4). It remains to showthat the �rst term is negative. We note that

d′(S1)

∫ ∞

0

xψ1e−dxdx

∫ ∞

0

ψ2e−dxdx− d′(S1)

∫ ∞

0

xψ2e−Ddx

∫ ∞

0

ψ1e−dxdxis positive. Hen e the �rst term is also negative. Therefore N is negative and f is de rea-sing. This proves the announ ed result.2.3 Linear stabilitySo far we have seen existen e and uniqueness of the steady states for (2.1)�(2.2) undersome onditions. Motivated by the asymptoti behavior of the solution of (2.1)�(2.2) westudy the linear stability of these steady states. The mathemati al theory of linear agestru tured population has started by Lotka and Sharpe (see [75℄, [99℄). In [107℄ Webb51

proved the `prin iple of linear stability', for ompleteness here we state this prin iple. Asteady state is exponentially stable if all the spe tral values of in�nitesimal generator ofthe linearized semi group has negative real part and is unstable if there is one spe tralvalue with positive real part. In [52℄ linear age stru tured population models served asmotivation for the theory of positive semi groups. Integrated semi groups are used toprove the prin iple of linearized stability for a variety of nonlinear age stru tured models(see [102℄). Unfortunately the instability part has not been proved for stru tures otherthan age (see [32℄).2.3.1 LinearizationIn this se tion we linearize the system (2.1)�(2.2) around a steady state U(.) (a solutionof (2.8)) and dedu e the hara teristi equation. This equation is vital be ause its rootsdetermine the linear stability (see [44℄). From this se tion onwards we assumeψ := ψ1 ≡ ψ2 (2.18)To be ompatible with this notation we de�neS := S1 = S2.Unless spe i�ed we assume (2.3)�(2.7), and by Remark 2, we have a unique nonzerosolution to (2.8). Here onwards the steady state means this unique solution. To beginwith we give solution of orresponding steady state equation whi h reads

U(x) = U0e−D(x,S) for D(x, S) =

∫ x

0

d(y, S)dy, (2.19)where S solves the equation∫ ∞

0

B(x, S)e−D(x,S)dx = 1, (2.20)and U0 is given byU0 =

S∫∞

0ψ(x)e−D(x,S)dx

. (2.21)De�ne w by u(t, x) = w(t, x)+U(x) for every x, t ≥ 0 and the linearization of (2.1)�(2.2)around the steady state U(.) leads to

∂∂tw(t, x) + ∂

∂xw(t, x) + d(x, S)w(t, x) + dS(x, S)U(x)Sw(t) = 0,

w(t, 0) =

∫ ∞

0

B(x, S)w(t, x)dx+ Sw(t)

∫ ∞

0

BS(x, S)U(x)dx,

Sw(t) =

∫ ∞

0

ψ(x)w(t, x)dx,

(2.22)52

for every x, t ≥ 0. To investigate the long time behavior of solutions of (2.22) we seek asusual a parti ular solutions of the form w(t, x) = eλtW (x) with λ ∈ C. Substituting thisansatz in (2.22) we arrive at the orresponding eigenvalue problem given by

ddxW (x) +

(d(x, S) + λ

)W (x) + dS(x, S)U(x)SW = 0,

W (0) =

∫ ∞

0

B(x, S)W (x)dx+ SW

∫ ∞

0

BS(x, S)U(x)dx,

SW =

∫ ∞

0

ψ(x)W (x)dx.

(2.23)One an easily solve (2.23), and re all (2.10) to getW (x) =

(W (0) − SWU0

∫ x

0

dS(y, S)eλydy)e−D(x,S)−λx. (2.24)We substitute W (.) the expression of SW in (2.23). We arrive at

SW = W (0)

∫ ∞

0

ψ(x)e−D(x,S)−λxdx

− SWU0

∫ ∞

0

(ψ(x)e−D(x,S)−λx

∫ x

0

dS(y, S)eλydy)dx.We introdu e the notations

K(λ) =

∫ ∞

0

ψ(x)e−D(x,S)−λxdx,

L(λ) = − SK(0)

∫ ∞

0

ψ(x)e−D(x,S)−λx( ∫ x

0

dS(y, S)eλydy)dx,

(2.25)

M(λ) =

∫ ∞

0

B(x, S)e−D(x,S)−λxdx,

N(λ)= SK(0)

∫ ∞

0

e−D(x,S)(BS(x, S) −B(x, S)e−λx

∫ x

0

dS(y, S)eλydy)dx.

(2.26)Be ause from (2.21), U0 = SK(0)

, after substituting W (.) in the boundary ondition of(2.23) we an write the onditions on SW and W (0) asW (0)K(λ) + SW (L(λ) − 1) = 0, (2.27)W (0)(M(λ) − 1) + SWN(λ) = 0. (2.28)The fa tor S/K(0) appears in the expression for L(λ) be ause of the equation (2.21).Similarly from the boundary ondition of (2.23) one an get where M, N are given byFor the existen e of a nontrivial solution eλtW (x) to the linear system of equations (2.27),(2.28) the ne essary and su� ient ondition is that the determinant of orrespondingmatrix vanishes, i.e.,

Γ(λ) := K(λ)N(λ) − L(λ)M(λ) + L(λ) +M(λ) = 1. (2.29)53

This is the hara teristi equation orresponding to (2.23). Immediately we observe from(2.12) that M(0) = 1 and thereforeΓ(0) = K(0)N(0) + 1

= S

∫ ∞

0

e−D(x,S)(BS(x, S) −B(x, S)DS(y, S)

))dx+ 1

= S(∫ ∞

0

e−D(x,S)B(x, S)dx)

S+ 1. (2.30)We on lude this se tion with the following hara terization of linearly stable steadystates.Lemma 7. The steady state U(.) is linearly asymptoti ally stable if all the roots (λ's) ofthe equations (2.20), (2.29) have negative real part and unstable if there exists at least oneroot whi h has positive real part.The main di� ulty to analyze this ondition is that the equation (2.29) is very intri ate.We an do this in spe ial ases and examples.2.3.2 Examples of linear stabilityIn this se tion our main aim is to study the hara teristi equation and dedu e some onditions for the stability of the steady state. In [84℄ P. Mi hel gave a ondition for globalnonlinear stability and for linear stability when mortality is just age dependent and thebirth rate is of a parti ular form. Now we improve this latter result by onsidering ageneral birth term in the followingLemma 8. (Stability) Assume (2.3), (2.6)�(2.7), and that the mortality solely dependson age, i.e., d(x, S) ≡ d(x) ∈ L∞(R) for all x, S ≥ 0. Then the steady state is linearlyasymptoti ally stable, in other words all the solutions to (2.20), (2.29) satisfy Re(λ) < 0,in the following ases.(i) There is a µ > 0 su h that

−1 ≤ S

∫ ∞

0

e−D(x)BS(x, S)dx < 0,

ψ(x) = µB(x, S), ∀ x > 0,

(2.31)(ii)Re

∫ ∞

0

ψ(x)e−D(x)−λxdx ≥ 0, ∀ Re(λ) > 0, (2.32)for instan e when there is a µ > 0 and r ≥ ‖d‖∞ su h thatψ(x)e−D(x) ≡ µe−rx ∀ x > 0. (2.33)54

Démonstration. First we observe that in this ase we have L(λ) ≡ 0. Therefore the ha-ra teristi equation (2.29) redu es to1 = Γ(λ) =

S

K(0)

∫ ∞

0

ψ(x)e−D(x)−λxdx

∫ ∞

0

e−D(x)BS(x, S)dx

+

∫ ∞

0

B(x, S)e−D(x)−λxdx. (2.34)Suppose that there exists a root λ = p+iq su h that p ≥ 0. Then by omparing real partswe obtain from (2.34)1 =

S

K(0)

∫ ∞

0

ψ(x)e−D(x)−px cos(qx)dx

∫ ∞

0

e−D(x)BS(x, S)dx

+

∫ ∞

0

B(x, S)e−D(x)−px cos(qx)dx, (2.35)0 =

S

K(0)

∫ ∞

0

ψ(x)e−D(x)−px sin(qx)dx

∫ ∞

0

e−D(x)BS(x, S)dx

+

∫ ∞

0

B(x, S)e−D(x)−px sin(qx)dx. (2.36)Proof of (i). For ψ(x) = µB(x, S) for µ > 0, we immediately noti e that thanks to(2.20), K(0) = µ. Therefore the relation be omes

1 =(1 + S

∫ ∞

0

e−D(x)BS(x, S)dx)∫ ∞

0

B(x, S)e−D(x)−pxcos(qx)dx. (2.37)From the assumption the �rst term in the right hand side of (2.37) lies in the interval[0,1). We know from (2.20) that the se ond term is always less than 1 for p ≥ 0. Hen eright hand side is stri tly less than 1, a ontradi tion. Therefore there exists no root for(2.29) su h that its real part is positiveProof of (ii). If we assume (2.32) then the real part of the �rst term in the right handside of (2.37) is negative and we know from the previous proof that the real part of these ond term is less than 1. We immediately get Re(Γ(λ)

)< 1.If we assume (2.33) then one an ompute that

∫ ∞

0

ψ(x)e−D(x)−pxcos(qx)dx = µ

∫ ∞

0

e−(r+p)xcos(qx)dx =µ(r + p)

q2 + (r + p)2, (2.38)whi h is positive. Therefore (2.32) holds true.

55

2.3.3 Examples of instabilityIn this subse tion �rst we onstru t an example in whi h the birth term B is in reasingwith S but still the steady state is linearly unstable. It is possible if the ompetition amongthe population at any time solely depends on the density of the population of a parti ularage.We begin with two �xed real numbers p > 0, q ∈ R r {0}. Consider ψ is of the form of aDira massψ(x) = δ(x = a), a > 0 (2.39)where we hoose a > 0 depending on p, q later. Moreover we assume

B(x, S) ≡ B(S), d(x, S) = d(x). (2.40)Introdu e notationsc(p, q) :=

∫ ∞

0

e−D(x)−px cos(qx)dx, s(p, q) :=

∫ ∞

0

e−D(x)−px sin(qx)dx,

A(p, q) :=B(S)c(p, q) − 1

B(S)s(p, q).A ru ial assumption with shorthand notations is

SBS = −B exp[pqarccot(A)

]√B2s2 +

(Bc− 1

)2. (2.41)At this stage we spe ify the quadrants to whi h the term ar ot(A) belongs to in orderto have a > 0, (we see later that a = 0 is not possible). Our hoi e highly depends on thesgn(s), sgn(q). And we an make this hoi e with the rule

sgn(q).sgn(arccot(A)

)= 1. (2.42)To do so, �rst we note that for �xed p, the fun tion A(p, .) is an odd fun tion of q be ause

c is even and s is odd. If s = 0, q > 0 then hoose ar ot(A) = π. If s = 0, q < 0 then wetake ar ot(A) = −π.Next we re all thatB(S)|c| < B(S)

∫ ∞

0

e−D(x)dx = 1,therefore we always have As < 0. And we omputes(p, q) =

1

q

∫ ∞

0

e−D(x)−px(d(x) + p)(1 − cos(qx))dx.Therefore we obtainsgn(s(p, q)).sgn(q) = 1. (2.43)If s > 0, q > 0 then ar ot(A) ∈ (π

2, π). If s < 0, q < 0 then ar ot(A) ∈ (−π,−π

2).56

One an easily ompute to obtainK(λ) = e−D(a)−λa = K(0)e−λa, M(λ) = B(S)

∫ ∞

0

e−D(x)−λx,

L ≡ 0, N(λ) =SBS(S)

K(0)

∫ ∞

0

e−D(x)dx =SBS(S)

K(0)B(S).The hara teristi equation (2.29), for λ = p+ iq be omes

1 = K(λ)N(λ) +M(λ) =SBS(S)

B(S)e−λa +B(S)

∫ ∞

0

e−D(x)−λx.Compare real and imaginary parts to obtain

1 =SBS(S)

B(S)e−ap cos(aq) +B(S)c,

0 =SBS(S)

B(S)e−ap sin(aq) +B(S)s.

(2.44)Use (2.41) to end up at1 − Bc = −

√B2s2 +

(Bc− 1

)2exp

[pqarccot(A) − ap

]cos(aq), (2.45)

Bs =

√B2s2 +

(Bc− 1

)2exp

[pqarccot(A) − ap

]sin(aq). (2.46)Sin e B(S)c − 1 < 0, we should have cos(aq) < 0 and a = 0 is not a solution for any hoi e of p > 0, q 6= 0. We an solve (2.45), (2.46) and obtain a parti ular solution

a(p, q) =1

qarccot

(A(p, q)

)=

1

qarccot

(B(S)c(p, q) − 1

B(S)s(p, q)

), (2.47)with the onvention already stated before we have a(p, q) > 0. We readily he k that

a(p, q) given by (2.47) indeed satis�es both (2.45)�(2.46). Therefore for this hoi e of a,though B(S) de reases with S, the steady state is unstable.Remark 3. Note that in the hypothesis we ex luded the ases a = 0, q = 0. If a = 0 thenwe from (2.35) we get stability of the steady state in even more general ase. The aseq = 0 is dis ussed in the Lemma 9.In the general ase where the death rate d ≡ d(x, S), birth rate B = B(x, S) we provethe following partial result. This is mentioned in [45℄ but we give here a simpler proof.Lemma 9. Assume (2.3)�(2.7) then there is no real positive root of the hara teristi equation (2.29). 57

Démonstration. More pre isely we prove Γ(λ) < 1 for all λ ≥ 0. From (2.30) it followsthat Γ(0) < 1. Assume that λ > 0 is given. Then note that M(λ) < 1 and this implies−L(λ)M(λ) < −L(λ) as L(λ) < 0. Also it is lear that K(λ) > 0, N(λ) < 0. Thereforewe have K(λ)N(λ) < 0. All these fa ts put together, we obtain

Γ(λ) = K(λ)N(λ) − L(λ)M(λ) + L(λ) +M(λ)

< 0 − L(λ) + L(λ) + 1 = 1This proves the announ ed result.We on lude this se tion by giving a ondition for instability of the steady state.Proposition 10. (Instability) Assume BS(x, S) > 0, dS(x, S) < 0,0 <

∫ ∞

0

B(x, 0)e−R x

0d(y,0)dydx < 1,

1 <

∫ ∞

0

B(x,∞)e−R x

0d(y,∞)dydx <∞,and dS(x, S) is bounded then the steady state exists and is linearly unstable.Démonstration. Existen e and uniqueness of nonzero steady state is straightforward fol-lowing the similar lines of proof of Proposition 9. To prove instability we prove that thereexists a positive root to the hara teristi equation (2.29). From (2.30) it is lear that

Γ(0) > 1. One an readily he k from (2.25), (2.26) thatlimλ→∞

K(λ) = limλ→∞

M(λ) = 0.Furthermore we havelimλ→∞

L(λ) = −U0 limλ→∞

∫ ∞

0

(ψ(x)e−D(x,S)

( ∫ x

0

dS(y, S)e−λ(x−y)dy))dx = 0.Similarly we have

limλ→∞

N(λ) = U0

∫ ∞

0

e−D(x,S)BS(x, S)dx,a �nite onstant. Combining these limits we arrive at Γ(λ) → 0 as λ → ∞. This provesthat there exists at least one positive root for the hara teristi equation (2.29).2.4 A model with onservation of total populationConservative age stru tured problems arise naturally in several ases where the unk-nown des ribes a probability density as in some models in neural networks [91℄. Thesimpler model whi h re�e ts this is

∂∂tu(t, x) + ∂

∂xu(t, x) + d(x, S(t))u(t, x) = 0, t ≥ 0, x ≥ 0,

u(t, 0) =

∫ ∞

0

d(x, S(t))u(t, x)dx,

u(0, x) = u0(x) ∈ L1[0,∞],

∫ ∞

0

u0(x)dx = 1,

(2.48)58

where again we takeS(t) =

∫ ∞

0

ψ(x)u(t, x)dx (2.49)and ψ is an integrable fun tion. In this ase, at any time the number of deaths is equalto the number of new borns. In other words the total population satis�es∫ ∞

0

u(t, x)dx = 1, for all t ≥ 0. (2.50)L∞ bound Assume ψ is bounded and that d(.,Σ) is bounded for all x, Σ ≤ ‖ψ‖∞. Thenfrom (2.49), (2.50) a priori we have

S(t) ≤ ‖ψ‖∞, u(t, 0) ≤ sup0≤x<∞

0≤Σ≤‖ψ‖∞

d(x,Σ) for all t > 0.Therefore the usual proof of existen e (see [91℄) gives that for any bounded initial datau0, we have uniformly bounded solution.2.4.1 Existen e and uniqueness of steady stateIn this subse tion we give some su� ient onditions for the existen e and the unique-ness of the steady state. We assume that there exists M > 0 su h that

‖d(.,∞)‖∞ ≤M,∂

∂Sd > 0, ∀ x, S ≥ 0; d(., S) /∈ L1(R+), ∀S ≥ 0. (2.51)Pro eeding in the same way as before we end up with a steady state

U(x) = U0e−D(x,S) for D(x, S) =

∫ x

0

d(y, S)dy,where (S, U0) solves

S = U0

∫ ∞

0

ψ(x)e−D(x,S)dx,

U0

∫ ∞

0

e−D(x,S)dx = 1,

(2.52)be ause we always have ∫ ∞

0

d(x, S)e−D(x,S)dx = 1. (2.53)Therefore from (2.52), the steady states S are hara terized by the solutions ofS =

∫∞

0ψ(x)e−D(x,S)dx∫∞

0e−D(x,S)dx

. (2.54)Therefore our quest for existen e and uniqueness of steady state redu es to �nding theroots of (2.54). As the solution S to this equation uniquely determines the steady stateU(x), sometimes we may refer S as the steady state. It should be understood a ording tothe ontext. Now we turn our attention to the usual question of existen e and uniquenessof the steady states. This is given in the following59

Proposition 11. Assume (2.51), ψ ∈ L1(R+) then the steady state problem orrespondingto (2.48) has a solution. Moreover ifψ(x) +

1

DS(x, S)≥ S, (2.55)for all x ≥ 0 and steady state S, then the steady state is unique.Démonstration. The proof follows along the similar lines of Proposition 9. The ondition(2.54) is rewritten as

f(S) := S

∫ ∞

0

e−D(x,S)dx−

∫ ∞

0

ψ(y)e−D(y,S)dy = 0. (2.56)For existen e we argue as follows, from (2.51) we havef(S) ≥ S

∫ ∞

0

e−D(x,∞)dx− ‖ψ‖1 ≥ S

∫ ∞

0

e−Mxdx− ‖ψ‖1.For su� iently large S0, (S0 > M‖ψ‖1) we have f(S0) > 0, and learly f(0) < 0. Thisgives the existen e of steady state.For uniqueness we show that f is an in reasing fun tion. To this end we omputef ′(S) =

∫ ∞

0

e−D(x,S)[1 − SDS(x, S) + ψ(x)DS(x, S)

]dx.From (2.55) we dedu e that f is in reasing. Therefore the steady state is unique.Remark 4. If ψ ∈ L∞[0,∞) then the existen e result in above proposition is straightfor-ward be ause (we re all the fun tion f from the Proposition 11)

f(S) =

∫ ∞

0

e−D(x,S)(S − ψ(x))dx > 0 for S > ‖ψ‖∞.2.4.2 Linear stability revisitedFollowing similar lines as we did in Se tion 2.3, �rst we �nd the steady state S andapproximate u(t, x) near the steady state S by introdu ing w(t, x) su h that u(t, x) =U(x) + w(t, x) for all x, t ≥ 0. Now the onservation of total population imposes anadditional onstraint ∫ ∞

0

w(t, x)dx = 0, for all t ≥ 0. (2.57)As usual we seek a parti ular solution of the form w(t, x) = eλtW (x). Just for the sake of ompletion we present the eigenvalue problem whose solution is W (x). This is

ddxW (x) +

(d(x, S) + λ

)W (x) + SWdS(x, S)U(x) = 0,

W (0) =

∫ ∞

0

d(x, S)W (x)dx+ SW

∫ ∞

0

dS(x, S)U(x)dx,

SW =

∫ ∞

0

ψ(x)W (x)dx,

∫ ∞

0

W (x)dx = 0.

(2.58)60

The �rst equation of (2.58) an be solved by the standard linear ODE te hniques. Un-like previous ase, we get two onditions depending on λ for the existen e of nonzeroW (0), SW . There are three equations left for two unknowns namely W (0), SW .We laim that for λ 6= 0 the se ond and the fourth equations of (2.58) are the same.To see this redundan y we integrate the �rst equation of (2.58) over positive real line toobtain

−W (0) + λ

∫ ∞

0

W (x)dx+

∫ ∞

0

d(x, S)W (x)dx+ SW

∫ ∞

0

dS(x, S)U(x)dx = 0. (2.59)Therefore we eliminate W (0), SW from the se ond and third equations of (2.58).In this pro ess we get new set of K, L, M, N , Γ analog to K, L, M, N,Γ whi h readsas for λ 6= 0,

K(λ) =

∫ ∞

0

ψ(x)e−D(x,S)−λxdx,

L(λ) = − SK(0)

∫ ∞

0

(ψ(x)e−D(x,S)−λx

∫ x

0

dS(y, S)eλydy)dx,

(2.60)

M(λ) =

∫ ∞

0

d(x, S)e−D(x,S)−λxdx,

N(λ) = SK(0)

∫ ∞

0

e−D(x,S)(dS(x, S) − d(x, S)e−λx

∫ x

0

dS(y, S)eλydy)dx

(2.61)and the hara teristi equation Γ is still given byΓ(λ) := K(λ)N(λ) − L(λ)M(λ) + L(λ) + M(λ) = 1, for λ 6= 0. (2.62)We turn our attention towards the ase λ = 0. In this ase we observe from (2.59) that the ondition on W (0) (2.58) is automati ally satis�ed. Hen e we an eliminate this onstraintand deal with the third and fourth equations of (2.58). Sin e we have the third onstraintas a ommon onstraint in the ases λ 6= 0, λ = 0, when we pro eed in the same mannerto eliminate W (0), SW we an dire tly get K(0), L(0) from (2.60). Whereas M(0), N(0) hanges and therefore Γ(0) too. They are give byM(0) =

∫ ∞

0

e−D(x,S)dx, N(0) =−S

K(0)

∫ ∞

0

e−D(x,S)DS(x, S) (2.63)and the hara teristi equation Γ is given byΓ(0) := K(0)N(0) − L(0)M(0) + M(0) = 0,whi h reads expli itly with shorthand notations as

0 = Γ(0) =

−

∫ ∞

0

ψ(x)e−D(x,S)dx

∫ ∞

0

e−D(x,S)DS(x, S)dx∫ ∞

0

e−D(x,S)dx

+

∫ ∞

0

ψ(x)DS(x, S)e−D(x,S)dx+

∫ ∞

0

e−D(x,S)dx. (2.64)61

Therefore to say that zero is a eigenvalue for the linearized system (2.58) one has to verifythe ondition (2.4.2). As an immediate onsequen e we give two examples where zero isnot an eigenvalue.Example 1. (i) If ψ ≡ µ > 0 we immediately see that Γ(0)=

∫ ∞

0

e−D(x,S)dx whi h is notequal to zero. Hen e in this ase zero is not an eigenvalue.(ii) Assume ψ is of the form

ψ(x) = µDS(x, S), ∀x > 0, (2.65)where we hoose µ later. With this hoi e of ψ we omputeΓ(0)=

[−( ∫∞

0ψ(x)e−D(x,S)dx

)2

µ( ∫∞

0e−D(x,S)dx

)2 +

∫∞

0ψ2(x)e−D(x,S)dx

µ∫∞

0e−D(x,S)dx

+ 1] ∫ ∞

0

e−D(x,S)dxSet the probability measuredν(x) :=

e−D(x,S)

∫∞

0e−D(x,S)

,then the ondition Γ(0) = 0 be omes( ∫ ∞

0

DS(x, S)dν(x))2

−

∫ ∞

0

D2S(x, S)dν(x) =

1

µ.Now it is easy to make a hoi e of µ su h that this ondition is violated.2.4.3 An exampleIn this subse tion we deal with a parti ular ase where we an expli itly ompute theroots of the hara teristi equation (2.62). Assume that the rate d is independent of ageand in reases with S, i.e.,

d ≡ d(S), d′(S) > 0. (2.66)Assume the ompetition weight is given byψ(x) = e−mx for some m > 0. (2.67)For this hoi e of ψ, from Remark 4 there exists at least one steady state with S < 1.Lemma 10. Assume (2.66)�(2.67) and

d′(S) <(d(S) +m)2

m, (2.68)for all steady states S ≥ 0 then the steady state is unique and is linearly stable. The steadystate S is linearly unstable if

d′(S) >(d(S) +m)2

m. (2.69)62

Démonstration. With this parti ular hoi e of ψ and d, we ompute from (2.54) thatS =

d(S)

d(S) +m. (2.70)First we prove the uniqueness of the steady state following the lines of the uniquenesspart of the Proposition 11. In order to do that we re all the fun tion f in Proposition 11and prove that it is in reasing at every steady state. A straightforward omputation gives

f(S) =S

d(S)−

1

d(S) +mandf ′(S) =

1

d(S)−Sd′(S)

d2(S)+

d′(S)

(d(S) +m)2=

1

d(S)

(1 −

md′(S)

(d(S) +m)2

).Therefore from the assumption (2.68) one an dire tly obtain the uniqueness of the steadystate.Now we turn our attention to the hara teristi equation. We verify whether zero iseigenvalue or not. In order to do that we ompute

K(0) =1

d(S) +m, L(0) =

−d(S)d′(S)(d(S) +m

)2 , M(0) =1

d(S), N(0) =

−d′(S)

d(S)andΓ(0) = K(0)N(0) + (1 − L(0))M(0) =

1

d(S)

[1 −

md′(S)(d(S) +m

)2].From our hypothesis Γ(0) 6= 0, therefore zero annot be an eigenvalue. Now we look forthe existen e of nonzero eigenvalues. To this end �rst we ompute

K =1

d(S) +m+ λ, L =

−Sd′(S)

d(S) +m+ λ, (2.71)

M =d(S)

d(S) + λ, N =

(d(S) +m)Sd′(S)λ

d(S)(d(S) + λ). (2.72)Now onsider

Γ(λ) = K(λ)N(λ) + L(λ)(1 − M(λ)) + M(λ)

=(d(S) +m)Sd′(S)λ+ d2(S)(d(S) +m+ λ) − Sd′(S)d(S)λ

d(S)(d(S) + λ)(d(S) +m+ λ)Therefore the hara teristi equation (2.62) is a quadrati equation and its nonzero rootisλ =

mSd′(S) − d(S)(d(S) +m)

d(S)< 0 (2.73)from (2.68) and (2.70). The instability part of lemma an be proved in similar way. Weobserve that the inequality (2.69) annot hold for all steady states S be ause f(0) < 0.Therefore if there exists at least one S satisfying (2.69) then we do not have uniqueness ofthe steady state. Moreover that S is unstable. Hen e we have proved announ ed result.63

2.5 Numeri al resultsIn this se tion we present some of the numeri al tests we have performed and showsome eviden es for o urren e of Hopf bifur ation. We refer to [77℄ for a mathemati alproof of the existen e of bifur ations.Consider a variant of a nonlinear renewal equation

ut(t, x) + ux(t, x) = 0, x > 0, t > 0,

u(t, 0) = g( ∫ ∞

0

u(t, x)dµ),

u(0, x) = u0(x)

(2.74)where dµ is a probability measure. Nonlinear stability part of this type of equation ispartially done in [73℄,[84℄. To get insight for the nonlinear stability we present some ofthe numeri al results for di�erent hoi es of g whi h do not fall in the ases dis ussed in[84℄. In parti ular we assume there is a unique nonzero �xed pointg′(u♯) = u♯, (2.75)and we dis uss the ase where g′(u♯) < −1. Consider the fun tion g given by

g(x) = Axe−αx A ≥ 1, α > 0. (2.76)From the analysis in [84℄ we have that the steady state is the �xed point of the fun tiong and is given by u♯ = 1

αlogA. For this fun tion g we ompute

g′(u♯) = 1 − ln(A).Now the ondition g′(u♯) < −1 is equivalent to A > e2. We depi t a typi al g whi hsatis�es this ondition in Figure 1.Consider the probability measure given bydµ = B(x)dx for B(x) =

1

1 − x0

11[x0,1], (2.77)and the initial datau0(x) =

1

211[0,2] x ≥ 0. (2.78)As the hara teristi s move with unit speed, in our omputations we take the CFL numberto be 1 and we use a �rst order upwind s heme (see [103℄ for the details of the s heme).We have taken 1000 points for omputation and the omputational domain is 0 ≤ x ≤ 4for Figure 2.3.In Figure 2.2 we show how the boundary term u(t, 0) varies with the parameter x0 involvedin the probability measure in (2.77). In this ase we have witnessed the Hopf bifur ation,i.e., for small values of x0 we observe that the solution is rea hing the steady state whe-reas for higher values it be omes periodi . In Figure 2.2(a) we observe that there is severedamping e�e t and �nally the solution onverges to the onstant steady state though weobserve several os illations in the beginning. This kind of behavior is observed in all the64

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y=x

y=g(x)

y=g(g(x))

Fig. 2.1 � The fun tion g (dotted line), gog (dashed line) in (2.76) with A = 10 and withα = 4. Note that there are 3 �xed points for gog. ases where the parameter x0 ≤ 0.4. When this parameter value in reases we observe thatthe damping e�e t is de reasing.In the Figure 2.2(b), as time progresses we observe that the frequen y of the os illationsin reases without any damping e�e t. In this ase we see that the solutions be ome per-iodi for large times.In the Figure 2.2( ) we noti e the similar e�e t happening with more intensity, i.e., withhigher frequen y and amplitude. In Figure 2.2(d) we have taken x0 = 0.9, i.e., the pro-bability measure is lose to Dira measure. In this ase also we get os illations but theyexhibit re tangular pro�les. It is evident as the parameter value us approa hing to 1, theperiod is monotoni ally in reasing to 2.This phenomena an be explained by the following omputation. Noti e that the equation an be redu ed, setting u(t) = u(t, 0), to

u(t) = g(∫ ∞

0

B(x)u(t− x)dx). (2.79)When B(x) = δ(x = 1) we �nd

u(t) = g(u(t− 1)

). (2.80)Lemma 11. There is a 2-periodi solution u(t) to the system (2.74) when B(x) = δ(x = 1) hara terized by the �xed points u+__ of gog with

u(t) =

{u− for 0 < t < 1,u+ for 1 < t < 2.

(2.81)This is lear from the following omputation, thanks to (2.80)u(t) = u(t+ 2) = g(u(t+ 1)) = gog(u(t)).65

0 5 10 15 200.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

(a) 0 5 10 15 20

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( ) 0 5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d)Fig. 2.2 � Numeri al solution to (2.74) with g given by (2.76) with the initial data givenby (2.78). We depi t the urve t → u(t, 0). The probability measure is given by (2.77)(a) : with x0 = 0.3 ; (b) : with x0 = 0.5 ; ( ) : with x0 = 0.72 ; (d) : with x0 = 0.9.In the present example A = 10, α = 4 we an expli itly ompute the �xed points of gogand they are given by v♯ = 0.233, w♯ = 0.917 (see Figure 2.1). The solution is 2-periodi and os illates indeed between these steady states as x0 → 1.In Figure 2.3 we present for �xed time (at t = 40) the pro�les of the solution at di�erentparameters x0 (with x0=0.6, 0.8, 0.9, 0.98). In this ase also we see re tangular pro�lesas x0 → 1.Our next main numeri al observation regarding nonlinear stability is when −1 < g′(u♯) <0. Linear stability of the steady state is proved in this ase in [84℄. For g given by (2.76)this ondition reads as e < A < e2. In this ase from our numeri al tests we got eviden esthat the steady state is nonlinear stable. Here we present some of them.In Figure 2.4 we take A = 5, α = 4. We an ompute that the steady state u♯ = 0.402.Now we take di�erent initial data and show that the pro�le of the solution eventually66

0 0.5 1 1.5 2 2.5 3 3.5 40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) 0 0.5 1 1.5 2 2.5 3 3.5 40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

0 0.5 1 1.5 2 2.5 3 3.5 40.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( ) 0 0.5 1 1.5 2 2.5 3 3.5 40.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d)Fig. 2.3 � Numeri al solution to (2.74) with g given by (2.76) with the initial data givenby (2.78). We depi t the urve x → u(40, x). The probability measure is given by (2.77)(a) : with x0 = 0.6 ; (b) : with x0 = 0.8 ; ( ) : with x0 = 0.9 ; (d) : with x0 = 0.98.67

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) 0 5 10 15 200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b)Fig. 2.4 � Numeri al solution to (2.74) with g given by (2.76) with A = 5, α = 4. Theprobability measure is given by (2.77) with x0 = 0.98. We depi t the urve t → u(t, 0)with the initial data given by (a) : with u0 = 0.811[0,2] ; (b) : u0(x) = | sin(x)|11[0,2].

0 5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a) 0 5 10 15 20 25 300.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

(b)Fig. 2.5 � Numeri al solution to (2.74) with g given by (2.76) with A = 6, α = 4. Theprobability measure is given by (2.77) with x0 = 0.98. We depi t the urve t → u(t, 0)with the initial data given by (a) : with u0 = 0.811[0,2] ; (b) : u0(x) = 0.5| sin(x)|11[0,2].68

rea hes the steady state. We take the probability measure is lose to Dira mass, i.e.,x0 = 0.98 to apture the os illations. In Figure 2.4(a) we take the initial data to be

u0(x) = 11[0,2].In Figure 2.4(b) we hange the initial pro�le tou0(x) = | sin(x)|11[0,2].In this �gure we see some dark regions it is be ause of os illations at the initial timewith high frequen y. But eventually we see that the solution rea hing the steady stateirrespe tive of the initial data in both ases.Next we take A = 6, α = 4 and keep the probability measure as it is, i.e. x0 = 0.98. InFigure 2.5(a) we take initial data asu0(x) = 0.811[0,2].In Figure 2.5(b) we take initial data as

u0(x) = 0.5| sin(x)|11[0,2].In this ase also we observe the boundary term u(t, 0) rea hing the onstant steady state.This suggests the nonlinear stability of the steady states for nonlinear renewal equationwith quasi-linear boundary data.A knowledgment Suman Kumar Tumuluri has been supported by Indo Fren h CEFI-PRA proje t no. 3401-2.

69

70

Chapitre 3Numeri al StudyCommuni ated to the journal Methods and Appli ations of Analysis.Abstra tThe renewal equation is one of the key players in the study of population dynami s arisingin various ontexts from modelling of ell y le, epidemiology, an er growth, e ology. Inthese equations the nonlinearity is introdu ed with ompetition weights depending on ageand the birth rate, death rate depends on the weighted population in the age-stru turednonlinear renewal equation. Analyti al properties then rely on numeri al integration andwe propose here a semi impli it s heme. We prove the onvergen e of the �rst order ver-sion of the s heme and design a se ond order s heme that we ompare numeri ally onspe i� omplex dynami s exhibited by this simple equation.Keywords Renewal equation, �nite volume s hemes, weak solution, nonlinear dynami s.AMS Subje t Classi� ation 35F30, 35L60, 35Q80, 92B05, 92B253.1 Introdu tionConsider an isolated population living in an invariant habitat and also assume thatthere is no sex di�eren e. Let u(t, x) be the population density of age x, at time t. Letd, B be the age spe i� death rate, birth rates respe tively depending on weighted totalpopulation. Let ψ1, ψ2 be the ompetition weights re�e ting the age dependen e of everyindividual in the ontribution and ompetition for survival. In this situation the populationdynami s are governed by the following nonlinear M Kendri k equations, that is

∂∂tu(t, x) + ∂

∂xu(t, x) + d(x, S1(t))u(t, x) = 0, t ≥ 0, x ≥ 0,

u(t, 0) =

∫ ∞

0

B(x, S2(t))u(t, x)dx,

u(0, x) = u0(x) ≥ 0, u0(x) ∈ BV (R+),

(3.1)with a oupling, whi h we take as simple as possibleSi(t) =

∫ ∞

0

ψi(x)u(t, x)dx for i = 1, 2. (3.2)71

Though we present (3.1)�(3.2) as an age stru tured model, this is very generi modelwhi h appears at several situations. For example a size stru tured model des ribing tumorgrowth (see [29℄, [65℄), a model for dynami s of viral infe tion in vitro in luding infe tion, ell death, produ tion of interferon and development of resistan e (see [50℄) and the list ontinues. Several people have extensively studied the linear version of this equation (see[58℄, [86℄, [93℄, [95℄, [107℄). Theoreti al aspe ts, like existen e, uniqueness, stability for thenonlinear system (3.1)�(3.2) are dis ussed in [95℄. Ex ept in few spe ial ases (see [58℄,[95℄), it is unfeasible to �nd the expli it analyti al solution of (3.1)�(3.2). Neverthelessseveral numeri al methods were proposed to integrate this model, for a omprehensivereview refer [1℄. Here we brie�y outline some of the s hemes. In the linear ase i.e., d, Bdepends only on age, Ito et al gave a s heme using spline fun tions and trapezoidal rule (see[64℄), Angulo et al used Runge-Kutta methods along the hara teristi s (see [4℄). Douglasand Milner proposed an impli it �nite di�eren e s heme when ψ1 = ψ2 ≡ 1 and d = d(S)(see [39℄). In this ase they had integrated along the hara teristi urves and obtained as heme whi h is of �rst order in L∞(R+ × R+). In the ase when ψ1 = ψ2 ≡ 1 and thedeath rate d is in a parti ular form Iannelli, Kim introdu ed a s heme alled `splittings heme' (see [59℄). This is also a �rst order s heme based on method of hara teristi sand they use this s heme to integrate SIS, SIR epidemi models.Using standard �nite di�eren e method an expli it upwind s heme for (3.1)�(3.2) hasbeen given and the stability onditions are dis ussed in [74℄. In [43℄ an impli it s heme alled `box s heme' has been proposed whi h is se ond order in time and spa e.Our main obje tive in this paper is to prove the onvergen e of the �rst order numeri als heme whi h we present and ompare the �rst order s heme with higher order ones onproblems exhibiting dis ontinuities. In our analysis the te hni al di� ulty omes fromthe boundary term whi h makes that dire t BV estimates in spa e are not possible. The�semi-group� is not a L1 ontra tion.Now we re all some of the assumptions brie�y. We always assume that the fun tionsd, B, ψ are ontinuous, nonnegative, lo ally bounded. Then, we also need some standardassumptions, there exists L > 0 su h that for all x, Σ, S ≥ 0 we have

|B(x,Σ) − B(x, S)| ≤ L|Σ − S|, |d(x,Σ) − d(x, S)| ≤ L|Σ − S|. (3.3)Assume that there exists d, d, B, ψ > 0 su h thatd < d(., .) < d, B(., .) < B, 0 ≤ ψ1(.), ψ2(.) < ψ. (3.4)The paper is organized as follows. In Se tion 3.2 we present a �rst order numeri al s hemeand prove a priori estimates for the solution of orresponding di�eren e s heme. In thelater part of the this se tion we prove that the solution of this numeri al s heme onvergesto the weak solution of (3.1)�(3.2). In Se tion 3.3 we present a se ond order s heme usingthe te hnique of slope re onstru tion. In Se tion 3.4 we ompare both the �rst and se ondorder s hemes and analyze the advantages of the higher order s heme over the �rst orders heme. 72

3.2 First order numeri al studyIn this se tion we prove the onvergen e of a �rst order numeri al s heme for (3.1)�(3.2). In this se tion our main aim is to prove its onvergen e. Later we mention standardhigher order s hemes in spa e as well as in time. We use BV bounds throughout thisse tion be ause we deal with a nonlinear problem. Passing to the limit requires strong onvergen e. As there is no a priori BV estimate in spa e, we begin with a time estimate.To this end �rst we introdu e some notation.3.2.1 NotationWe dis retize the positive quarter plane by hoosing an uniform mesh width △x > 0and an uniform time step △t > 0. A typi al mesh point (tn, xj) := (n△t, j△x) forn, j = 0, 1, . . . . Assume the CFL ondition

λ :=△t

△x≤ 1. (3.5)For the sake of ompletion we introdu e the following standard notation, we denote thenodal points xj+ 1

2

:= xj + △x2

for j ≥ 0, and the ell averageuj(t) :=

1

△x

∫ xj+ 1

2

xj− 1

2

u(t, y)dy. (3.6)Furthermore for n ≥ 0 we de�neu0j =

1

△x

∫ xj+ 1

2

xj− 1

2

u0(x)dx, Sni = △x∞∑

j=1

ψi(xj)unj , for i = 1, 2, (3.7)

dj(S) =1

△x

∫ xj+1

2

xj− 1

2

d(y, S)dy, Bj(S) =1

△x

∫ xj+ 1

2

xj− 1

2

B(y, S)dy, (3.8)dnj = dj(S

n1 ), Bn

j = Bj(Sn2 ). (3.9)For n ≥ 0, we de�ne the l1− norm of (unj )1≤j≤∞ as

‖un‖1 :=∞∑

j=1

|unj |△x. (3.10)Let M be the number of spa ial points we use whenever we do numeri al omputation.Therefore M△x is the maximum numeri al age. For theoreti al study we take M = ∞.The same analysis we present here is still valid for �nite M .3.2.2 A �rst order numeri al s heme and a priori estimatesIn this se tion we present a �rst order s heme and give some of its a priori estimateswhi h are vital to prove the onvergen e of the s heme. We re all the notation introdu ed73

in the previous se tion. As the hara teristi s move with positive speed, one an simplyuse the following semi�impli it upwind s heme. For n ≥ 0, we set

un+1j = unj − λ(unj − unj−1) −△tdnj u

n+1j for 1 ≤ j ≤M, n ≥ 0,

un0 = △xM∑

j=1

Bnj u

nj ,

Sni = △xM∑j=1

ψi(xj)unj , for i = 1, 2.

(3.11)As well known in the numeri al analysis of hyperboli equations, this is a �rst order s heme(see [13℄, [70℄). When d, B are su� iently smooth, this numeri al s heme onverges to theweak solution of (3.1)�(3.2). For ompleteness we de�ne the weak solution of the system(3.1)�(3.2) in the followingDe�nition 2. A fun tion u ∈ L1

loc(R+ × R+) satis�es the nonlinear renewal equation(3.1)�(3.2) in weak sense if ∫∞

0B(x, S2(t))|n(t, x)|dx ∈ L1

loc(R+) and for all T > 0, forall test fun tions Ψ ∈ C1

comp

([0, T ] × [0,∞)

) su h that Ψ(T, x) ≡ 0, we have−

∫ T

0

∫ ∞

0

u(t, x){ ∂∂t

Ψ(t, x) +∂

∂xΨ(t, x) − d(x, S1(t))Ψ(t, x)

}dx dt

=

∫ ∞

0

u0(x)Ψ(0, x)dx+

∫ T

0

Ψ(t, 0)

∫ ∞

0

B(x, S2(t))u(t, x)dx dt.Now we state the onvergen e result in the followingTheorem 16. We assume (3.3)�(3.5), u0(x) ∈ BV (R), assume △xk → 0, △tk → 0 ask → ∞. Then the full sequen e uk := u△xk

of solutions to (3.11) satis�esuk

k→∞−→ u, in C([0, T ];L1(R+)

)∩ L∞

([0, T ];L∞(R+) ∩ L1(R+)

)and a.e.,for some fun tions u, b, S1, S2 satisfying

u ≥ 0, u ∈ L∞([0, T ];L1(R+) ∩ L∞(R+)

),

b(t) =

∫ ∞

0

B(x, S2(t))u(t, x)dx ∈W 1,∞([0, T ]

), (3.12)

Si(t) =

∫ ∞

0

ψi(x)u(t, x)dx ∈W 1,∞([0, T ]

) for i = 1, 2 (3.13)and u is a distributional solution to equation (3.1)�(3.2).The main properties of the �rst order s heme are given in the followingProposition 12. We assume (3.3)�(3.5), u0(x) ∈ BV (R) and de�ne u0 by (3.7), thenfor n△t = T we have(i)(a) ‖un‖1 ≤ ‖u0‖1e

(B−d)n△t := K(T ), 74

(i)(b) ‖un+1‖∞ ≤ max(‖u0‖∞, BK(T )

),

(ii)∞∑j=1

|un+1j − unj | ≤

λ(1 + △tA)n

(1 + △td)n+1

[ ∞∑j=1

|u0j − u0

j−1| + d‖u0‖1

], ∀n ≥ 0,where A := B + 2ψLK(T ), (see the equation (3.3) for the de�nition of L),

(iii)∞∑j=1

|unj − unj−1| ≤(1 + △tA)n

(1 + △td)n+1

[ ∞∑j=1

|u0j − u0

j−1| + d‖u0‖1

]

+d‖u0‖1e(B−d)(n+1)△t.Moreover assume xu0 ∈ L1(R+) then

(iv)∞∑j=1

(j − 1)|un+1j | ≤

∞∑j=1

(j − 1)u0j +

n∑l=1

∞∑j=1

|ulj|.Démonstration. We rewrite the �rst equation of the s heme (3.11) as(1 + △tdnj )u

n+1j = (1 − λ)unj + λunj−1. (3.14)For (i)(a), we simply sum up on j ≥ 1 the equation (3.14) after taking absolute value.We �nd

(1 + △td)

∞∑

j=1

|un+1j | ≤

∞∑

j=1

|unj | + λ|un0 |and observe that|un0 | = |△x

∞∑

j=1

Bnj u

nj | ≤ B‖un‖1. (3.15)Thus we obtain

(1 + △td)

∞∑

j=1

|un+1j | ≤

∞∑

j=1

|unj |(1 + △tB).By iteration one an get△x

∞∑

j=1

|un+1j | ≤

(1 + △tB

1 + △td )n+1

△x

∞∑

j=1

|unj |whi h gives the announ ed inequality (i)(a). The se ond part of (i) follows from thefollowing straightforward omputation using (3.14). We havesupj≥1

|un+1j | ≤ sup

j≤0|unj |, ombining with (3.15) and (i)(a) to obtain

supj≥0

|un+1j | ≤ max

{supj≥0

|unj |, BK(T )}.75

Iteration of this estimate gives (i)(b). For (ii), �rst we subtra t (3.14) for the labels n+ 1and n,un+1j − unj + △t(dnj u

n+1j − dn−1

j unj ) = (1 − λ)(unj − un−1j ) + λ(unj−1 − un−1

j−1 ),therefore(un+1

j − unj )(1 + △tdnj ) = (1 − λ)(unj − un−1j ) + λ(unj−1 − un−1

j−1 ) −△tunj (dnj − dn−1

j ).Take absolute values and sum over j ≥ 1 to �nd(1 + △td)

∞∑

j=1

|un+1j − unj | ≤

∞∑

j=1

|unj − un−1j | + λ|un0 − un−1

0 |

+ △t∞∑

j=1

|unj ||dnj − dn−1

j |.We noti e thatun0 − un−1

0 = △x∞∑

j=1

(Bnj u

nj − Bn−1

j un−1j )

= △x

∞∑

j=1

Bn−1j (unj − un−1

j ) + △x

∞∑

j=1

unj (Bnj − Bn−1

j ).Thus we have(1 + △td)

∞∑

j=1

|un+1j − unj | ≤

∞∑

j=1

|unj − un−1j | + △tB

∞∑

j=1

|unj − un−1j |

+[|Sn1 − Sn−1

1 | + |Sn2 − Sn−12 |

]△tL

∞∑

j=1

|unj |.We also noti e that|Sni − Sn−1

i | ≤ △xψ

∞∑

j=1

|unj − un−1j | for i = 1, 2.We use (i)(a) to obtain

(1 + △td)

∞∑

j=1

|un+1j − unj | ≤

[1 + △tB + 2△tψL‖u0‖1e

(B−d)n△t]×

∞∑

j=1

|unj − un−1j |.Therefore we have

∞∑

j=1

|un+1j − unj | ≤

(1 + △tA)n

(1 + △td)n

∞∑

j=1

|u1j − u0

j |. (3.16)76

for A := B + 2ψLK(T ). Now from (3.14) we write(1 + △td0

j)(u1j − u0

j) = −λ(u0j − u0

j−1) −△td0j u

0j .Take absolute values and sum over j to obtain

∞∑

j=1

|u1j − u0

j | ≤λ

1 + △td[ ∞∑

j=1

|u0j − u0

j−1| + d‖u0‖1

]. (3.17)Combine equations (3.16), (3.17) to get

∞∑

j=1

|un+1j − unj | ≤

λ(1 + △tA)n

(1 + △td)n+1

[ ∞∑

j=1

|u0j − u0

j−1| + d‖u0‖1

]and the result (ii) follows.For (iii), we rewrite the equation (3.14) asunj − unj−1 =

1

λ

(un+1j − unj + △tdnj u

n+1j

).Therefore,

∞∑

j=1

|unj − unj−1| ≤1

λ

∞∑

j=1

|un+1j − unj | + △x

∞∑

j=1

dnj |un+1j |.We use (i)(a), (ii), onsequently we obtain the result (iii). For (iv), we sum up on theequation (3.14) after taking absolute value and multiplying with (j − 1). We obtain

∞∑

j=1

(j − 1)|un+1j | ≤ λ

∞∑

j=2

(j − 1)|unj−1| + (1 − λ)∞∑

j=2

(j − 1)|unj |

≤ λ

∞∑

j=2

(j − 2)|unj−1| + λ

∞∑

j=2

unj−1 + (1 − λ)

∞∑

j=2

(j − 1)|unj |

≤∞∑

j=2

(j − 1)|unj | +∞∑

j=1

|unj |.The result (iv) follows by iteration.One an easily translate the Proposition 12 to the ontinuous level and this helps us tounderstand better, the a priori estimates dis ussed in previous proposition. In order to do77

that �rst we need to introdu e the notation at the ontinuous level. We setu0△x(x) :=

∞∑

j=1

u0j1I{xj−1<x<xj},

Si,△t(t) :=

∞∑

n=1

Sni 1I{tn≤t<tn+1}, for i = 1, 2,

B△x(x, t) :=

∞∑

j=1

Bnj 1I{xj−1<x<xj}, for tn ≤ t < tn+1,

d△x(x, t) :=∞∑

j=1

dnj 1I{xj−1<x<xj}, fortn ≤ t < tn+1,

u△x(t, x) :=

∞∑

j=1

unj 1I{xj−1<x<xj}, for tn ≤ t < tn+1,

β△x(t) :=

∫ ∞

0

B△x(x, S2,△t(t))u△x(t, x)dx, for tn ≤ t < tn+1.With the help of these notation Proposition 12 turns out to be equivalent to theProposition 13. Under assumptions of Proposition 12 the fun tion u△x(t, x) satis�esfor all t ∈ [0, T ] the estimates,(i)(a)

∫ ∞

0

|u△x(t, x)|dx ≤

∫ ∞

0

|u0△x|e

(B−d)n△tdx,

(i)(b) ‖u△x(t)‖∞ ≤ max(‖u0

△x‖∞, Bmaxs≤t

∫ ∞

0

|u△x(s, x)|dx),

(ii) ‖∂

∂tu△x(t, x)‖M1(R+) ≤ λe(A−d)T

[‖∂

∂xu0△x(x)‖M1(R+) + d

∫ ∞

0

|u0△x(x)|dx

]

(iii) ‖ ∂∂xu△x(t, x)‖M1(R+) ≤ e(A−d)T

[‖ ∂∂xu0△x(x)‖M1(R+) + d

∫ ∞

0

|u0△x(x)|dx

]

+de(B−d)(n+1)△t∫∞

0|u0

△x(x)|dx.Moreover if we assume xu0 ∈ L1(R+) then(iv)

∫ ∞

△x

(x−△x

2)|u△x(t, x)|dx ≤

∫ ∞

△x

(x−△x

2)|u0

△x(x)|dx+

∫ t

0

∫ ∞

0

|u△x(τ, x)|dxdτ.Having these BV estimates for u△x, we are able to prove the following onvergen e resultfor subsequen es of (u△x).Proposition 14. Under the hypothesis of Proposition 12, for △x, △t→ 0 we have b△t ∈C([0, T ]), Si,△t ∈ C[0, T ], for i = 1, 2, ∀ T > 0. There exists u ∈ L∞

([0, T ]; L1(R+) ∩78

L∞(R+)), b ∈ C([0, T ]) and Si,△t ∈ C([0, T ]) for i=1,2 su h that for a subsequen e

u△x∗⇀ u weak- *, in L∞

([0, T ];L1(R+) ∩ L∞(R+)

), a.e., (3.18)

Si,△t(t)−→Si(t) in C([0, T ]) for i=1,2, (3.19)β△x(t) −→ b(t) in C([0, T ]), (3.20)

d△x(x, t)−→d(x, S1(t)) in C([0, T ];Lplo (R+)), for 1 ≤ p <∞, (3.21)

B△x(x, t)−→B(x, S2(t)) in C([0, T ];Lplo (R+)

) for 1 ≤ p <∞. (3.22)Démonstration. From the Proposition 13 it follows that we have uniform bounds inspa es : u△x ∈ L∞([0, T ];L1(R+) ∩ L∞(R+)

). First we noti e from (i)(a) of Proposi-tion 13 that for i = 1, 2, Si,△t(t) ∈ L∞([0, T ]). Moreover one an noti e from (iv) ofProposition 13 that

∣∣ ddtSi,△t(t)

∣∣ ≤ ψ‖∂

∂tu△t(t, x)‖M1(R+) ≤ α(T ) <∞where α is independent of △t. Hen e we get Si,△t ∈W 1,∞[0, T ]. Therefore using Morrey'stheorem, for i = 1, 2, {Si,△t} is an equi ontinuous and bounded family of fun tions. ByAs oli-Arzela ompa tness theorem there exists Si ∈ C([0, T ]) and a subsequen e (we stilldenote by Si,△t) su h that for all t ∈ [0, T ]

Si,△t −→ Si in C([0, T ]) for i = 1, 2.Similarly one an prove (3.20).Next we onsider a ompa t set Ccpt ⊂ R+, therefore there exists m > 0 su h thatCcpt ⊂ [0, xm]. We re all the de�nition of d△x(x, t) and take Lp norm of following quantityover Ccpt to obtain

‖d△x(x, t) − d(x, S1(t))∥∥Lp ≤‖

m∑

j=1

(dnj − dj(S1(t))

)11x∈(xj−1,xj)

∥∥Lp

+∥∥

m∑

j=1

dj(S1(t))11x∈(xj−1,xj) − d(x, S1(t))∥∥Lp

≤∥∥

m∑

j=1

1

△x

∫ xj+ 1

2

xj− 1

2

(d(y, S1,△t) − d(y, S1(t))

)dy

×11x∈(xj−1,xj)

∥∥Lp

+∥∥

m∑

j=1

1

△x

∫ xj+ 1

2

xj− 1

2

d(y, S1(t))dy − d(x, S1(t))∥∥Lp

≤ |Ccpt|1/pL|S1,△t(t) − S1(t)|

+∥∥

m∑

j=1

1

△x

∫ xj+ 1

2

xj− 1

2

d(y, S1(t))dy − d(x, S1(t))∥∥Lp.From (3.19) the �rst term in the above inequality is of order O(△xk). To ontrol these ond term we re all the Lebesgue theorem whi h states that if a fun tion is absolutely79

integrable on a set X then almost every point in X is a Lebesgue point of the fun tion.Therefore we have1

△x

∫ xj+ 1

2

xj− 1

2

d(y, S1(t))dy −→ d(x, S1(t)) a.e. x ∈ Ccpt.As we are dealing with ompa t sets and d is bounded, Lebesgue dominated onvergen etheorem implies the se ond term is of order O(△xk) and hen e we have (3.21). Similarmethod an be used to prove (3.22).3.2.3 Convergen e of the s heme and a proof of the main resultThanks to these a priori estimates, we are in the situation of �nite volume methods in onservation laws and we an apply bounded onvergen e results, under bounded variationestimates (see [13℄, [72℄). In this subse tion we are in a position to prove the onvergen eof solution of numeri al s heme (3.11) to a weak solution of (3.1)�(3.2).Démonstration. of Theorem 16 To prove this theorem we use standard te hniques to provethe onvergen e for �nite volume s hemes. The main novelty is that we need a spe i� treatment near boundary x = 0.Step 1. In this step we prove that the limit fun tions b, S1, S2 we obtained in Proposition3 are the proper limits satisfying (3.12), (3.13). First we prove (3.12). Re all the de�nitionof weak-* onvergen e in (3.18) to get

∫ T

0

∫ ∞

0

σ(t)β(x, t)uk(t, x)dxdt −→

∫ T

0

∫ ∞

0

σ(t)β(x, t)u(t, x)dxdt, (3.23)for all σ ∈ L∞[0, T ], β ∈ L1(R+ × [0, T ]). We set βX(x, t) = B(x, t)11{x<X}. Using theproperty of �nite speed of propagation of hara teristi s, B(., .) < B, u0 ∈ L1(R+), weobtain ∫ T

0

∫ ∞

X

|σ(t)|B(x, S(t))|uk(t, x)|dxdt = O(1

X).Therefore we obtain

∫ T

0

σ(t)β△xk(t)dt =

∫ T

0

∫ ∞

0

σ(t)βX(x, S(t))uk(t, x)dxdt+O(1

X).In the limit k → ∞, by weak-* onvergen e we have

∫ T

0

σ(t)b(t)dt =

∫ T

0

∫ ∞

0

σ(t)βX(x, S(t))u(t, x)dxdt+O(1

X)and in the limit X → ∞ we obtain

∫ T

0

σ(t)b(t)dt =

∫ T

0

∫ ∞

0

σ(t)B(x, S(t))u(t, x)dxdt.80

This proves (3.12). Using the same te hnique one an easily dedu e (3.13).Step 2. In this step we give brief sket h of a proof of u(t, x) being a distributional solutionfor (3.1)�(3.2). To begin with we take an arbitrary test fun tion Ψ ∈ C∞

([0, T )× [0,∞)

)su h that for some R > 0, Ψ(T, x) = 0 and Ψ(t, x) = 0 for x > R. We setΨnj :=

1

△x△t

∫ tn+1

tn

∫ xj+1

xj

Ψ(t, x)dxdt. (3.24)Now we test the s heme (3.11) against Ψnj and be ause Ψ has ompa t support we endup at

∞∑

n=0

∞∑

j=1

(un+1j

Ψnj − Ψn+1

j

△tk+ unj

Ψnj − Ψn

j+1

△xk+ dnj u

n+1j Ψn

j

)

=1

△tk

∞∑

j=1

u0jΨ

0j +

1

△xk

∞∑

k=0

uk0Ψk1.Using ontinuous notation we an rewrite it as

∫ T

0

∫ ∞

△xk

(uk(t+ △tk, x)

Ψ(t, x) − Ψ(t+ △tk, x)

△tk

+uk(t, x)Ψ(t, x) − Ψ(t, x+ △xk)

△xk+ d△xk

(x, S1,△tk(t))u(t+ △tk, x)Ψ(t, x))

=

∫ ∞

△xk

uk(0, x)

∫ △tk

0

Ψ(s, x)

△tkdsdx+

∫ T

0

β△xk(t)

∫ 2△xk

△xk

Ψ(t, y)

△xkdydt.Now we an pass to the limit using (3.12), (3.13), (3.19), (3.20), (3.21), (3.23) to obtain

−

∫ T

0

∫ ∞

0

u(t, x){ ∂∂t

Ψ(t, x) +∂

∂xΨ(t, x) − d(x, S1(t))Ψ(t, x)

}dx dt

=

∫ ∞

0

u0(x)Ψ(0, x)dx+

∫ T

0

Ψ(t, 0)

∫ ∞

0

B(x, S2(t))u(t, x)dx dt.

Step 3. As uniqueness of the solution to (3.1)�(3.2) is assured (see [95℄), all the sequen es onverge to the same limit and thus the full family of solutions uk onverges. Hen e weproved the announ ed result.3.3 A se ond order s hemeIn this se tion we present a �high resolution� s heme. This term applies to methods thatare at least se ond order a urate on smooth solutions and thus give better resolution,in parti ular on the dis ontinuities we onsider here. In parti ular we study a methodwhi h is se ond order in age. One of the main on lusions of the analysis arried out inthe late 70's gave the se ond order s hemes for the hyperboli onservation laws. This is81

related to the te hniques of �slope re onstru tion�. The basi idea is to generalize upwinds hemes by repla ing pie ewise onstant representation by pie ewise linear representation.S hemes of this type were introdu ed by van Leer (see [105℄) and they have developed theMUSCL (Monotone Upstream- entred S heme for Conservation Laws).To this end, at ea h time level tn for n ≥ 0 using (3.11) we re onstru t a pie ewise linearapproximation of the solution of (3.1)-(3.2) on the intervals on (xj− 1

2

, xj+ 1

2

) for j ≥ 1 andis given byv(tn, x) = Wj(t

n, x) := vj(tn) + (x− xj)σ

nj , xj− 1

2

< x < xj+ 1

2

, (3.25)where σj is the numeri al derivative of v in the interval (xj− 1

2

, xj+ 1

2

) at time tn. We de�neapproximation near the boundary without re onstru tion usingv(tn, x) = v0(t

n), for 0 ≤ x ≤△x

2. (3.26)Therefore we annot expe t the se ond order a ura y at the boundary. Now we turn ourattention towards the slope σnj whi h is very ru ial in the re onstru tion. We de�ne theso- alled min-mod re onstru tion as

σnj = minmod { vj(tn) − vj−1(tn)

△x,vj+1(t

n) − vj(tn)

△x

}, (3.27)with minmod(a, b) =

{0, ab ≤ 0sgn(a) min(|a|, |b|), ab > 0.

(3.28)With this re onstru tion the standard se ond order s heme in spa e reads as, for 1 ≤ j ≤M, n ≥ 0,

vn+1j = vnj − λ(V n

j+ 1

2

− V nj− 1

2

) −△td(xj , Sn1 )vn+1

j ,

V n1

2

= △xM∑

j=1

B(xj , Sn2 )vnj ,

Sni = △xM∑j=1

ψi(xj)vnj , for i = 1, 2.

(3.29)Regarding numeri al �uxes at ell interfa es V n

j+ 1

2

we hoose upwind s heme. For 1 ≤ j ≤

M , we setV nj+ 1

2

= vnj +△x

2σnj . (3.30)Hen e (3.29)�(3.30) onstitute the se ond order s heme. Note that the s heme we havepresented here has higher order spa ial a ura y but only �rst order temporal a ura y.Time a ura y also an be in reased by introdu ing semi-dis rete s hemes and usinghigher order Runge-Kutta type of s hemes in time instead of just forward Euler timedis retization (see [67℄, [92℄, [100℄). We apply this kind of te hniques to study the stabilityof steady state for the system (3.1)�(3.2) (see[104℄).82

3.4 Numeri al results and analysisFor the simple age stru tured model we have presented, one an use a CFL numberequal to 1. Then �rst order s hemes are very ompetitive, but in many appli ations thespeed varies in time and in the stru turing variable x. Then high order s hemes are needed(see [29℄, [10℄ and the referen es therein). In this se tion we show some numeri al resultsillustrating the advantages of having se ond order in spa e s heme (3.29)�(3.30) over the�rst order s heme (3.11) presented in Se . 3.2.2. Here we restri t ourself to the se ondorder s heme in spa e. For di�erent hoi es of d, B, ψ1, ψ2 and the initial data we have omparered the numeri al solutions obtained using both s hemes. We have got eviden esthat in many of the ases qualitative behavior of this higher order s heme is as the higherorder s hemes for hyperboli onservation laws.In numeri al tests we present here, we take △x = 0.1, △t = 0.05 for the �rst order andthe se ond order s hemes whi h satis�es CFL onditions of respe tive s hemes. We makesure that all the examples we present have nonzero steady states and the zero steady stateis unstable. To this end we begin with a simple ase in whi h we have age independentmortality term given byd(x, S1) =

c1S1 + ε

1 + S1

, (3.31)where c1, ε > 0. Noti e that d is in reasing with S1 if and only if c1 > ε. To keep thesystem to be simple, the birth term has been taken by (to be onsistent with previousnotation)B(x, S2) :=

2

1 + S2. (3.32)In this type of examples we take ε 6= 0 be ause we need the net reprodu tion rate (denotedby R(S)) is �nite even there is no ompetition, (see [58℄). In other words we need

R(0) =

∫ ∞

0

B(x, 0)e−R x

0d(y,0)dydx <∞.Note that the ondition R(0) > 1 implies that the zero steady state is unstable (see [95℄).Further we take the ompetition weights ψ1, ψ2 to be

{ψ1 = x

311{0≤x≤1.5} + (2 − x) 11{1.5≤x≤2},

ψ2 = x 11{0≤x≤1} + (2 − x) 11{1≤x≤2}.(3.33)We omputed the solution with this data using �rst and se ond order s hemes for di�erentvalues of the parameters c1, ε with di�erent initial data we illustrate now. We begin withpie ewise onstant initial data given by

u0(x) := 11{0≤x≤12.5} for x ≥ 0. (3.34)It is very easy to he k that there is a jump dis ontinuity at the origin. In �rst examplewe start with the parameters c1 = 0.1, ε = 0.01 and the initial pro�le given by (3.34)and in this ase we observe the better onvergen e rate (see Figure 3.1(a), Figure 3.1(b)).Another phenomena we observe here is that the se ond order s heme is not as dissipative83

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

(a) 0 5 10 15 20 250

2

4

6

8

10

12

(b)Fig. 3.1 � Solution at time t = 10 (dotted line), 15 (dashed line), 20 (dash-dotted line),25 ( ontinuous line) with the initial data given by (3.34), c1 = 0.1, ε = 0.01 using, (a) :the �rst order s heme, (b) : the se ond order s heme.as the �rst order s heme.In Figure 3.2, for �xed initial data give by (3.34) at di�erent times (namely at t = 5, 10)we ompare both �rst order and se ond order s hemes. In this ase also we observe lessos illations and better onvergen e rate for higher order s heme. Now we take anotherpie ewise onstant initial data given byu0(x) := 0.511{0≤x<7.5} + 11{7.5≤x≤12.5} for x ≥ 0. (3.35)The numeri al solution with the data (3.35) by �rst and se ond order s hemes are givenin Figure 3.3(a),Figure 3.3(b) respe tively. In this ase also we observe less os illationsand better onvergen e rate for higher order s heme. Now we onsider a ombination ofpie ewise linear and pie ewise onstant initial data given byu0(x) := x11{0≤x<7.5} + 1011{7.5≤x≤12.5} for x ≥ 0. (3.36)This ase is given in Figure 3.4(a), Figure 3.4(b). Here we an learly see some of the lassi al features of se ond order s hemes of hyperboli equations. One among them isprevention of lo al extrema and sharp edges in numeri al solution. Consequently in this ase we see that the total variation in the se ond order s heme is less than the totalvariation in the �rst order s heme.Now we turn our attention towards the initial data with pie ewise linear pro�le given by

u0(x) := x11{0≤x<7.5} + (3 + x)11{7.5≤x≤12.5} for x ≥ 0. (3.37)Both Figure 3.5(a), Figure 3.5(b) represents the solution of (3.1)�(3.2) with initial data(3.37) using �rst and se ond order s hemes. From the �gures it is lear that the total va-riation of numeri al solution is less in se ond order ase. Moreover prevention of formationof lo al extrema is also taken are in se ond order s heme.84

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

(a) 0 5 10 15 20 250

2

4

6

8

10

12

(b)Fig. 3.2 � Solutions obtained using the �rst order s heme ( ontinuous line), se ond orders heme (dotted line) with the initial data given by (3.34), c1 = 0.1, ε = 0.01 using, (a) :at t = 5, (b) : at t = 10.

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

(a) 0 5 10 15 20 250

2

4

6

8

10

12

(b)Fig. 3.3 � Solution of (3.1)�(3.2) at time t = 10 (dotted line), 15 (dashed line), 20 (dash-dotted line), 25 ( ontinuous line) with the initial data given by (3.35), c1 = 0.1, ε = 0.01using, (a) : the �rst order s heme, (b) : the se ond order s heme.85

0 10 20 30 40 500

5

10

15

20

25

30

35

(a) 0 10 20 30 40 500

2

4

6

8

10

12

14

16

(b)Fig. 3.4 � Solution at time t = 10 (dotted line), 15 (dashed line), 20 (dash-dotted line),25 ( ontinuous line) with the initial data given by (3.36), c1 = 0.1, ε = 0.01 using, (a) :the �rst order s heme, (b) : the se ond order s heme.

0 10 20 30 40 500

5

10

15

20

25

30

35

40

(a) 0 10 20 30 40 500

2

4

6

8

10

12

14

16

18

(b)Fig. 3.5 � Solution at time t = 10 (dotted line), 15 (dashed line), 20 (dash-dotted line),25 ( ontinuous line) with the initial data given by (3.37), c1 = 0.1, ε = 0.01 using, (a) :the �rst order s heme, (b) : the se ond order s heme.86

A knowledgement The author would like to a knowledge Prof. Perthame for numerousdis ussions and the suggestions for the improvement of this arti le. The author has beensupported by Indo Fren h CEFIPRA proje t no. 3401-2.

87

88

Chapitre 4Perspe tives4.1 Non- onstant growth ratesSo far we have dis ussed age-stru tured models with linear onstant growth rate (uni-tary growth). We indi ate here several problems that extend naturally the nonlinear agestru tured equations studied in this thesis. It is interesting to study the behavior of thesolutions where the growth rate is non onstant but depends on the stru ture and is timedependent. Therefore the stru tural variable in these models is no longer the hronologi- al age. In many su h ases there is an additional variable alled 'maturity'. This kind ofmodels arise in several ontexts in biology and medi ine, for instan e see [2℄, [3℄, [7℄, [81℄,[98℄. Assume that g(t, x) > 0 is the speed at whi h the ells grow. A generi model of thiskind is given by

∂∂tu(t, x) + ∂

∂x

(g(t, x)u(t, x)

)+ d(x, S1(t))u(t, x) = 0, t ≥ 0, x ≥ 0,

g(t, 0)u(t, 0) =

∫ ∞

0

B(x, S2(t))u(t, x)dx,

u(0, x) = u0(x) ≥ 0, u0(x) ∈ BV (R+),

(4.1)as usual the oupling equation reads,Si(t) =

∫ ∞

0

ψi(x)u(t, x)dx for i = 1, 2. (4.2)Classi al questions like existen e and uniqueness of positive global solution to the system(4.1)�(4.2) are still to be answered. Existen e and uniqueness of steady states and largetime dynami s are ompletely unknown. Numeri al study of this system is also hallengingbe ause of its time dependent growth rate besides la k of BV estimates in x. One anexpe t that higher order s hemes are needed in order to apture the os illations anddis ontinuities due to g.4.2 Size stru tureAnother main dire tion to work further is the size stru tured equations. These equa-tions generally arise in the modeling of phenomena involving ell division. A typi al equa-89

tion des ribing dynami s of the ell division where a mother ell of size x into two daughter ells of sizes y, x− y is given for t ≥ 0, x ≥ 0 by

∂∂tu(t, x) + ∂

∂xu(t, x) +B(x, S(t))u(t, x) =

∫ ∞

0

b(y, x, S(t)

)u(t, y)dy,

u(t, 0) = 0,

u(0, x) = u0(x) ≥ 0, u0(x) ∈ BV (R+),

(4.3)where we take S from (4.2). The kernel b typi ally satis�esb ≥ 0, b(x, y, .) = 0 for y > x,

∫ ∞

0

b(., y, .)dy = 2B(., .) (4.4)∫ ∞

0

yb(x, y, .)dy = xB(x, .), b(x, y, .) = b(x, x− y, .). (4.5)We an re over the renewal equation from (4.3) by settingb(x, y, .) = B(x, .)

[δ(y = x) + δ(y = 0)

]. (4.6)There are several appli ations of these models in neuros ien e (see [18℄, [19℄) and ellbiology (see [56℄). General theory for the linear version of (4.3) an be found in [85℄, [86℄,[93℄. An interesting endeavor an be to look for GRE inequality to this equation with ap-propriate hoi e of B, b, ψ su h that the equation (4.3) redu es to a quasi-linear equationanalogous to what P. Mi hel did in [84℄.4.3 Nonlinear �uxesAnother perspe tive an be the study of the equations with highly nonlinear growthterm [from the dis ussions with Prof. Adimurthi℄. The mathemati al motivation is toin lude both the theory of onservation laws with that of renewal type of equations.Assume the growth �ux is given by F (x, u) > 0 then the typi al equation reads as

∂∂tu(t, x) + ∂

∂xF(x, u(t, x)

)+ d(x, S(x))u(t, x) = 0, t ≥ 0, x ≥ 0,

u(t, 0) =

∫ ∞

0

B(x, S(t))u(t, x)dx,

u(0, x) = u0(x) ≥ 0, u0(x) ∈ BV (R+),

(4.7)withS(t) =

∫ ∞

0

ψ(x)u(t, x)dx. (4.8)This problem is ompletely open. In purely mathemati al point of view, it is already inter-esting to onsider the problem (4.7)�(4.8) for simple ases (at least for d ≡ 0, F (x, u) =u2) and pro eed further. 90

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