Nonlinear Renewal Equations

Benoıt PERTHAME123 and Suman Kumar TUMULURI12

1 Departement de Mathematiques et Applications, Ecole Normale Superieure,75230 Paris, France. perthame@dma.ens.fr, suman@dma.ens.fr

2 INRIA-Rocquencourt B.P. 105, Projet BANG, F78153, Le Chesnay Cedex3 Institut Universitaire de France

Summary. The renewal equation plays a central role in modeling population biol-ogy and appears in various domains ranging from cell proliferation, tumor growthto epidemiology and ecology. Being simple and with a direct interpretation, it canbe considered as a first step towards more elaborate mathematical descriptions.

The linear renewal equation is well understood and there are several mathemat-ical ways to express the main behavior of its solutions: they exhibit exponentialgrowth or decay of the population with a rate and profile that can be entirely char-acterized, hence possible applications to cancer therapy.

However, the theory for nonlinear models is much more complicated. Severalbehaviors are possible (chaotic, periodic, or stable steady states). In this Chapterwe give an introduction to this theory with a special interest on cases where thereis an exponentially attractive steady state.

1 Introduction

In this chapter we consider an usual nonlinear age structured populationmodel which arises in many different contexts. One of them is the descriptionof cell proliferation and thus tumor growth. It can be written as the PartialDifferential Equation (PDE in short) on the unknown function n(x, t) ≥ 0which represents 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, t) = n0(x) ≥ 0.

(1)

The vector valued function S(t) =(S1(t), S2(t), · · · , Sk(t)

), represents the

environmental factors which depend on the solution n(x, t) itself, with a cou-pling, which we take as simple as possible

Si(t) =∫ ∞

0

ψi(x)n(t, x)dx, 1 ≤ i ≤ k. (2)

Also B ≥ 0, d ≥ 0 represent the birth and death rates respectively. Through-out this article we consider a single population living isolated, in an invarianthabitat, all of individuals being equal and there is no sex difference.

The model (1)–(2) arises in many examples issued from population biology.Historically, it is the first PDE introduced in biology and the linear equation(when d and b do not depend upon S) is usually known after the name ofMcKendrick [29] who introduced it for epidemiology, and Feller [19] made anextensive study through Markov processes. The linear equation is also knownas the VonFoerster equation because he was the first one to use it for modelingcell cultures. It is well understood and there are several mathematical waysto express the main behavior of its solutions: they exhibit exponential growthor decay of the population with a rate and profile that can be entirely charac-terized. For example, using the General Relative Entropy (GRE) inequalities([34, 35, 38]), one can prove that solutions to the linear model satisfy the longtime asymptotics ∫

φ(x)|n(x, t) e−λt − ρN(x)|dx −−−−→t→∞ 0, (3)

for some real number ρ > 0 and appropriate functions φ and N (see Sect. 3).Hence depending on the sign of λ, we conclude that either the population willgrow for ever or get extinct.

As for nonlinear models, the most famous was proposed by Kermack andMcKendrick for epidemiology with continuous state (age in the disease), [27].Nowadays, these models are used in various domains ranging from epidemi-ology to ecology, medicine and cell cultures. We give several examples andreferences in the Sect. 2. The first mathematical study of such nonlinear equa-tions is due to Gurtin, MacCamy, [21], and thus (1)–(2) is sometimes referredto as the Gurtin-MacCamy model in the case ψ1 = ψ2 ≡ 1 at least. Exis-tence, uniqueness, stability results of solutions of this model was discussed in[10, 20, 21]. Afterwards it has been vastly studied by several mathematiciansusing various techniques as semigroup theory, entropy GRE methods, Laplacetransforms. To deal with this model, the basic technique which many people(including Gurtin and MacCamy) used was to apply the method of charac-teristics to convert this problem to system of Volterra integral equations (see[21, 24, 25, 43]). The papers [12, 13] and the book [41] contain a recent ac-count of the theory. Here we will try to avoid this artefact and deal with PDEmethods, some of them can be extended to more elaborate models as, e.g.,size structured models [31, 38, 39, 34, 33].

An important aspect in the linear model leading to the behavior (3) is thatit does not take resources into account. This is the main drawback of the linear

2

model. In the system (1)–(2) we overcome this and considered the consump-tion of resources like nutrients, by introducing nonlinearity in birth and deathterms. In many of the examples we will consider later, these are limitatingthe possible growth and induce extra-growth when the solution gets to small.In other words, these models contain the classical assumption of Verhulst (fora mere ODE) as a basic ingredient. However due to the delay induced bythe boundary condition in (1), this limitation can have dramatic effects asexistence of chaotic solutions, periodic solutions, ‘oscillating’ solutions, see[16, 36, 32]. Traveling waves were also studied [15]. Of course stability stud-ies are also a major issue, studied by many authors, see also [7, 30]. In thisChapter we pay a special interest on global apriori bounds and cases wherewe can prove that there is an exponentially attractive non-zero steady state.

We begin with several examples of nonlinearities taken from the literature(Sect. 2). Then, we introduce our main assumptions (Sect. 3). Existence theoryand uniform bounds on the solutions require some work which we perform inSects. 4, 5. Our first original result concerns a case where we can prove theasymptotic behaviour as n(x, t) ∼ N(x)n(t), see Sect. 6. Then, we recallin Sect. 7, the long time asymptotics result of Ph. Michel [32] on ’concavetype’ nonlinearities on the birth term. After these general cases we concludein Sect. 8 with special nonlinearities for which we can reduce the system toODE systems and prove again their exponential stability.

2 Examples of Age Structured Models

Many variants of the system (1)–(2) have been proposed in the literature invarious area of biology, leading to different choices of the model parametersd, B and ψ. In this section we present some specific examples. Of course thispresentation is incomplete but give a general view of the broad use of suchmodels. We begin with examples from epidemiology, ecology, bacterial cellculture and conclude with the cell population models which arise in medicalapplications.

Epidemiology As mentioned earlier, the first age structured model in epi-demiology goes back to Kermack and McKendrick, [27, 29]. It has been widelystudied by many mathematicians as well as biologists (see [24, 25, 43]). Herewe briefly introduce the model which describes the propagation of a virus ina population. Let Σ(t), n(t, x), R(t) denote the total susceptible population,infective population and total recovered population at time t respectively, re-flecting the effect of virus. Here the age structure is incorporated into n, thedensity of infective population. Let BΣ , dΣ denote birth rate, mortality rateof susceptible population, BR(x), dn denote the rate of conversion of infectedcells into recovered cells, mortality rate of infected cells. It is very naturalto assume that dn > dΣ . Another main assumption in this model is that

3

individuals are infected from encounters between susceptibles and infected in-dividuals with age x in the disease with the rate ψ(x). Therefore the totalinfection rate is S(t) defined by

S(t) =∫ ∞

0

ψ(x)n(t, x)dx,

and thus the Kermack-Mckendrick model is defined, for t > 0, x > 0, by

d

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 [14] one can find a recent account of the subject. At this pointwe would like to notice that if we make the quasistatic hypothesis on Σ(t),we arrive at

0 = BΣ − dΣΣ(t)− S(t)Σ(t), Σ(t) =BΣ

dΣ + S(t).

This model falls in the class studied by Ph. Michel [32] that we recall in Sect. 7.

Cannibalism Cannibalism is one of the most interesting phenomena in pop-ulation studies (see [11] for an evolutionary point of view). F. Bekkal Brikciet al. analyzed theoretically a population model with cannibalism (see [5]). Inthat paper the model is age structured,

∂∂tn(t, x) + ∂

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

n(t, 0) = Φ(S(t))∫∞0B(x)n(t, x)dx,

n(0, t) = n0(x) ≥ 0,

and S(t) is given by the equation

S(t) =∫ ∞

0

n(t, x)dx

and Φ(·) is continuous, bounded, positive, decreasing, locally Lipschitz func-tion which vanishes at infinity.

Ecology Our next examples concern models in ecology. To begin with, werefer the reader to the book [37] which contains a full analysis of crocodilespopulation based on age structured equations. Bees et al (see [4]) studied

4

Deroceras reticulatum population dynamics (this species of slugs cause themajority of the damage to agricultural crops which turn out to be a pestof global economic importance. Another example of application in ecologycan be found in [18] where a similar model describes the density n(t, x) of aparticular tree (Pinus Cembra) in a forest. Let S(t) describes the populationof a certain bird (Nucifraga Caryocatactes) that helps disseminating the seeds.They arrive to a variant where the equation on S(t) is

d

dtS(t) + µ(S(t))S(t) = S(t)

∫ t

−∞k(t− x)P (s, t)ds

where P (s, t) is the total population of Pinus Cembra trees at time t whichwere born at time s, s ≤ t. If the time scale for birds reproduction is fasterthan that of Pinus Cembra (these trees don’t produce seeds in first fortyyears!) then we arrive to a model as (1)–(2), transforming this delay integralinto an age structured equation.

Cell proliferation As mentioned earlier, modeling cell cultures by age struc-tured equations is an old subject, see [42, 9, 3] for instance. Gyllenberg pro-posed a nonlinear age structured model for bacterial culture growth in a con-tinuous fermentation process (see [23]). He studied the existence, uniqueness,positivity, and boundedness of solutions, the existence of equilibrium solutionsand the stability of equilibria of the model. In his model, the growth, deathand fission rates of the cells are nonlinear functions of the substrate concen-tration in the reactor tank.

More general and realistic age-structured models were obtained by differ-ent authors. We recall now some of them because they use PDEs in higherdimensions. The first one was proposed by Rotenberg [40], still for cells, whointroduces a maturation velocity variable µ ∈ [0, 1]. It is the ratio betweenbiological age and physical age. Let n(t, x, µ) be the density of population,then it satisfies

∂∂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 change of maturation velocity from µto µ′. For stability and long time asymptotics results via GRE method andexistence of periodic solutions see [36].

Medical sciences, tumor growth Another model was proposed by Mackeyand Rey [28] to study the production of red blood cells (hematopoiesis) hasbeen attracted by several people. In this model the main biological assumptionis the life period of any cell is divided into the quiescent and proliferating

5

phases. The cells in quiescent phase can’t divide, but they mature and if theydon’t die, then they will enter the proliferating phase. There, when they don’tdie by apoptosis, they will divide and give birth to two daughter cells whichare in quiescent phase. Hence this age structured model takes into accountmaturity m and is the following coupled system of two nonlinear equations.For t ≥ 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 proliferatingcells and resting cells at time t, having age x, with maturity m, with mortal-ity rates d1, d2 respectively. Cell division rate or birth rate is b1. The mainassumptions in this model is V (0) = 0 and ∀m,

∫m

0dm′

V (m′) = ∞, V (·) is in-creasing. Second assumption is G : R+ → R+ such that G ∈ C1(R+), G isincreasing, G(0) = 0 and G(m) < m. Moreover b2(., N) is decreasing in N.This model is related to cancer modeling. It has nonvanishing steady stateand possesses periodic solutions (which might represent some classical blooddisease called Chronic Myeloid Leukemia). Adimy, Pujo-Menjouet simplifiedthis model by assuming that all cells in proliferating phase divide exactly atage τ (see [2]) and obtained existence, uniqueness results and global expo-nential stability. A more general case was considered by Adimy and Craustein which cells in proliferating phase divide at age distributed between τ andτ with continuous density function x 7→ k(x,m) supported in [τ , τ ] (see [1]).An important observation made in [1, 2] is if the time of replication is largeenough then the distruction of population of stem cells affects total popula-tion in greater extent and total population go extinct in finite time.

A further example in higher dimension is a coupled nonlinear model whichtakes the content of cyclin/cyclin dependent kinases(CDKs) into account hasbeen studied by Perthame et al. in [6] for comparing healthy tissues andtumoral tissues. Any tissue comprises two compartments namely prolifera-tive and quiescent compartments. The first one represents complete cell cy-cle with G1, S, G2, M phases. Cyclins/CDKs complexes are the most cru-cial control molecules in phase transitions. Each phase has its particular cy-clins/CDKs. The main idea behind the modeling is the proliferating cells growand divide whereas quiescent cells don’t possess physiological evolution. Letp(t, x, c), q(t, x, c) be the densities of proliferating and quiescent cells at timet with age a and content c in cyclin/CDKs. Let L(x, c), F (x, c) denote demo-bilisation rate from proliferation to quiescence and the rate of cell division.

6

Let d1, d2 be the rates of apoptosis of proliferating cells and quiescent cellsrespectively. Let ! Γ1(x, c) be the evolution speed of cyclin/CDKs and Γ0 > 0be a constant. With this information the model reads as an age structuredmodel

∂∂t p(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′,

∂∂t q(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 which quiescent cells reenter the proliferative phase.S(t) denotes total weighted population given by

S(t) =∫ ∞

0

∫ ∞

0

ψ1(x, c)p(t, x, c) + ψ2(x, c)q(t, x, c)dxdc.

Typical boundary data and appropriate initial data has been given to closethe model. Finally let us mention an original model describing the ovulatoryprocess. Clement et al. [17] have derived an age and maturity structuredmodel, with either proliferative or differentiated phases, depending on the cellmaturity level and cells differentiated phase will never re enter proliferativephase. In [17], nonlocal nonlinearities also arise through hormones production.For instance Follicule–Stimulating Hormone acts on follicular cells and aredescribed by a biochemical dynamical model.

3 Assumptions and Eigenelements

In this section, we present our basic assumptions and definitions which will beused from now on. We always assume that the functions d, B, ψ are contin-uous in S, nonnegative, locally bounded. Then, we also need the assumptions

B(., 0) ∈ L∞(0,∞) ∩ L1(0,∞), B(x, .) ∈ L∞loc(0,∞) for all x ≥ 0, (4)

1 <∫ ∞

0

B(x, 0)e−R x0 d(y,0)dydx <∞, (5)

0 <

∫ ∞

0

B(x,∞)e−R x0 d(y,∞)dydx < 1, (6)

1 < lima→∞

∫ ∞

0

B(x,∞)eax−R x0 d(y,∞)dydx = β∞ <∞. (7)

There exists L > 0 such 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|. (8)

7

∂d(., .)∂S

> 0, (9)

d(.,∞) ∈ L∞(0,∞), d(x, .) ∈ L∞loc(0,∞) for all x ≥ 0, (10)∂B(., .)∂S

< 0. (11)

There exist two maps D0, D∞ : R+ → R defined by

D0(S) = infx

{(d(x, 0)− d(x, S) +

φ0(0)φ0(x)

(B(x, S)−B(x, 0)

)}, (12)

D∞(S) = supx

{(d(x,∞)− d(x, S) +

φ∞(0)φ∞(x)

(B(x, S)−B(x,∞)

)}. (13)

Finally for the competition weight ψ(·), we assume that there are two positiveconstants C0

min and C0max such that

C0minφ0(x) ≤ ψ(x) ≤ C0

maxφ0(x). (14)

Also there are two positive constants C∞min and C∞max such that

C∞minφ∞(x) ≤ ψ(x) ≤ C∞maxφ∞(x). (15)

Our last two assumptions use notations that are introduced later on. Howeverwe give them now in order to gather all the assumptions. The eigenelementsλ0, λ∞, N0, N∞, φ0, φ∞ are defined as follows.

Consider the eigenvalue problem corresponding to S = 0ddxN0(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+).

(16)

The corresponding adjoint equations for above equations are{− d

dxφ0(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+).

(17)

The eigenvalue problem corresponding to S = ∞ is also written asddxN∞(x) + (d(x,∞) + λ∞)N∞(x) = 0, ∀ x ≥ 0,

N∞(0) =∫∞0B(y,∞)N∞(y)dy = 1,

N∞(·) ≥ 0, N∞ ∈ L∞loc(R+).

(18)

Similarly for this set of equations at S = ∞, we have the following adjointequation

8

{− d

dxφ∞(x) + (d(x,∞) + λ∞)φ∞(x) = φ∞(0)B(x,∞), ∀ x ≥ 0,∫∞0φ∞(x)N∞(x)dx = 1, φ∞(·) ≥ 0, φ∞ ∈ L∞(R+).

(19)

We conclude this section with a result concerning existence and uniqueness ofsolutions to eigenvalue problems, corresponding adjoint problems and signs ofeigenvalues. A simple application of Lebesgue monotone convergence theoremleads to the following result.

Theorem 1. Under assumptions (4)–(7), the problems (16)–(19) have uniquesolutions, and we have the inequalities λ0 > 0, λ∞ < 0. Moreover there existsC > 0, for all r > 0 we have

N0(x) ≤ e−λ0x, N∞(x) ≤ e−λ∞x, φ0(x) ≤ C, φ∞(x) ≤ β∞φ∞(0)e(λ∞−r)x

for all x ≥ 0.

Proof. We first consider the direct problems on N0 and N∞. A direct compu-tation shows that N0(x) is given by

N0(x) = e−R x0 (λ0+d(y,0))dy.

Consider the map λ 7−→∫ ∞

0

B(x, 0)e−λxe−R x0 d(y,0)dydx and observe that it

is integrable for all λ > 0, continuous and decreasing. Moreover by (5)

limλ→0

∫ ∞

0

B(x, 0)e−λxe−R x0 d(y,0)dydx > 1,

limλ→∞

∫ ∞

0

B(x, 0)e−λxe−R x0 d(y,0)dydx = 0.

Therefore there exists a unique λ0 satisfying (16) which is positive.

A similar argument using (6) and (7) proves that λ∞ is negative. Next weturn to the adjoint problem. One can also compute φ0 explicitly to get

φ0(x) =φ0(0)

e−R x0 (λ0+d(y,0))dy

∫ ∞

x

B(y, 0)e−R y0 (λ0+d(x′,0))dx′dy, (20)

φ∞(x) =φ∞(0)

e−R x0 (λ∞+d(y,∞))dy

∫ ∞

x

B(y,∞)e−R y0 (λ∞+d(x′,∞))dx′dy. (21)

Clearly φ0, φ∞ ≥ 0. Using (4) we have B(., 0) ∈ L1(0,∞), therefore φ0 ∈L∞(0,∞). Moreover we can choose φ0(0) in order to normalize∫ ∞

0

N0(x)φ0(x)dx = φ0(0)∫ ∞

0

xB(x, 0)e−R x0 (λ0+d(x′,0))dx′dx = 1.

This is normalization is possible because B(., 0) ∈ L∞(0,∞) and λ0 > 0. Nowwe prove similar result on φ∞. According to (11), we denote an upper boundby L′, d(x,∞) ≤ L′ <∞. From the explicit formula (21) for φ∞(x), we get

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φ∞(x) ≤ φ∞(0)e(λ∞+L′)x

∫ ∞

x

B(y,∞)e−R y0 (λ∞+d(x′,∞))dx′dy

and thus

φ∞(x) ≤ φ∞(0)e(λ∞−r)x

∫ ∞

x

e(−λ∞+L′+r)yB(y,∞)e−R y0 d(x′,∞)dx′dy.

Now thanks to (7), we obtain φ∞(x) ≤ β∞φ∞(0)e(λ∞−r)x, for any r > 0, withλ∞ < 0. Again it can be normalized because N∞ has growth as e−λ0x.

4 Existence of Solutions to the Linear Problem

In this section we consider S(t) is a given locally bounded function. We recallbasic facts 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, t) = n0(x).

(22)

We define the weak solution as follows

Definition 1. A function n ∈ L1loc(R+ × R+) satisfies the renewal equation

(22) in weak sense if∫∞0B(t, x)|n(t, x)|dx ∈ L1

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

all test functions Ψ ∈ C1comp

([0, T ]× [0,∞)

)such that Ψ(T, x) ≡ 0, we have

−∫ T

0

∫ ∞

0

n(t, x){ ∂∂tΨ(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.

Theorem 2. Let there exist M > 0 such that B(., .), d(., .) < M , S(t) ≥0, S(t) ∈ L∞loc(R+),n0 ∈ L∞(0,∞) ∩ L1(0,∞) then there is a unique weaksolution n ∈ C(R+;L1(R+)) solving (22). Moreover n(t, x) ≥ 0 whenevern0 ≥ 0, and ∫ ∞

0

|n(t, x)|dx ≤ e||(B−d)+||∞t

∫ ∞

0

|n0(x)|dx. (23)

We refer to [38] for proofs of such results. In particular the GRE methodallows for more precise results as the long time asymptotics (3).

10

5 Nonlinear Problem, uniform bounds

In this section we prove existence and uniqueness along with a priori boundson S(t). First we state our main theorem as

Theorem 3. Assume (4)–(15), n0 ∈ L∞(0,∞) ∩ L1(0,∞), also assume∫∞0φ0(x)n0(x)dx > 0, then there exists a unique weak or distributional solu-

tion n ∈ C(R+;L1(R+)) to (1)–(2). Moreover, because λ0 > 0, λ∞ < 0 (seeTheorem 1) there exists two positive constants m, M such that

m ≤ S(t) ≤M, ∀ t > 0.

This theorem is a consequence of Theorem 4 and Proposition 1 below. In thehypothesis,

∫∞0φ0(x)n0(x)dx > 0 is a technical one, it tells us that the initial

population density should be positive on a subset having nonzero measure,of [0,∞). Under this condition the estimate we present here tells us that theweighted population continue for ever and it neither blows up, nor go extincteven at infinite time. Before proving the main theorem we prove a theoremwith stronger hypothesis on B, d.

Theorem 4. Let us assume (8), there exists M > 0 such that B(., .), d(., .),ψ(·) < M and n0 ∈ L∞(0,∞) ∩ L1(0,∞), then there exists a unique weaksolution n ∈ C(R+;L1(R+)) to (1)–(2).

Proof. We prove this result by the Banach fixed point theorem. First we setX = C([0, T ]) with the sup norm, later we choose T (very small). Let X+

be the set of all nonnegative continuous functions on [0, T ] and define Λ =||(B − d)+||∞. For S(t) ∈ C([0, T ]), thanks to Theorem 2, 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, t) = n0(x) ≥ 0.

(24)

Now define a map T : X+ → X+ by

S(t) 7−→∫ ∞

0

ψ(x)n(t, x)dx. (25)

To prove the Theorem 4, it is enough to prove T is a contraction map.

Let n1(t, x), n2(t, x) be the solutions of (24) corresponding to S1(t), S2(t),then n := n1 − n2 satisfies

11

∂∂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, t) ≡ 0.

Now |n(t, x)| satisfies∂∂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, t)| ≡ 0.(26)

Therefore by integration in age, we have

d

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)|). (27)

From this we deduce that

12

sup0≤t≤T

|T S1 −T S2| = 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)|).

Now we choose T small enough such that T becomes a contraction map.Hence we proved existence and uniqueness of solution of (1)–(2).

Proposition 1. Under assumptions (4)–(7), (9)–(15) and∫∞0φ0(x)n0(x) >

0 there exists m,M > 0 for S(t) which is an unknown in the coupled system(1)–(2) such that m ≤ S(t) ≤M ∀t > 0.

Proof. To prove this proposition we use a technique developed Carrillo et al.in [8] based on adjoint problem. First we treat the lower bound. We define anauxiliary function

S0(t) =∫ ∞

0

φ0(x)n(t, x)dx. (28)

A duality computation (related to the GRE method ) using (1), (17) leads to

d

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). (29)

Last inequality holds because D0 is decreasing with D0(0) = 0. As long asC0

maxS0(t) is smaller than D−10 (−λ0), S0 is increasing. Therefore we obtain

S0(x) ≥ min{S0(0),

D−10 (−λ0)C0

max

}, ∀t ≥ 0.

Finally we exploit the assumption (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 technique by defining anotherauxiliary function

S∞(t) =∫ ∞

0

φ∞(x)n(t, x)dx. (30)

We follow the same methodology to get an upper bound for S(·). A particularchoice of M, we get after repeating similar exercise is given by

M = max{C∞maxS∞(0),

C∞max

C∞min

D−1∞ (−λ∞)

}.

13

6 A Case with asymptotic Decoupling

We consider particular variants of above birth rate and death rate. In thiscase we reduce the renewal equation to a single O.D.E. where we can easilyconclude about the convergence of the solution to the steady state. It is when

d(x, S) = d1(x) + d2(S), d1(x) ≥ 0, (31)B(x, S) ≡ B(x) ≥ 0, (32)∫ ∞

0

B(x)e−D(x) = 1, D(x) =∫ x

0

d1(x)dx. (33)

Here mortality rate d is the sum of the mortality rate due to inherent speciesaging d1 and fluctuations (extra birth or death) d2. In particular d2 has nosign but it is natural to keep d1(x) + d2(S).

Our method relies on the study of the following linearized state given byddxN(x) + d1(x)N(x) = 0, x > 0,

N(0) =∫ ∞

0

B(x)N(x)dx = 1.(34)

In other words N = e−D(x). As usual we introduce the adjoint problem for(34) is

− ddx φ(x) + d1(x)φ(x) = φ(0)B(x), x > 0,∫ ∞

0

φ(x)N(x)dx = 1.(35)

6.1 A particular solution

Before going to the main convergence result which we prove in next subsectionwe prove the following

Proposition 2. With assumptions (32)–(33), the system (1)–(2) admits theparticular solution n(t, x) = n(t)N with n(t) given by the differential equation

n(t) + d2(kn(t))n(t) = 0, (36)

withk =

∫ ∞

0

ψ(x)N(x)dx. (37)

Therefore the nonlinear problem admits this particular solution to (36) whichcan be solved by standard O.D.E. methods.

In particular from this example we can insight the long time asymptotics ofthe solution. The conditions λ0 > 0, λ∞ < 0 here means d2(0) < 0, d2(∞) >

14

0. Assuming also d′2(·) > 0 as usual we see that n(t) −→ n > 0 as t −→ ∞with n the unique solution to d2(kn) = 0.

Now let us turn our attention towards a convergence result. In this case weget exponential convergence to the solution obtained by variables separablemethod.

6.2 Long time decoupling

Now let us consider a related case where we describe long time asymptotics forthe system (1)–(2). In this case we prove L1 convergence with proper weightfunction which will be introduced later. This can be done for instance whenn(t) =

∫∞0n(t, x)φ(x), then it satisfies

d

dtn(t) + n(t)d2(S(t)) = 0. (38)

In this subsection we explain why these solutions built previously attract allother trajectories. Here we adapt a classical argument for parabolic systems,which can be found for instance in [26]. Namely, we prove the long timeasymptotics result for this model:

Theorem 5. Under the assumptions (31)–(33) and if there exists µ, δ > 0such that B(x) ≥ µφ(x), µ+ d2(S(t)) ≥ δ then the solution of age structuredequation (1)–(2) satisfies∫ ∞

0

|n(t, x)− n(t)N(x)|φ(x)dx ≤ e−δt

∫ ∞

0

|n0(x)− n(0)N(x)|dx

with n(t) =∫∞0n(t, x)φ(x).

Proof. We prove this convergence result with the help of a combination of aperturbation argument and of the duality method. One can easily compute toarrive at

∂

∂tn(t)N(x) +

∂

∂xn(t)N(x) +

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

)n(t)N(x) = 0.

We subtract this expression from (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 byconstruction ∫ ∞

0

h(t, x)φ(x)dx = 0, (39)

15

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|.

We integrate against φdx to and use (39) arrive at

0 =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)|φdx

From hypothesis we have µ+ d(S(t)) ≥ δ, therefore

d

dt

∫ ∞

0

|n− nN |φdx+ δ

∫ ∞

0

|n− nN |φdx ≤ 0.

From this the announced result follows.

7 A concave nonlinearity on birth term

Here we recall the result of Ph. Michel [32] based on ideas from the GREmethod. He considers a particular nonlinearity in the age structured equationbut that still contains several interesting applications (see Sect. 2), and in par-ticular the Kermack-McKendrick model of epidemiology. His result describesthe long time asymptotics of solutions: the nontrivial steady state is globallyattractive.

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, t) = n0(x) ≥ 0.

(40)

Here and as usual, d and B are nonnegative functions and g(·) is a continuousand nonlinear function that satisfies assumptions which are described later

16

on. Now our goal is to obtain that the solution to (1)–(2) converges to thenonzero steady state.

Changing the notation for g(·) by g(r·) if necessary, we may assume∫ ∞

0

B(x)e−D(x)dx = 1, with D(x) =∫ x

0

d(y)dy.

Moreover let us make the following assumption on the nonlinearity g (that ismore general but contains increasing and concave functions)

∃ N0 > 0, g(N0) = x0,

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

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

(41)

With this hypothesis it is clear that the corresponding steady state problemddxN(x) + d(x)N(x) = 0, x ≥ 0,

N(0) = g( ∫∞

0B(x)N(x)dx

),

(42)

admits the unique solution N(x) = N0e−D(x). As an immediate consequence

we obtain that

g(∫ ∞

0

B(x)N(x)dx)

= N0 =∫ ∞

0

B(x)N(x)dx.

Once interpreted in this way, the corresponding adjoint problem is given by{− d

dxφ(x) + d(x)φ(x) = φ(0)B(x), x ≥ 0,∫∞0φ(x)N(x)dx = 1, φ(·) ≥ 0.

(43)

We are now able to state the nonlinear stability result

Theorem 6. Under the assumption (41), as t → ∞, any solution n(t, x) to(40), with nonzero initial data, converges to the solution N(x) to (42), i.e.,∫ ∞

0

∣∣n(t, x)−N(x)∣∣φ(x)dx −→ 0 as t −→∞.

Proof. As we did in previous sections we try to get a contraction inequalityand write

∂

∂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)]

+.

17

After integrating against φ(x)dx to obtain

d

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 consider the times t where∫∞0B(x)n(t, x)dx ≥

∫∞0B(x)N(x)dx, then

from (41) we notice 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 inequality is straightforward and hence we have obtained

d

dt

∫ ∞

0

[n(t, x)−N(x)]+φdx ≤ 0.

Using similar arguments we get

d

dt

∫ ∞

0

[n(t, x)−N(x)]−φdx ≤ 0.

Finally we getd

dt

∫ ∞

0

|n(t, x)−N(x)|φdx ≤ 0. (44)

Using standard compactness argument (for instance see [33], [36], [38]) weobtain the convergence of the solution to (40) to the steady state N(x).

To conclude this section, we point out that the assumption (41) is nearlyoptimal for the above stated stability result. In [32], the reader may find manycounterexamples.

18

8 Stability for nonlinearities reducing to ODE systems

In general, the problem of long time asymptotics for system (1)–(2) is notcompletely solved. Any approach takes advantage of inbuilt special propertiesof the system and proceeds by giving sufficient conditions to conclude the longtime behavior. One of the well known techniques to deal with such kind ofproblems is to reduce the original equation or system to a system of ODE. Forinstance Iannelli, in [25] reduced both linear and nonlinear renewal equationshaving particular structure. In the nonlinear case he studied local behavioraround nonzero steady state. In this section first we discuss a case in whichwe can get nonlinear stability and the solution of (1)–(2) converges to steadystate. Finally we give some examples in which we reduce (1)–(2) to a 2 × 2system of ODE. We pay attention to this problem and we would like to usethis path because it serves as a simplified background before studying themore general problem.

8.1 Reduction to a system of ODE (Example 1)

A classical case that can be reduced to a differential system has been studiedby Iannelli (see [21, 25]). Here in Example 1, we slightly modify that to studyits linear and nonlinear stability. It is when for some α > 0,

B(x, S) = b1(S)e−αx + b2(S) (45)

withbi > 0,

dbidS

(·) < 0 for i = 1, 2. (46)

Also we assume the death rate is independent of age

d(x, S) = d(S),d

dSd(·) > 0. (47)

Then we define two quantities

S(t) =∫ ∞

0

n(t, x)dx, Q(t) =∫ ∞

0

e−αxn(t, x)dx. (48)

Which means that the competition weight is assumed to be constant

ψ ≡ 1. (49)

Now (1) becomes∂∂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, t) = n0(x) ≥ 0.(50)

19

Integrating (50) against dx and e−αxdx, we arrive to the system on S(·) andQ(·)

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).

(51)

Remark 1. With this notation if α = 0, then (1)–(2) reduces to S = Q and

dS

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 classicalMalthusian equation with parameter d − b. If d(S) − b(S) = k1 − k2S forconstants k1, k2 > 0, then our model (1)–(2) turned out to be another classicalmodel of Verhulst with intrinsic growth constant k1 and carrying capacityk1k2

(see Sect. 1). In general from (46),(47) one can see that S(t) −→ S ast −→ ∞, with S being unique solution of b(S) = d(S) if it exists, moreovern(t, x) −→ b(S)Se−d(S)x as t −→ ∞. Even though the system (1)–(2) isreduced to a single equation we can notice that it is still more elaborate thanthe example of Sect. 6.1.

In order to prove that S(t) and Q(t) are comparable, we have the

Lemma 1. If S(t), Q(t) are solutions of (51) then there exists ρ > 0 depend-ing on (S(0), Q(0)), such that ρS(t) < Q(t) < S(t).

Proof. One can easily compute

d

dt

(Q(t)S(t)

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

Q

S

)(1− Q

S)− α

Q

S. (52)

As b2(0) > 0, for fixed initial data (S(0), Q(0)), from (52) it follows that

the quantityQ(t)S(t)

cannot be arbitrary close to zero. This implies there exists

ρ such that ρS(t) < Q(t). In (52) we observe that as soon as Q(t)S(t) > 1, the

quantity Q(t)S(t) starts decreasing and since initial data satisfies 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 (51).

Lemma 2. Assume, similar to the conditions λ0 > 0, λ∞ < 0 (see Theorem1), that

b2(0)d(0)

+b1(0)

d(0) + α> 1,

b2(∞)d(∞)

+b1(∞)

d(∞) + α< 1, (53)

then there exists a unique positive steady state (S, Q).

20

Proof. It is enough to prove that there exists (S, Q) 6= (0, 0) which 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 finally reduce these equations toQ =

d(S)− b2(S)b1(S)

S,

b2(S)d(S)

+b1(S)

d(S) + α= 1.

(54)

First we show that there exists a unique positive solution to the second equa-tion in (54). Now consider F : R+ → R+ defined by

F (S) =b2(S)d(S)

+b1(S)

d(S) + α. (55)

Observe that F is a decreasing function. By (53) there exists a unique S > 0satisfying second equation of (54). Again from second equation of (54) wehave b2(S) < d(S), this assures Q > 0.

8.2 Stability of the linearized system (Example 1)

By first order Taylor’s expansion around (S, Q) we get the linearized systemassociated with (51). The Jacobian 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

(56)

We compute

Tr(J) = b1(S) + b2(S)− 2d(S)− α+ (b′2(S)− d′(S))S + b′1(S)Q.

From (54) we have

b2(S) < d(S), b1(S) < d(S) + α. (57)

Combining this with (46),(47) we conclude that

21

Tr(J) < 0. (58)

On the other hand we can compute

Det(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 (54) we can reduce 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 (46), (47) and (57) we have

Det(J) > 0. (59)

Therefore both eigenvalues of linearized version of (51) have negative realpart. Hence (51) is linearly stable around the steady state (S, Q).

8.3 Nonlinear stability (Example 1)

We recall the definition of steady state in (54) and assume

θ :=b2(S)d(S)

>b1(S)

d(S) + α:= 1− θ. (60)

Theorem 7. Under the assumptions (46)–(48), (53), (60), the steady state(S, Q) is globally and exponentially attractive for the system (51).

Our proof is based on a Lyapunov function for the system which shows acontraction inequality. We do not have a global Lyapunov function in the casewhen the opposite inequality in (60), however the numerical results in Sect.8.6 seem to indicate that global exponential stability can hold true in general.

Proof. First let us fix initial point (S(0), Q(0)). As a first step, from (51) wecompute

d

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 gives

22

d

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,

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) orsgn(S − S) 6= sgn(Q− Q). In both the cases by (54), (64) we have

d

dt

(θ|S − S|+ (1− θ)|Q− Q|

)< 0. (61)

Therefore (θ|S(t)− S|+ (1− θ)|Q(t)− Q|

)↘ L ≥ 0 as t −→∞. (62)

As a second step we notice that for fixed initial data (S(0), Q(0)), the curve(S(t), Q(t)) cannot come arbitrary close to zero. Indeed (61), together withlemma (1) imply that S(t), Q(t) increase if they are close enough to zero.Hence we get

S(t), Q(t) ≥ Σmin > 0.

In third step we prove exponential convergence. We notice that the solutionsare bounded thanks to (62) and thus there is a γ depending on (S(0), Q(0))such that

|d(S)− d(S)|+ |b2(S)− b2(S)| ≥ γ|S − S|. (63)

Now we choose θ such that

1− θ < θ < θ, θ(1− θ) <(1− θ +

γΣmin

d(S))θ. (64)

Finally we compute 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 convergence(θ|S(t)−S|+(1−θ)|Q(t)−Q|

)< −c

(θ|S(0)−S|+(1−θ)|Q(0)−Q|

)e−ct (65)

Hence we proved the announced result.

23

8.4 Nonlinear stability (Example 2)

With the notations (S,Q) used in last example, we now consider another casein which we can reduce the equation (1) to a 2×2 system. This is when

B(x, t) = b1

(Q(t)S(t)

)e−αx + b2

(Q(t)S(t)

), α > 0, (66)

bi > 0,d

dPbi(P ) < 0 for i = 1, 2. (67)

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

), (68)

withd

dSd(S, .) > 0,

d

dQd(., Q) > 0. (69)

Finally the competition weight is chosen as

ψ ≡ 1. (70)

Now let us define a new variable P (t) := Q(t)S(t) . Notice that P (·) < 1. With

this notation (1) becomes with the help of system in previous exampled

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).

(71)

Lemma 3. Assume (67)–(69),

d(0, 0) ≤ b2(1) and d(∞,∞) ≥ b1(0) + b2(0),

then the system (71) has a unique nonzero steady state (S, P ).

Proof. First let us focus on second equation of (71). We prove that there existsunique positive root to the function

F (P ) = (1− P )(Pb1(P ) + b2(P ))− αP

We have F (0) = b2(0) > 0, F (1) = −α < 0, therefore F vanish at least oncein (0,1). We can write as

F (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.

24

Zeros of F are precisely the zeros G and G is decreasing. Observe that G(P )is positive for small values of P and G(1) = F (1) which is negative. Thisproves the existence and uniqueness of zero of G, consequently the same forthe function F . As a conclusion we characterize the steady state by

b1(P )(1− P )− b2(P ) +b2(P )P

= α. (72)

Now we turn towards first equation of (71). Observe d(S, PS) increases as Sincreases. This fact together with our hypothesis on d confirms the existenceand uniqueness of steady state S, i.e.,

b2(P )− d(S, P S

)+ P b1(P ) = 0. (73)

This proves our claim.

Now we introduce the notations b1 = b1(P ) and b2 = b2(P ), in order to avoidbig expressions. Now we are ready to prove the stability result for (71).

Theorem 8. Under the assumptions of Lemma 3, the steady state (S, P ) for(71) is globally and exponentially attractive.

Proof. Again we prove a contraction inequality. First we fix initial valuesS(0) > 0, P (0) > 0. We begin in the same manner as we did for in Example1, by considering the Lyapunov functional

d

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 (72) we haved

dt|P − P | < − b2

P|P − P |. (74)

and thus|P (t)− P | ≤ |P (0)− P | e−βt. (75)

where β = b2P

. This shows that P (t) −→ P as t −→ ∞. Next we treat theunknown S(t). One can easily calculate using (73) to get

d

dtS(t) = S

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

)+ S

(d(S, P S)| − d(S, PS)

).

For sufficiently large t, sign ofd

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

because P (t) −→ P as t −→ ∞. Since we have uniqueness of steady state S

25

and d(S, PS) is increasing with S, therefore using (67), (69), (75) one can findC1, C2 > 0 such that

d

dt|S − S| ≤ C1e

−βt − C2|S − S|, ∀t > 0.

With this we get the required result.

8.5 Nonlinear stability (Example 3)

In this example we consider more general situation. Here the death term de-pends only on total population S, birth term depends on total population Sand P , which is the ratio between weighted total population Q, total popula-tion S. As usual we assume death term increases with its argument. Regardingmonotonicity of birth term, our assumption in this example is different fromthe assumptions we had in previous examples. That is,

B(x, S(t)) = b1(S(t), P (t))eαt + b2(S(t), P (t)), (76)b3(S(t), P (t)) := P (t)b1(S(t), P (t)) + b2(S(t), P (t)), (77)

b3(S(t), P (t)) > 0,d

dSb3(S, .) > 0,

d

dPb3(., P ) > 0, (78)

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

dSd(S) > 0. (79)

As usual we choose the competition weight

ψ ≡ 1.

For future use we introduce the notations

S := logS, P := − log(1− P ). (80)

Thanks to (71), we getd

dtS(t) = b3

(S(t), P (t)

)− d(S(t)),

d

dtP (t) = b3

(S(t), P (t)

)− α P (t)

1−P (t) .

(81)

Lemma 4. Assume (76)–(79) and there exists Σ > 0 such that

b3(Σ,

d(0)d(0) + α

)< d(0) < b3(0,

d(0)d(0) + α

)

then the system (81) has unique steady state (S, P ) > (0, 0).

26

Proof. Let (S, P ) be a steady state. It is characterised 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

. Hence d(S) = b3(S,d(S)

d(S)+α) which

has unique solution from the hypothesis. Positivity and uniqueness of steadystate are straightforward. Hence we proved our assertion.

Theorem 9. Under the assumptions of Lemma 4 the steady state (S, P ) forthe system (81) is globally exponentially attractive.

Proof. We build a Lyapunov functional as follows. After a small computationone can get

d

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 thatS(t), P (t) are bounded. Therefore there exists δ > 0 depending on (S(0), P (0))such that δ ≤ d′(S(t)) <∞. Finally we have proved the contraction inequality

d

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.

8.6 Some numerical results

In this section we present some numerical evidences concerning the stabilityresults we have discussed in the previous section and in particular the firstexample discussed in subsection 8.1. As stated in subsection 8.3, the proof ofthe global convergence result relies on the condition (60). However we haveproved in Sec. 8.2 that the non-zero steady state is always stable under moregeneral conditions. Therefore the question left open is to know if condition(60) is really necessary for exponential convergence to the non-zero steady

27

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. Solutions of (51) with the choice of b1, b2, d given by (82); Left: S(t)(continuous line) and Q(t) (dashed line) with parameters k = 0.2, α = 0.58, Right:S(t) (continuous line) and Q(t) (dashed line) with parameters k = 0.4, α = 0.48.These indicate that condition (60) might be too strong, and global convergence tothe steady state holds true more generally.

state. Our numerical results indicate that this condition might not be neces-sary and are based on two examples we describe below.

As a first example, we choose

b1(S) =1

1 + S, b2(S) =

k

1 + S, d(S) =

S

1 + S, (82)

with k > 0 a constant. One can easily check that a choice of α which violatesthe condition (60) is given by

0 < α < min{1− 2k

k,k − 2 +

√(k + 2)2 + 42

}. (83)

We have computed numerically the trajectories of solutions of (51), whereb1, b2, d are given by (82) with various choices of parameters α, k satisfying(83). All the cases we have computed turned out to exhibit global convergence.Two cases are depicted in the Fig. 1.

As a second example

b1(S) =1 + k

1 + S, b2(S) =

11 + S

, d(S) =S

1 + S. (84)

A straightforward computation shows that the condition

α <2k3

(85)

is sufficient to violate (60). To conclude this section we show two more caseswith different choices of the parameters k, α satisfying the condition (85), seeFig. 2.

28

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. 2. Solutions of (51) with the choice of b1, b2, d given by (84); Left: S(t)(continuous line) and Q(t) (dashed line) with parameters k = 1, α = 0.1, Right:S(t) (continuous line) and Q(t) (dashed line) with parameters k = 10, α = 1.

Acknowledgement SKT has been supported by CEFIPRA Project 3401-2.

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