Content uploaded by Yukihiko Nakata
Author content
All content in this area was uploaded by Yukihiko Nakata on Jun 27, 2015
Content may be subject to copyright.
Content uploaded by Yukihiko Nakata
Author content
All content in this area was uploaded by Yukihiko Nakata on Jun 27, 2015
Content may be subject to copyright.
Content uploaded by Yukihiko Nakata
Author content
All content in this area was uploaded by Yukihiko Nakata on Jun 27, 2015
Content may be subject to copyright.
SIAM J. APPL. MATH .c
2014 Society for Industrial and Applied Mathematics
Vol. 74, No. 4, pp. 1266–1297
STABILITY ANALYSIS OF A RENEWAL EQUATION FOR CELL
POPULATION DYNAMICS WITH QUIESCENCE∗
TOM ´
AS ALARC ´
ON†, PHILIPP GETTO‡,AND YUKIHIKO NAKATA§
Abstract. We propose a model to analyze the dynamics of interacting proliferating and quiescent
cell populations. The model includes age dependence of cell division, transitions between the two
subpopulations, and regulation of the recruitment of quiescent cells. We formulate the model as
a pair of renewal equations and apply a rather recent general result to prove that (in)stability of
equilibria can be analyzed by locating roots of characteristic equations. We are led to a parameter
plane analysis of a characteristic equation, which has not been analyzed in this way so far. We
conclude with how quiescence of cells as well as two submodels for cell division may influence the
possibility of destabilization via oscillations.
Key words. quiescence, cell population model, age structure, renewal equation, Hopf bifurca-
tion, characteristic equation
AMS subject classifications. 37N25, 45D05, 45G15, 45M10, 92C37
DOI. 10.1137/130940438
1. Introduction. Cells in many types of tissue in the human body are in a
quiescent state, i.e., they are under cell cycle arrest [11]. For blood cells the ability
to enter and exit the quiescent state seems essential for preventing the supply of
mature blood cells from becoming too large or too small [30]. In treatment of cancer
a major obstacle is acquired resistance by cancer cells to chemotherapy [12]. It is
an accepted hypothesis that cancer stem cells are the factory of cancer cells in solid
tumors as well as in hematological disorders such as leukemia [32, 40]. The cancer
stem cell hypothesis states that quiescent cells are far less sensitive to drugs and
thus drive the increase of resistance [12, 9]. Based on the analysis of a mathematical
model it is indeed suggested in [3] that the quiescent population provides a buffer
for a hostile environment for the whole population, i.e., mediates the survival of the
population.
Individual cells base appropriate responses, such as proliferation and cell death,
largely on their processing of both internal signals and signals from their environment
[4]. A modeling technique used to describe these response-generating mechanisms is
that of physiologically structured population modeling which incorporates the dynam-
ics of the internal state of the cell. In general one also needs to take the environmental
∗Received by the editors October 8, 2013; accepted for publication (in revised form) May 29,
2014; published electronically August 28, 2014.
http://www.siam.org/journals/siap/74-4/94043.html
†Centre de Recerca Matem`atica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona),
Spain, and Departament de Matem`atiques, Universitat Auton`oma de Barcelona, 08193 Bellaterra
(Barcelona), Spain (tomasalarc@gmail.com, talarcon@crm.cat). This author was supported by the
Spanish Ministry for Science and Innovation (MICINN) under grant MTM2011-29342 and the Gen-
eralitat de Catalunya under grant 2009SGR345.
‡TU Dresden, Fachrichtung Mathematik, Institut f¨ur Analysis, 01062 Dresden, Germany, and
BCAM–Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain
(getto@bcamath.org, philipp.getto@tu-dresden.de). This author was supported by the DFG and the
Spanish Ministry of Economy and Competitiveness (MINECO) under project MTM 2010-18318.
§Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi v´ertan´uk tere 1, Hungary
(nakata@math.u-szeged.hu). This author was supported by MINECO under project MTM 2010-
18318 and by European Research Council StG 259559.
1266
STABILITY OF A RENEWAL EQUATION 1267
conditions into account as well as the way the population of cells does impact these
conditions. This feedback cycle makes models nonlinear.
There is an abundance of interesting linear and nonlinear structured models that
incorporate transitions between proliferating and quiescent cell populations and are
formulated as partial differential equations, e.g., [5, 7, 8, 19, 21, 24, 33]. Many of
these postulate regulation of one or both transition rates by the population. It is a
common feature that a positive equilibrium is possible if and only if there is a strong
enough regulation of the transition processes.
In the tumor model in [24] the behavior of both proliferating and cancer cells is
dependent on cell size. The authors elaborate conditions for asynchronous exponential
growth of the population (meaning roughly that population size grows exponentially
while the cell size distribution stabilizes) and for the stability of a trivial equilibrium,
which means extinction of the tumor. In [33] the behavior of cells in both stages
depends on a variable called “age.” The authors develop a numerical scheme and
use this to compute the time development of the population density. In [5] a very
general model that incorporates dependence of the cell’s behavior on age and cyclin
content is developed. The paper contains a mathematical analysis of an unregulated
variant of the model. Moreover it is shown that there exists a positive equilibrium
for a regulated variant and convergence to this equilibrium is numerically simulated.
In [7, 8] cyclin content structured versions of the model in [5] are considered. In
[7] well-posedness is established and the existence of equilibria is studied. In [8] the
authors show numerically that, apart from convergence to an equilibrium, oscillations
are also possible. In [21] a general model for cell population dynamics that includes
cell size structure, spatial structure, as well as density-dependent transitions to and
from quiescence is developed. The authors establish well-posedness for a porous media
type single compartment model derived from the general model via a limiting process
and simulate spatial dynamics.
In [19] the authors analyze the model developed in [33]. They use a (formally
derived) characteristic equation to compute stability boundaries for a nontrivial equi-
librium in a parameter plane. Stability of a nontrivial equilibrium means roughly
that a population can be expected to persist. From an interpretation of the stability
boundaries the authors conclude that both increasing the growth rate of the stem
cell population and decreasing the rate of differentiation can be responsible for a
destabilization of the equilibrium.
On the other hand there are many models [1, 6, 20, 22, 23, 29], in particular the
work of Mackey and collaborators, that show the importance of modeling to explain
the interplay of quiescence and clinically observed oscillations at the population level.
Many of these use delay differential equations [18, 36] as the basic modeling tool.
In [1, 20, 23] cell population models that include explicitly feedback, division, and
quiescence are analyzed and oscillations are detected and related to quiescence. Some
of these models are very general; e.g., [20] incorporates interactions with a stem cell
population and considers, in addition to age, maturity of cells.
One of our aims here is to show how (in)stability of a positive equilibrium can
be analytically proved for models with an explicitly incorporated cell cycle. We an-
alyze possibilities for the emergence of oscillations at the population level and try
to identify at the cell level some biological mechanisms that trigger the oscillations.
These aims are facilitated by our formulation of the dynamics with renewal equations
or Volterra functional equations. Linear Volterra equations have been used, e.g., in
[37] to analyze an epidemiological problem. The results in [14] provide our basis for
proving linearized stability results and the Hopf bifurcation theorem for nonlinear
1268 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
Volterra functional equations. There are few linearized stability results for structured
proliferation-quiescence models and we hope to advertise renewal equations as a useful
tool for related problems.
We start with a model where the cell cycle is incorporated via age dependence as
a continuous process. We consider transitions between quiescent and proliferating cell
populations, with age dependence in division and mortality processes. As a result,
one difference with the models in [1, 20, 23] is that we include the two mechanisms
of dividing and going quiescent in a more probabilistic way. We incorporate a control
of the recruitment from quiescence by the population with contributions weighted
according to whether cells are quiescent. We keep the number of parameters low by
fixing a point, age zero of the proliferation phase, in the cell cycle state space at
which cells start after a transition. The flow of cells through this point can then be
described as a population level “birth” rate. For the resulting model we elaborate
sharp conditions for the stability of a positive equilibrium and its destabilization
by way of growing oscillations. We explicitly verify conditions for some linearized
stability theorems and find relations between (in)stability of the equilibrium and the
different ways of modeling the division process. Moreover we relate (in)stability to
the regulation mode of the recruitment process.
A key point is the analysis of characteristic equations. Such equations can be
visualized by defining stability boundaries in planes of parameters. The characteristic
equation that we find here has to our knowledge not been analyzed in a parameter
plane before, and also here we hope that our work can be useful for related problems
in the future.
The remainder of the paper is organized as follows. In section 2 we introduce
assumptions and ingredients of the model and formulate the population dynamics as
a renewal equation. In section 3 we prove that for the renewal equation the principle
of linearized stability holds for any equilibrium. In section 4 we elaborate conditions
for the behavior of an individual cell that lead to (in)stability of the zero equilibrium
and conditions under which there exists a unique positive equilibrium. We also give
a dissipativity result. In section 5 we specify modeling ingredients in more detail and
use these specifications to analyze the stability of the positive equilibrium. We also
exploit the fact that we can allow for age dependence of the per capita division rate.
In particular we introduce two parameterizations: one that describes cell division at a
constant rate and one in which division is concentrated in a point of the age axis such
that there is a fixed delay between two divisions in the absence of a quiescent phase.
We show how different ways of modeling the control of recruitment influence stability
and destabilization. In section 6 we discuss the biological motivation of the model,
comparisons with the literature, interpretations and mathematical results. Finally we
refer to the appendix for proofs of our results.
2. Model formulation.
2.1. Biological ingredients. For the model we assume that cell division is a
moment at which the cell dies and gives birth to two daughters. Immediately after
birth, each daughter either goes into quiescence, with probability 1 −α, or commits
itself to proliferation, with probability α; see also Figure 1, Table 1, and Assump-
tion 3.1. So in a sense we neglect the duration of the G1-phase, which is when cells
usually go quiescent. By the age of a cell we mean the time elapsed since the cell
was born, irrespective of whether it went quiescent. Quiescent cells can be recruited,
which means that they become proliferating cells. By the proliferation age of a pro-
liferating cell we mean the time it lived as a proliferating cell. We define β(a)asthe
STABILITY OF A RENEWAL EQUATION 1269
Tabl e 1
Model ingredients and symbols used in the main text with interpretations and references to
sections and equations in which they are introduced or specified.
Symbol Short description References
aage, proliferation age section 2.1
ai,aij characteristic equation in μ-c-space Lemma 5.2
αprobability of going quiescent after birth section 2.1
b(t) population birth rate at time tsection 2.2
Bdivision probability, concentrated division section 5
β(a) individual division rate sections 2.1, 5
βdivision rate, constant division section 5.1.3
cmaximum recruitment rate section 5
c∗(ν)c-component of bifurcation curve (5.7)
Ckbifurcation curve section 5.1.1
ηproportionality factor, constant division (5.11)
F(ϕ, ψ) operator defining renewal equation section 3.1
ˆ
f(λ) Laplace transform section 5
F(a) survival in proliferation if no division section 2.1
˜
F(a) survival in quiescence if no recruitment section 2.1
Fβ(a) (unconditioned) survival in proliferation section 2.2
G(N) individual recruitment rate sections 2.1, 5
H(μ, c, λ) characteristic equation, concentrated division (5.5)
I(t) individual recruitment rate at time tsection 2.2
Ikdomain of bifurcation curve section 5.1.1
Jkdomain of curve in existence region Lemma 5.7
li(q), i=1,2 representation of bifurcation curves section 5
mij entries of characteristic matrix (3.9)–(3.12)
m(ν) term in bifurcation curves section 5.1.1
M(λ) characteristic matrix (3.8)–(3.9)
μmortality rate sections 4, 5
μ∗(ν)μ-component of bifurcation curves (5.6)
N(t) weighted total population at time t(2.1)
νparameter for bifurcation curves section 5.1.1
νkν-value defining JkLemma 5.7
P(t) total proliferating population at time tsection 2.1, (2.9)
pi(μ, c), i=1,2 characteristic equation, constant division (5.10)
qrelative weight of quiescent population section 2.1
q(r0)q-value defining bifurcation curves section 5.1.1
ΩS,Ω
Uregions of (in)stability section 5.1.1
Q(t) total quiescent population at time tsection 2.1, (2.10)
R(ψ)operatorinb-component of renewal equation (2.4)
r0expected daughter production in proliferation section 4
R0(I) expected reproduction number (4.1)
ρ(q) auxiliary function to define νkLemma 5.7
S(ψ)operatorinI-component of renewal equation (2.5)
ξproportionality factor in existence boundary (4.9)
individual division rate of a cell at proliferation age a. Next, we denote by F(a)the
probability for a cell to survive in the proliferation age interval [0,a), given that it
does not divide, and by ˜
F(a) the probability for a cell to survive in quiescence in the
age interval [0,a), given that it does not get recruited. The probability per unit of
time that a quiescent cell is recruited we call the individual recruitment rate G.We
assume that, at time t,Gdepends on the weighted total population, i.e.,
G=G(N(t)),N(t):=(1−q)P(t)+qQ(t),(2.1)
where P(t)andQ(t) are the respective numbers of proliferating and quiescent cells
and qand 1 −qare relative weights.
1270 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
Fig. 1.Cycle of cells that enter proliferation directly or experience quiescence and recruitment
before. Dashed lines refer to instantaneous events, straight lines to processes that take time.
2.2. Individual dynamics and population bookkeeping. We denote by
Fβ(a):=F(a)e−a
0β(α)dα the (unconditioned) survival probability for a cell in pro-
liferation. We use the notation xt(θ):=x(t+θ), θ≤0, as usual in the theory of
functional differential equations; see, e.g., [26]. By b(t)wedenotethepopulation birth
rate. Moreover, we introduce I(t):=G(N(t)). We formulate the population dynamics
by the system of renewal equations
b(t)=∞
0
b(t−a)R(It)(a)da,(2.2)
I(t)=G∞
0
b(t−a)S(It)(a)dawith(2.3)
R(ψ)(a):=2αβ(a)Fβ(a)(2.4)
+2(1−α)a
0
˜
F(a−θ)ψ(−θ)e−a
θψ(−σ)dσβ(θ)Fβ(θ)dθ,
S(ψ)(a):=(1−α)(1 −q)a
0
˜
F(a−θ)ψ(−θ)e−a
θψ(−σ)dσFβ(θ)dθ(2.5)
+q˜
F(a)e−a
0ψ(−σ)dσ+α(1 −q)Fβ(a).
In the following we explain how this system is constructed. First, 2β(a)Fβ(a)isthe
expected rate of giving birth of a mother at age a, given that she has not been
quiescent. Next, e−a
θI(t−σ)dσ is the probability for not getting recruited in the time
interval [t−a, t−θ]. Hence, ˜
F(a−θ)I(t−θ)e−a
θI(t−σ)dσ for 0 <θ<ais the expected
rate of recruitment at age a−θand time t−θof a cell that has gone quiescent at
time t−a; see Figure 1. Then
2a
0
˜
F(a−θ)I(t−θ)e−a
θI(t−σ)dσβ(θ)Fβ(θ)dθ(2.6)
is the expected rate of giving birth of a mother at age aand time t, given that she
has gone quiescent at time t−aand was recruited at some time t−θin [t−a, t).
From the definition of Rit follows that
(2.7)
R(It)(a)=2αβ(a)Fβ(a)+2(1−α)a
0
˜
F(a−θ)I(t−θ)e−a
θI(t−σ)dσβ(θ)Fβ(θ)dθ.
STABILITY OF A RENEWAL EQUATION 1271
So we can interpret R(It)(a) as the expected rate of giving birth by a mother cell at
age aand time t. Since already a simple survival probability of the form F(a)=e−μa,
μ>0, is nonzero for no matter how large a, we have to integrate up to infinity. Now
(2.2) follows as a consistency relation from the interpretation of b.Next,αFβ(a)is
the probability for a newborn to become proliferating and survive to age aand
(1 −α)a
0
˜
F(a−θ)I(t−θ)e−a
θI(t−σ)dσFβ(θ)dθ(2.8)
is the probability for a newborn to become quiescent, get recruited, and survive to
age aat time t. Hence,
P(t)=∞
0
b(t−a)αFβ(a)(2.9)
+(1−α)a
0
˜
F(a−θ)I(t−θ)e−a
θI(t−σ)dσFβ(θ)dθda,
Q(t)=(1−α)∞
0
b(t−a)˜
F(a)e−a
0I(t−σ)dσda.(2.10)
By definition of S, one has
S(It)(a)=(1−α)(1 −q)a
0
˜
F(a−θ)I(t−θ)e−a
θI(t−σ)dσFβ(θ)dθ(2.11)
+q˜
F(a)e−a
0I(t−σ)dσ+α(1 −q)Fβ(a).
If we compute Nvia (2.1), (2.9), and (2.10) and use (2.11) we get
N(t)=∞
0
b(t−a)S(It)(a)da.(2.12)
Now, (2.3) follows as I(t)=G(N(t)). Moreover, S(It)(a) in (2.11) can now be in-
terpreted as the weighted probability for a newborn to survive to age aat time t,
where “weighted” refers to whether at age athe cell is quiescent or proliferating.
In summary we can interpret (2.2)–(2.3) as an equation for the reproduction of the
population coupled to a law for the feedback via recruitment.
3. The principle of linearized stability. It is shown in [14, 15] that for equa-
tions of the type x(t)=F(xt) the principle of linearized stability holds if Fis con-
tinuously Fr´echet differentiable for short C1. In the remainder of the section, we
guarantee continuous Fr´echet differentiability and conclude the principle of linearized
stability.
3.1. Continuous differentiability. We here work with weighted L1-spaces.
One reason for the weight is that a constant function, say, b>0, such as a steady
state solution, is not integrable on (−∞,0], but the weighted function a→ e−ρab,
ρ>0, is. We therefore define for some ρ≥0 to be specified
ϕ1,ρ := ∞
0
e−ρa|ϕ(θ)|dθ
whenever the integral converges. Note that if survival probabilities reach zero in finite
time, one may simplify the setting by choosing ρ= 0. Then, for m∈N
L1,m
ρ:= {ϕ:R−→Rmmeasurable,ϕ1,ρ <∞}
1272 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
becomes a Banach space with norm ·
1,ρ. Its dual space can be represented as
L∞,m
ρ:= {k:R+→Rm,k∞,ρ <∞},
where
k∞,ρ := sup esss∈R+{eρs|k(s)|<∞},
via the pairing
ϕ, k:= ∞
0
ϕ(−τ)k(τ)dτ, ϕ ∈L1,m
ρ,k∈L∞,m
ρ.
For (ϕ, k )∈L1,1
ρ×L∞,m
ρwe introduce an m-vector via
ϕ, k:= (ϕ, ki)i=1...m.
Assumption 3.1. The survival probabilities are nonincreasing, nonnegative,
F(0) = ˜
F(0) = 1,and
∃κ, K > 0such that F(a)≤Ke−κa,˜
F(a)≤Ke−κa.
The division rate βis bounded and nonnegative. Moreover α, q ∈[0,1] and G:R+→
R+is continuous.
For κgiven in this way we can specify ρ.
Assumption 3.2. 0<4ρ<κ.
For the following result we use only 0 <2ρ<κ, but in the differentiability proof
the full property will be used. Next, we define the positive cones
L1,m
ρ,+:= {ϕ∈L1,m
ρ:ϕ(θ)≥0 for almost all θ∈(−∞,0]}
and similarly L∞,m
ρ,+.
Lemma 3.3. R,S:L1,1
ρ,+→L∞,1
ρ,+are well-defined operators.
Proof. We show the statement for S, the statement for Rcan be shown similarly.
First, α(1 −q)Fβ∈L∞,1
ρ,+by Assumptions 3.1 and 3.2. Next, similarly, for ψ∈L1,1
ρ,+
we have
eρa|q˜
F(a)e−a
0ψ(−σ)dσ|≤Ke−(κ−ρ)a≤K.
Hence, a→ q˜
F(a)e−a
0ψ(−σ)dσ is an element of L∞,1
ρ,+. Finally for some K1,K2≥0
eρa a
0
˜
F(a−θ)ψ(−θ)e−a
θψ(−σ)dσFβ(θ)dθ
≤K1e−(κ−ρ)aa
0
ψ(−θ)dθ
≤K1e−(κ−2ρ)aa
0
e−ρθψ(−θ)dθ ≤K1e−(κ−2ρ)aψ1,ρ ≤K2ψ1,ρ .
Thus
sup essa∈[0,∞)eρa a
0
˜
F(a−θ)ψ(−θ)e−a
θψ(−σ)dσFβ(θ)dθ<∞.
Hence, the statement follows
STABILITY OF A RENEWAL EQUATION 1273
Now we can define
F:L1,2
ρ,+→R2
+;F(ϕ, ψ)=(ϕ, R(ψ),G(ϕ, S (ψ))),(3.1)
set x=(b, I), and rewrite (2.2–2.3) as
x(t)=F(xt),t>0,x(t)=(ϕ0(t),ψ
0(t)),t∈(−∞,0]
for given functions ϕ0,ψ
0∈L1,1
ρ,+.NotethatasFis nonnegative, if there exists a
solution for nonnegative initial conditions, it is necessarily nonnegative. Our next aim
is to show that Fis continuously Fr´echet differentiable. As the domain of Fis the
positive cone we use the concept of relative Fr´echet differentiability, where the point
at which it is differentiated and the perturbation are required to be elements of the
domain (see, e.g., Definition 2.1 in [35]), which here is the positive cone. The next
result we prove in the appendix.
Proposition 3.4. The operators Rand Sare C1with
DR(ψ)ψ(a)=2(1−α)a
0
˜
F(a−θ)Fβ(θ)β(θ)e−a
θψ(−σ)dσ
(3.2)
·ψ(−θ)−ψ(−θ)a
θ
ψ(−σ)dσdθ,
DS(ψ)ψ(a)=(1−α)(1 −q)a
0
˜
F(a−θ)Fβ(θ)e−a
θψ(−σ)dσ
(3.3)
·ψ(−θ)−ψ(−θ)a
θ
ψ(−σ)dσdθ −q˜
F(a)e−a
0ψ(−σ)dσa
0
ψ(−σ)dσ
.
Differentiability of Fis now a straightforward combination of the previous result
and the chain rule.
Theorem 3.5. Suppose that for an element (ϕ, ψ)∈L1,2
ρ,+the map Gis C1in
a neighborhood of ϕ, S(ψ);thenFis C1in a neighborhood of (ϕ, ψ)with derivative
DF(ϕ, ψ)(ϕ, ψ)T=(D1F(ϕ, ψ)ϕ, D2F(ϕ, ψ)ψ),where
D1F(ϕ, ψ)ϕ=ϕ, (R(ψ),G
(ϕ, S(ψ))S(ψ)),(3.4)
D2F(ϕ, ψ)ψ=ϕ, (DR(ψ),G
(ϕ, S(ψ))DS(ψ))ψ(3.5)
with Rand Sdefined in (2.4)–(2.5) and DR and DS as computed in (3.2)–(3.3).
3.2. Linearized stability and characteristic equation. To establish the
principle of linearized stability we first specify derivatives for steady states. Let (ϕ, ψ)
be a steady state, i.e., a constant solution of (3.1). Suppose moreover that Gis in-
vertible at ψ; then it holds that ϕ, S(ψ)=G−1(ψ) and if the function Gis easy
to invert, the right-hand side may be easier to deal with than the left-hand side. In
particular the expressions given in (3.4)–(3.5) can then be simplified to
D1F(ϕ, ψ)ϕ=ϕ, (R(ψ),G
(G−1(ψ))S(ψ)),(3.6)
D2F(ϕ, ψ)ψ=ϕ, (DR(ψ),G
(G−1(ψ))DS(ψ))ψ.(3.7)
Now we formulate a linearized stability result that is a corollary of Theorem 3.15
in [15].
1274 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
Theorem 3.6. Suppose that there exists an equilibrium solution (ϕ, ψ)of (3.1)
and that Gis invertible at ψand C1in a neighborhood of G−1(ψ); then the stability
of (ϕ, ψ)is determined by the location of the roots of the characteristic equation
det (M(λ)−id)=0,with M(λ):=(mij )1≤i,j≤2,(3.8)
m11 := 2 ∞
0
e−λaαβ(a)Fβ(a)(3.9)
+(1−α)ψa
0
e−(a−θ)ψ˜
F(a−θ)β(θ)Fβ(θ)dθda,
m12 := 2ϕ(1 −α)(3.10)
·∞
0a
0
˜
F(a−θ)e−ψ(a−θ)1−ψ1−e−λ(a−θ)
λFβ(θ)β(θ)e−λθ dθda,
m21 := G(G−1(ψ))(1 −q)α∞
0
e−λaFβ(a)da(3.11)
+(1−α)ψ∞
0
e−λa a
0
˜
F(a−θ)e−ψ(a−θ)Fβ(θ)dθda
+q(1 −α)∞
0
e−(λ+ψ)a˜
F(a)da,
m22 := G(G−1(ψ))ϕ(1 −α)(1 −q)∞
0a
0
Fβ(θ)(3.12)
·e−λθ ˜
F(a−θ)e−ψ(a−θ)1−ψ1−e−λ(a−θ)
λdθda
−q∞
0
˜
F(a)e−ψa 1−e−λa
λda.
In particular, if all roots have negative real parts, then (ϕ, ψ)is locally exponentially
stable. If there exists a root with positive real part, then (ϕ, ψ)is unstable.
4. Linearized stability of the trivial equilibrium and existence of a pos-
itive equilibrium. To simplify the discussion of existence and stability of equilibria,
the following notation is useful. A crucial role will play the compound parameter
r0:= 2 ∞
0
β(a)Fβ(a)da
that gives the expected lifetime production of daughter cells of a cell, given that it
has not been quiescent. Note that r0<2 by what we have assumed. Moreover we
define
R0(I):=r0α+(1−α)I∞
0
˜
F(a)e−Iada
(4.1)
as the expected lifetime reproduction number of a cell in constant environmental
conditions as specified by I. The notion of reproduction number is widespread in the
literature on analysis of population dynamical models [31, 16]. Next, we can derive
equilibrium conditions as
STABILITY OF A RENEWAL EQUATION 1275
b=b∞
0
R(I)(a)da,(4.2)
I=Gb∞
0
S(I)(a)da.(4.3)
If b= 0 there is exactly one equilibrium, (b, I)=(0,G(0)), which we call the trivial
equilibrium. We give conditions for the (in)stability of this equilibrium with classical
interpretations at the individual level. We will apply the well-known identity
∞
0
e−λa a
0
f(θ)g(a−θ)dθda =∞
0
e−λθf(θ)dθ ∞
0
e−λag(a)da.(4.4)
Lemma 4.1. If R0(G(0)) <1holds, then the trivial equilibrium is stable; if
R0(G(0)) >1, then the trivial equilibrium is unstable.
Proof. If we incorporate λinto the notation, the characteristic equation for the
trivial equilibrium becomes m11 (λ) = 1, where by (4.4)
m11(λ)=2α+(1−α)G(0) ∞
0
e−θ(λ+G(0)) ˜
F(θ)dθ∞
0
e−λθβ(θ)Fβ(θ)dθ.
Note that m11(0) = R0(G(0)). Suppose that R0(G(0)) <1 and that there is a root λ=
x+iy,x≥0. Then |m11 (λ)|≤m11(0) = R0(G(0)) <1, which yields a contradiction,
and stability follows. Next, suppose that R0(G(0)) >1; then m11 (0) >1, m11(x)↓0,
x∈Ras x↑∞. Hence there exists some x>0 such that m11 (x) = 1 and instability
follows.
Next note that we can apply (4.4) to see that ∞
0R(I)(a)da =R0(I).If b>0,
and I≥0, we call the equilibrium positive. Hence, there exists a positive equilibrium
if and only if there exists some (b, I ) with b>0andI≥0 such that
1=R0(I)(4.5)
and (4.3) hold. An equivalent condition is the existence of a positive root Nof
1=R0(G(N)).(4.6)
If α= 1, then (2.2) is independent of I. Hence, the model is linear and in general only
the trivial equilibrium exists. In the following we will ignore this case by assuming
that α∈[0,1). We will guarantee monotonicity of the function N→ R0(G(N)) via
the following assumption.
Assumption 4.2. Either
(i) the function Gis strictly monotonously decreasing or
(ii) there exists some y∈(0,∞)such that Gis strictly monotonously decreasing
on [0,y)and constant on [y, ∞).
Note that by the nonnegativity property of Gthere exists a finite limit G(∞)and
that in case (i) G−1(G(∞)) is empty and in case (ii) G−1(G(∞)) = [y, ∞). We now
elaborate sufficient conditions for the existence of a positive equilibrium.
Theorem 4.3. If R0(G(0)) <1, then there is no positive equilibrium. If
R0(G(0)) >1and R0(G(∞)) <1, then there exists a unique positive equilibrium.
In this case there exists a unique positive root Iof (4.5),theI-component of the
positive equilibrium, G−1(I)is uniquely defined, and the b-component of the positive
equilibrium has the representation
b=r0IG−1(I)
(1 −q)I∞
0Fβ(a)da +q(1 −αr0).(4.7)
1276 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
Proof. First note that if I≥0, then
I∞
0
˜
F(a)e−Iada =1+∞
0
e−Iad˜
F(a),(4.8)
where the right-hand side should be understood as a Stieltjes integral. Now consider
the quantity in (4.8) as a function of I. From the left-hand side it follows that this
function is zero in zero. Considering the right-hand side, one sees that the derivative
of the function with respect to Iis positive, since ˜
Fis nonincreasing and nonconstant.
Hence, the function and thus R0(I) is strictly increasing in I. Then we can conclude
that N→ R0(G(N)) is nonincreasing. Hence it is clear that if R0(G(0)) <1, then
there cannot be a positive equilibrium. Now suppose that R0(G(0)) >1andthat
R0(G(∞)) <1. Then there exists some N>0 such that (4.6) holds and the existence
of a positive equilibrium follows. Moreover there exists a root I:= G(N) of (4.5) and
uniqueness and positivity of this root follow from the strict monotonicity of R0(I).
To understand the uniqueness of the b-component, first assume that Assumption
4.2(i) holds. Then there can be only one Nsatisfying I=G(N). Now suppose that
Assumption 4.2(ii) holds and assume that there is some N=Nsuch that G(N)=I.
Then G(N)=G(N), hence y≤N,thusG(N)=G(∞), and hence
1>R
0(G(∞)) = R0(G(N)) = R0(G(N)) = 1,
which is a contradiction. We can conclude that the b-component is uniquely defined by
b=G−1(I)
∞
0S(I)(a)da .
If we use (2.5), (4.4), and (4.5), we can deduce that the b-component has the repre-
sentation that is claimed.
Suppose now that R0(G(∞)) <1 and, when some parameter is changed, R0(G(0))
increases from below one to above one. From what we have shown so far, we know
that at the critical value the trivial equilibrium loses its stability and the positive
equilibrium emerges. We thus have a transcritical bifurcation and according to the
principle of the exchange of stability [27] we can expect the positive equilibrium to
be stable just above the critical value. That this is indeed the case, we will see below.
As soon as we identify two parameters we can use the equation R0(G(0)) = 1 that
corresponds to the transcritical bifurcation to define a curve in the two parameter
plane. A picture of the curve yields biological insight: at a glance we can see how
parameters influence the persistence of a cell population. In the following we specify
the survival probability for quiescent cells as ˜
F(a)=e−μa,μ>0, and use μas one
of the two parameters. Note first that for this particular ˜
Fwe get
R0(I)=r0α+(1−α)I
μ+I,sign(R0(I)−1) = sign(I(r0−1) −μ(1 −αr0)).
Moreover, we can solve R0(I) = 1 with respect to I, which yields
I=ξμ, with ξ=ξ(α, r0):= 1−αr0
r0−1.(4.9)
Note that I>0 if and only if
r0∈1,min 1
α,2.(4.10)
Then, the statements of Lemma 4.1 and Theorem 4.3 lead to the following corollary.
STABILITY OF A RENEWAL EQUATION 1277
Corollary 4.4. If either r0<1or both (4.10) and G(0) <ξμhold, then
the trivial equilibrium is stable and no positive equilibrium exists. If (4.10) holds but
G(0) >ξμ, then the trivial equilibrium is unstable; if additionally G(∞)<ξμ,then
there exists a unique positive equilibrium with Igiven as in (4.9) and
b=μr0G−1(I)
(1 −q)μ∞
0Fβ(a)da +q(r0−1).
In conclusion of the section, we prove that the system is dissipative, i.e., there
exists a bounded set that attracts solutions, when either the trivial equilibrium is
locally asymptotically stable or the positive equilibrium exists.
Theorem 4.5. Assume that there exists Msuch that G(N)N≤Mfor any
N∈R+.Ifq∈(0,1] and αr0<1,then
lim sup
t→∞
b(t)≤M
q
r0
1−αr0
.
Proof. Recall that I(t)=G((1 −q)P(t)+qQ(t)). Since Gis a decreasing function,
one has an estimation for q>0:
I(t)Q(t)≤G(qQ(t)) Q(t)≤M
q.
It is now convenient to write
b(t)=2∞
0
(αb(t−a)+I(t−a)Q(t−a)) β(a)Fβ(a)da.
Thus,
b(t)≤2∞
0
αb(t−a)β(a)Fβ(a)da +M
qr0.
Assume that lim supt→∞ b(t)=∞. Then there exists a sequence {tn}∞
n=1 such that
b(t)≤b(tn),t≤tnand lim
n→∞ b(tn)=∞.
One obtains b(tn)≤αr0b(tn)+ M
qr0, which implies
b(tn)≤M
q
r0
1−αr0
for each n. Thus we get a contradiction. It then holds that lim supt→∞ b(t)<∞.The
same estimation shows the conclusion.
For the case q= 0 we need to consider the behavior of two components Pand
Qto estimate IQ =G(P)Q. Since, in general, this does not seem straightforward
and analysis of the global behavior of solutions is not in the scope of this manuscript,
we here leave this as an open problem. We remark that, by a simple comparison
argument, one can show that if αr0>1, then b(t) tends to infinity.
1278 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
5. Stability boundaries for the positive equilibrium. Our next aim is to
study the stability of the positive equilibrium. We reduce the generality by assuming
that F(a):=e−μa or, in other words, that also proliferating cells have age-independent
mortality rate μ. The main benefit is parameter reduction. For this choice, we write
in Lemma 5.2 below, as a first step, the characteristic equation such that the type of
dependence on the complex variable is clearly visible. In this context it is useful to
introduce two functions l1(q):= 3q−1
2and l2(q):=1−2q.Notethatr0l1(q)+l2(q)>0,
since the left-hand side equals (1 −q)(1 −1
2r0)+q(r0−1), which is positive as q∈[0,1]
and r0<2. In the following we shall often omit the argument qof l1and l2.To
simplify representations we use the usual notation ˆ
f(λ)=∞
0e−λaf(a)da for the
Laplace transform of an appropriate function f. In the appendix we prove the next
proposition.
Proposition 5.1. The characteristic equation for the positive equilibrium can be
represented as
2
βFβ(λ)αλ +G(G−1(I))G−1(I)l1(r0−1)
r0l1+l2
+μ(1 −α)
r0−1
(5.1)
−λ+G(G−1(I))G−1(I)l2(r0−1)
r0l1+l2
−μr0(1 −α)
r0−1=0.
In the remainder of this paper, we specify G(N):=max{c(1 −N),0},where
cis a positive parameter. Note that G,F,and ˜
Fsatisfy Assumptions 3.1 and 4.2.
Moreover, G(0) = cand G(∞) = 0 such that Corollary 4.4 yields the line shown
in Figure 2(a). We now elaborate (5.1) for the specific recruitment function. The
proof of the next result is straightforward and we omit it. The notation is designed
to deduce stability boundaries in the μ-c-plane, which we shall do below.
Lemma 5.2. For G(N)=c(1 −N)the characteristic equation becomes
2
βFβ(λ)(αλ +a1(μ, c)T)−λ+a2(μ, c)T=0,a
i:= (ai1,a
i2),i=1,2,
a11 := 1−α
r0−1+(1 −αr0)l1
l1r0+l2
,a
12 := −(r0−1) l1
l1r0+l2
,
a21 := −(1 −α)r0
r0−1+(1 −αr0)l2
l1r0+l2
,a
22 := −(r0−1) l2
l1r0+l2
.
Below we would like to analyze a submodel where division is concentrated in a
point in the cell cycle. Note first that we cannot specify an essentially bounded β
such that for division probability densities one has
β(a)e−a
0β(α)dα =Bδ1(a),
where δ1is a Dirac-measure concentrated in one and B∈[0,1] denotes the probability
that a cell that has reached one divides. We can, however, generalize the characteristic
equation to
mb(λ)(αλ +a1)−λ+a2=0,(5.2)
abbreviating ai=ai(μ, c)T,i=1,2, where mb(λ)=∞
0e−λamb(da)isaLaplace
transform generalized to (positive) measures. Then, with
mβ(ω):=2ω
β(a)Fβ(a)da,(5.3)
mD(ω):=δ1(ω)2Be−μ
STABILITY OF A RENEWAL EQUATION 1279
we recover the old setting, i.e., mβ(λ)=2
βFβ(λ), as well as include concentrated
division with mD(λ)=2Be−(λ+μ). We remark that the case of division concentrated
at arbitrary τ>0 can be scaled to τ=1. Notefirstthatr0=mb(0) and in case of
concentrated division
r0=mD(0) = 2Be−μ.(5.4)
In the following we will first analyze (5.2) for general mb,thenformD, and finally
combine the two results to conclude (in)stability properties for approximated concen-
tration of division.
Lemma 5.3.
(i) c=ξμ ⇔(r0a1+a2)(μ, c)T=0.
(ii) If c=ξμ,thenλ=0solves (5.2) and a1(μ, c)T>0.
Proof. (i) This can be shown by filling in ξ,a1,anda2. Next, (5.2) in λ=0is
(r0a1+a2)(μ, c)T= 0, which is true by (i).
In the following let x,y∈R.
Lemma 5.4.
(i) If λ=x+iy,x≥0,then|mb(λ)|≤r0.Ifx>0,then|mb(λ)|<r
0.
(ii) If c=ξμ and λ=x+iy solves (5.2),thenx>0cannot be and x=0if and
only if λ=0.
Hence, for c=ξμ there are no roots in the right half plane and the only root on
the imaginary axis is λ=0.
Lemma 5.5. If at c=ξμ for fixed μthe parameter cincreases sufficiently little,
then the root λ=0 moves into the left half plane and there are no roots in the right
half plane.
Note that for absolutely continuous measures that can be expressed via (5.3) we
have now shown that at c=ξμ there is an exchange of stability such that for c>ξμ
locally the trivial equilibrium destabilizes and the nontrivial equilibrium stabilizes.
The next result makes sure that roots can enter the right half plane only through
a compact subset of the imaginary axis and not from infinity; see, e.g., Chapter XI
in [18].
Lemma 5.6 (a priori estimate). For any K>0there exists some L=L(K)>0
such that if λ=x+iy,x≥0, solves (5.2) and c,μ≤K,thenx,y≤L.
5.1. Stability boundary of the positive equilibrium. Fo r a n a nalys i s i n a
parameter plane we consider μand cas variable and q,α,andr0as fixed. Now, in the
μ-c-parameter plane the existence boundary for the positive equilibrium is defined as
the straight line c=ξμ and the existence region for the positive equilibrium is given
as {(μ, c)|c>ξμfor μ>0}; see Figure 2(a). In the following we present and analyze
respective characteristic equations for the positive equilibrium for two submodels of
concentrated and constant cell division.
5.1.1. Concentrated cell division. For concentrated cell division, we have
mb(λ)=e−λr0and can derive the characteristic equation for the positive equilib-
rium as
H(μ, c, λ) = 0 with H(μ, c, λ):=e−λr0(αλ +a1(μ, c)T)−λ+a2(μ, c)T.(5.5)
In the following we will analyze (5.5) in the μ-c-plane. We have seen already that there
are no roots with positive real part slightly above the existence boundary. Next, we
investigate the possibility that there are purely imaginary roots for parameter values
1280 TOM ´
AS ALARC´
ON, PHILIPP GETTO, AND YUKIHIKO NAKATA
in the interior of the existence region. For every nonnegative integer kwe define a
curve via
Ck:= {(μ∗,c
∗)(ν)|ν∈Ik},I
k:= (2kπ, (2k+1)π),
μ∗(ν):= ν
sin ν
r0−1
(1 −α)r0
m(ν),(5.6)
c∗(ν):= ν
sin ν1+αr0