www.biomathforum.org/biomath/index.php/biomath

REVIEW ARTICLE

One and two-phase cell cycle models
Katarzyna Pichór∗, Ryszard Rudnicki†

∗Institute of Mathematics, University of Silesia
Katowice, Poland

katarzyna.pichor@us.edu.pl
†Institute of Mathematics, Polish Academy of Sciences

Katowice, Poland
rudnicki@us.edu.pl

Received: 19 February 2019, accepted: 26 May 2019, published: 1 June 2019

Abstract—In this review paper we present deter-
ministic and stochastic one and two-phase models
of the cell cycle. The deterministic models are given
by partial differential equations of the first order
with time delay and space variable retardation. The
stochastic models are given by stochastic iterations
or by piecewise deterministic Markov processes. We
study asymptotic stability and sweeping of stochastic
semigroups which describe the evolution of densities
of these processes. We also present some results con-
cerning chaotic behaviour of models and relations
between different types of models.

Keywords-cell cycle; transport equation; stochas-
tic operator and semigroup; asymptotic stability;
Markov process

I. INTRODUCTION

The cell cycle is a series of events that take place
in a cell leading to its replication. It is regulated
by a complex network of protein interactions. For
example, a relatively simple mathematical model
of mammalian cell-cycle control consists of eigh-
teen differential equations [32]. Usually the cell
cycle is divided into four phases [2], [18], [31].

The first one is the growth phase G1 with synthesis
of various enzymes. The duration of the phase G1
is highly variable even for cells from one species.
The DNA synthesis takes place in the second phase
S. In the third phase G2 a significant protein syn-
thesis occurs, which is required during the process
of mitosis. The last phase M consists of nuclear
division and cytoplasmic division. We consider
also G0 phase (quiescence). A cell can enter the
G0 phase from G1 and may remain quiescent for a
long period of time, possibly indefinitely, or after
some period of time it can go back to the G1 phase.
The schematic model of the cell cycle is given in
Fig. 1.

There are several mathematical models of the
cell cycle. One can consider four-phase models [6],
but the most popular are one or two-phase models.
In one-phase models, we put together phases G1,
S, G2, and M and neglect the phase G0. The
second category are two-phase models. Biologists
used to divide the whole cycle into interphase,
which consists of G1, S and G2, and the mitotic
phase M. From a mathematical point of view it is

Copyright: c©2019 Pichór et al. This article is distributed under the terms of the Creative Commons Attribution License
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Citation: Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models, Biomath 8
(2019), 1905261, http://dx.doi.org/10.11145/j.biomath.2019.05.261

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

G0G1A

S

G2

B

M

Fig. 1. Schematic model of the cell cycle.

better to divide the cell cycle into the resting (or
growth) phase A = G1 with a random duration tA
and the proliferating phase B which consists of the
phases S, G2 and M, and has an almost constant
duration tB. In these models the phase G0 is also
neglected. There are also two-phase models with
cells in the proliferating state (phases G1, S, G2
and M) and in the quiescent state [4], [41].

The mathematical models of the cell cycle are
based on the concept of maturity, also called the
physiological age. The maturity can be the size of
a cell, its volume or DNA content. The core of
the theory was formulated in the late sixties [24],
[43]. A lot of new models appeared in the eighties
and we can divide them into two groups. The first
group contains discrete-time models (generational
models) which describe the relation between the
initial maturity of mother and daughter cells. In
this group we have one-phase models [20] and
two-phase models [57], [58].

The second group is formed by continuous-
time models characterizing the time evolution of
distribution of cell maturity. We consider two types
of continuous-time one-phase models: with the
division of a cell when the cell maturity has
a given level [24], [42], [43], or with division
at a random maturity (including size structured
models) [5], [7], [11], [15], [29], [34], [52], [59].
The second type consists of two-phase models [1],
[8], [27], [39], [56]. The above mentioned models
describe the distribution of maturity in the whole
population. We also investigate models given by

piecewise deterministic Markov processes [54],
which describe the evolution of maturity of con-
secutive descendants of a single cell [22], [29],
[38]. Such models seem to be the most suitable
for the description of the cell cycle because they
do not include environmental components.

This paper provides an introduction to one and
two-cycle models, with particular emphasis on
models given by piecewise deterministic Markov
processes. We also formulate some results con-
cerning their long-time behaviour and compare
asymptotic properties of discrete and continu-
ous time models. The evolution of distribution
of maturity in the discrete time models and in
models given by piecewise deterministic Markov
processes is described by stochastic (Markov) op-
erators and semigroups [21], [47]. The results
concerning asymptotic stability and sweeping of
stochastic semigroups are based on papers [35],
[36], [37]. We also present results concerning
chaotic properties of some maturity structured
models [49], [50].

II. ASYMPTOTIC PROPERTIES OF STOCHASTIC
OPERATORS AND SEMIGROUPS

Now we introduce the notion of a stochastic
operator and a stochastic semigroup. Then we
present some results concerning asymptotic stabil-
ity, sweeping and Foguel alternative for stochastic
semigroups. These results will be applied to mod-
els of cell cycle presented in the next sections.

Let a triple (X, Σ,µ) be a σ-finite measure
space. Denote by D the subset of the space L1 =
L1(X, Σ,µ) which consists of all densities

D = {f ∈ L1 : f ≥ 0, ‖f‖ = 1}.

A linear operator P : L1 → L1 is called stochastic
if P(D) ⊆ D. A family {P(t)}t≥0 of linear
operators on L1 is called a stochastic semigroup
if it is a strongly continuous semigroup and all
operators P(t) are stochastic. Now, we intro-
duce some notions which characterize the asymp-
totic behaviour of iterates of stochastic operators
Pn, n = 0, 1, 2, . . . , and stochastic semigroups
{P(t)}t≥0. The iterates of stochastic operators
form a discrete-time semigroup and we can also

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

use the notation P(t) = Pt for their powers so that
we formulate most of the definitions and results for
both types of semigroups without distinguishing
them. A stochastic semigroup {P(t)}t≥0 is asymp-
totically stable if there exists a density f∗ such that

lim
t→∞
‖P(t)f −f∗‖ = 0 for f ∈ D. (1)

From (1) it follows immediately that f∗ is invari-
ant with respect to {P(t)}t≥0, i.e. P(t)f∗ = f∗
for each t ≥ 0. A stochastic semigroup {P(t)}t≥0
is called sweeping with respect to a set B ∈ Σ if
for every f ∈ D

lim
t→∞

∫
B
P(t)f(x) µ(dx) = 0.

In order to formulate a result concerning asymp-
totic stability of a stochastic semigroup we need
the following definition. A stochastic semigroup
{P(t)}t≥0 is called partially integral if there exists
a measurable function q(t, ·, ·) : X ×X → [0,∞)
called kernel such that∫

X

∫
X
q(t,x,y) µ(dx) µ(dy) > 0

for some t > 0 and

P(t)f(y) ≥
∫
X
q(t,x,y)f(x) µ(dx) for f ∈ D.

(2)

Theorem 2.1 ([35]): Let {P(t)}t≥0 be a
continuous-time partially integral stochastic
semigroup. Assume that the semigroup {P(t)}t≥0
has a unique invariant density f∗. If f∗ > 0 a.e.,
then the semigroup {P(t)}t≥0 is asymptotically
stable.

The next result concerns the Foguel alterna-
tive [21], that is, when a stochastic semigroup
{P(t)}t≥0 is asymptotically stable or sweeping
from all compact sets. We assume additionally that
(X,ρ) is a separable metric space, Σ = B(X) is
the σ-algebra of Borel subsets of X and that the
semigroup {P(t)}t≥0 is partially integral with the
kernel q which satisfies the following condition:
(K) for every x0 ∈ X there exist ε > 0, t > 0
and a measurable function η ≥ 0 such that

∫
η(y) µ(dy) > 0 and

q(t,x,y) ≥ η(y)1B(x0,ε)(x) for x,y ∈ X, (3)

where 1B(x0,ε) is the characteristic function of
B(x0,ε) = {x ∈ X : ρ(x,x0) < ε}.

We define condition (K) for a stochastic opera-
tor P in the same way, remembering the notation
P(t) = Pt. Condition (K) is satisfied if, for exam-
ple, for every point x ∈ X there exist t > 0 and
y ∈ X such that the kernel q(t, ·, ·) is continuous
in a neighbourhood of (x,y) and q(t,x,y) > 0.
We need an auxiliary definition. We say that a
stochastic semigroup {P(t)}t≥0 overlaps supports
if for every f,g ∈ D there exists t > 0 such that

µ(supp P(t)f ∩ supp P(t)g) > 0.

The support of any measurable function f is
defined up to a set of measure zero by the formula

supp f = {x ∈ X : f(x) 6= 0}.

Theorem 2.2: Assume that {P(t)}t≥0 satisfies
(K) and overlaps supports. Then {P(t)}t≥0 is
sweeping or {P(t)}t≥0 has an invariant density
f∗ with a support A and there exists a positive
linear functional α defined on L1(X, Σ,µ) such
that
(i) for every f ∈ L1(X, Σ,µ) we have

lim
t→∞
‖1AP(t)f −α(f)f∗‖ = 0, (4)

(ii) if Y = X\A, then for every f ∈ L1(X, Σ,µ)
and for every compact set F we have

lim
t→∞

∫
F∩Y

P(t)f(x) µ(dx) = 0. (5)

In particular, if {P(t)}t≥0 has an invariant density
f∗ with the support A and X \ A is a subset of
a compact set, then {P(t)}t≥0 is asymptotically
stable.

The proof of Theorem 2.2 is based on theorems
on asymptotic decomposition of stochastic oper-
ators [36, Theorem 1] and stochastic semigroups
[36, Theorem 2] and it is given in [38]. Another
consequence of [36, Theorem 2] it the following.

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

Corollary 1: Assume that a continuous-time
stochastic semigroup {P(t)}t≥0 satisfies condition
(K) and has no invariant densities. Then {P(t)}t≥0
is sweeping from compact sets.

III. RUBINOW-TYPE MODELS

In all models in this section we consider a
sequence of consecutive descendants of a single
cell in a single line. One of the oldest models of
cell cycle, introduced by Rubinow [43], is based
on the concept of maturity and maturation velocity.
Maturity is a real variable m from the interval
[0, 1] which describes the position of a cell in the
cell cycle. A new born cell has maturity 0 and a
cell splits at maturity 1. In Rubinow’s model m
grows according to the equation m′ = v, where
the maturation velocity v can depend on m and
also on other factors such as time, the size of
the population, temperature, light, environmental
nutrients, pH, etc. If we neglect environmental fac-
tors, resource limitations, and stochastic variation,
then we can assume that v depends only on m and
all cells have identical cell cycles, in particular,
they have the same cell cycle length l.

However, experimental observations concerning
cell populations, cultured under identical condi-
tions for each member, revealed high variability of
l in the population. It means that the population
is heterogeneous with respect to cell maturation
velocities and therefore mathematical models of
the cell cycle should take into account maturation
velocities. A model of this type was proposed
by Lebowitz and Rubinow [24]. In their model
the cell cycle is determined by its maturation
velocity v, which is fixed at the birth of the cell
and constant during the cell cycle. The relation
between the maturation velocities of mother’s v
and daughter’s cells v̄ is given by a probability
transition function P(v,dv̄). Denote by tn the
time, when a cell from the nth-generation splits.
Then the maturity and the maturation velocity of
a cell from the nth-generation are described by
a stochastic process ξ(t) = (m(t),v(t)), t ∈
[tn−1, tn). The process ξ(t), t ≥ 0, has jumps
at points t0, t1, t2, . . . and between jumps the

pair (m(t),v(t)) satisfies the following system of
differential equations{

m′(t) = v(t),

v′(t) = 0.
(6)

At jump points we have m(tn) = 0 and P(v(tn) ∈
B |v(t−n ) = v) = P(v,B) for each n ∈ N
and each Borel subset B of (0,∞). Since v(t)
is constant in the interval (tn−1, tn), we have
tn − tn−1 = 1/v(tn−1). It is not difficult to check
that ξ(t), t ≥ 0, is a homogeneous Markov pro-
cess. Observe that there is an increasing sequence
of random times (tn), called jump times, such that
the sample paths (trajectories) of ξ(t) are defined
in a deterministic way in each interval (tn, tn+1).
A process which has such properties is called
piecewise deterministic.

The Lebowitz and Rubinow model can be iden-
tified with a one-dimensional stochastic billiard on
the interval [0, 1]. Namely, consider a particle mov-
ing in the interval [0, 1] with a constant velocity.
We assume that when the particle hits the bound-
ary points 0 and 1, it changes its velocity according
to the probability measures P0(−v,B) = P(v,B)
and P1(v,−B) = P(v,B), respectively, where
v > 0 and B is a Borel subset of (0,∞). Observe
that the PDMP defined in the Lebowitz–Rubinow
model is given by

ξ(t) =

{
(m(t),v(t)), if v(t) > 0,
(1 −m(t),−v(t)), if v(t) < 0,

where m(t) and v(t) represent position and veloc-
ity of the moving particle at time t.

Asymptotic properties of the general one-
dimensional stochastic billiard were studied in
[30]. Based on that paper we briefly present
properties of the Lebowitz–Rubinow model: for
v, v̄ ∈ (0, 1], P(v,dv̄) = q(v, v̄) dv̄, where∫ 1
0
q(v, v̄) dv̄ = 1. The process ξ(t) =

(m(t),v(t)) induces a stochastic semigroup
{P(t)}t≥0 on the space L1(X, Σ,µ), where X =
(0, 1]2, Σ = B(X), and dµ = dm × dv.
The infinitesimal generator A of the semigroup

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

{P(t)}t≥0 is given by the formula

Af(m,v) = −v
∂f

∂m
(m,v).

The functions f from the domain of the operator
A satisfy the boundary condition:

v̄f(0, v̄) =

∫ 1
0
vq(v, v̄)f(1,v) dv.

Observe that if this semigroup has an invariant
density f∗, then Af∗ = 0. Thus f∗ does not
depend on m. Set h(v) = vf∗(v). Then h is a
fixed point of the stochastic operator K on L1[0, 1]
given by

Kh(v̄) =

∫ 1
0
q(v, v̄)h(v) dv. (7)

Since the boundary condition contains a ker-
nel operator, one can check that the semigroup
{P(t)}t≥0 is partially integral. We assume that K
is irreducible which is equivalent to say that∫

(0,1]\B

(∫
B
q(v, v̄) dv

)
dv̄ > 0

for each measurable set B ⊆ (0, 1] of the Lebesgue
measure 0 < |B| < 1, see [55, p. 334]. From
irreducibility it follows that if an invariant density
f∗ exists then it is unique and f∗(v) > 0 for v-
a.e. The question of the existence of an invariant
density is nontrivial. If for example we assume that
there exist C > 0 and γ > 0 such that

q(v, v̄) ≤ C|v̄|γ for v, v̄ ∈ (0, 1], (8)

then an invariant density exists [30]. It means
that irreducibility of K and condition (8) imply
asymptotic stability of the semigroup.

Now we consider the case when the semigroup
{P(t)}t≥0 has no invariant density. Assume that
the kernel q is continuous and bounded. Then the
semigroup satisfies condition (K) and from Corol-
lary 1 it follows that the semigroup is sweeping
from compact sets. It means that

lim
t→∞

∫ ε
0

∫ 1
0
P(t)f(m,v) dmdv = 1 (9)

for every density f and every ε > 0. The sweeping
property in this case means that the length of the

cell cycle tends to infinity in the sense of distribu-
tion, which is not so a rare phenomenon in tissue
cells. For example if q ≡ 1, then the semigroup has
no invariant density, and consequently is sweeping
from compact sets. It is interesting that in this
example we have

P(t)f(m,v) ∼
c

|v|
(log t)−1 as t →∞

for v ≥ ε and m ∈ [0, 1], where c is some constant.
The Lebowitz–Rubinow model is a special case

of the Rotenberg model [42]. In the Rotenberg
model the maturation velocity can also change
during the cell cycle. A new born cell inherits
the initial maturation velocity from its mother
according to a transition probability P(v,dv̄), as
in the Lebowitz–Rubinow model. During the cell
cycle it can change its maturation velocity with
intensity ϕ(m,v), i.e., a cell with parameters
(m,v) can change the maturation velocity in a
small time interval of length ∆t with probability
ϕ(m,v)∆t + o(∆t). We suppose that if (m,v) is
the state of the cell at the moment of changing
of the maturation velocity, then a new maturation
velocity is drawn from a distribution P(m,v,dv̄).
The process ξ(t) = (m(t),v(t)) describing con-
secutive descendants of a single cell is a PDMP
which has jumps when cells split and random
jumps during their cell cycles. Between jumps the
pair (m(t),v(t)) satisfies system (6). If a jump
is at the moment of division, then it is given by
the same formula as in the Lebowitz–Rubinow
model. If a jump is during the cell cycle, then
m(tn) = m(t

−
n ) and

P(v(tn)∈B |m(t−n ) =m, v(t
−
n ) =v) =P(m,v,B)

for each Borel subset B of (0,∞). Sample graphs
of maturity in the Rubinow, Lebowitz-Rubinow
and Rotenberg models are presented in Fig. 2.

IV. BELL-ANDERSON-TYPE MODELS

Now we consider one-phase models which are
based on two main assumptions: the maturity m
grows with velocity g(m) and a cell can splits at
a rate ϕ(m), i.e. a cell with maturity m divides
during a small time interval of length ∆t with

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

1
m

t
a)

1
m

t
b)

1
m

t
c)

Fig. 2. Sample graphs of maturity in the models: a) Rubinow,
b) Lebowitz-Rubinow, c) Rotenberg.

probability ∆P = ϕ(m)∆t + o(∆t). The matu-
rity of the daughter cell m is a function of the
maturity of the mother cell m, i.e. m = h(m). We
assume that g : [0,∞) → (0,∞) is a C1 function
which increases sublinearly, ϕ: [0,∞) → [0,∞)
is a continuous function, and h is a positive C1

function such that h′(m) > 0. For example if m
is the volume of a cell, then h(m) = m/2.

First we consider a discrete-time model which
describes the relation between the initial maturities
of the mother and daughter cells. Since

∆P = ϕ(m) ∆t+o(∆t), ∆m = g(m) ∆t+o(∆t),

we have

∆P =
ϕ(m)

g(m)
∆m + o(∆m),

while

G(m) = exp

{
−
∫ m
m0

ϕ(s)

g(s)
ds

}
is the survival function, where m0 is the initial cell

maturity. Let Q(m) =

m∫
0

ϕ(r)

g(r)
dr. Then

G(m) = eQ(m0)−Q(m).

We assume that limm→∞Q(m) = ∞, which
guaranties that each cell splits with probability
one. Let ξ be the maturity of the cell at the moment
of division and let η be a positive random variable
with density e−x. Since

P(ξ>m) =eQ(m0)−Q(m) = P(η>Q(m)−Q(m0)),

we have

P(Q(ξ) > Q(m)) = P(Q(m0) + η > Q(m))

and therefore the random variables Q−1(Q(m0) +
η) and ξ have the same distribution. It is easy to
check that the random variable ξ has the density

λ′(m)Q′(λ(m))eQ(m0)−Q(λ(m)) for m ≥ h(m0),

where λ(m) = h−1(m). If we assume that the
distribution of the initial maturity of mother cells
has a density f, from the above formula we infer
that the initial maturity of the daughter cells has
the density

Pf(m) =

∫ λ(m)
0

λ′(m)Q′(λ(m))eQ(y)−Q(λ(m))f(y)dy.

(10)
Then P is a stochastic operator on the space
L1[0,∞).

In the continuous version of the above model
we consider a sequence of consecutive descendants
of a single cell. The maturity of cells can be de-
scribed by the following homogeneous piecewise
deterministic Markov process ξ(t). Let tn be the
time when a cell from the nth-generation divides.
If tn−1 ≤ t < tn, then the maturity satisfies the
equation ξ′(t) = g(ξ(t)). The process ξ(t) has a
jump at the moment of the division of the cell:
ξ(tn) = h(ξ(t

−
n )). If ξ(tn−1) = m0, then the

cumulative distribution function F of tn − tn−1
is given by

F(t) = 1 −G(π(t,m0)),

where π(t,m0) is the solution at time t of the
equation m′ = g(m) with the initial condition
m(0) = m0.

Maturity structured models have been investi-
gated in many papers. Usually, we are interested
in the behaviour of the density of the maturity

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

u(t,m). We should underline that u(t, ·) is not
density in our probabilistic sense because the inte-
gral of u with respect to m may be different than
one. Since in these models we consider the whole
population, beside the rate of division ϕ there is
also the rate of death µ. These models coincide
with the model given by the process ξ(t), when
ϕ = µ and the rate of division in the definition of
ξ(t) equals 1

2
ϕ.

We briefly present some examples of these
models. The basic model was proposed by Bell
and Anderson [7]. They assume that the size
of the cell is a number m ∈ (mmin, 1), 0 <
mmin < 1. In order not to cross 1, it is assumed

that
∫ 1
mmin

ϕ(m) = ∞. This model was studied
and generalized, for example, in [11], [15], [59].
Versions of this model with unequal division were
investigated in [3], [16], [19], [52]. The papers [5],
[33] are devoted to a general model that includes
also age structure. There were also considered
versions with unbounded growth of cells [29],
[53].

V. TWO-PHASE MODEL

We start with a short biological description of
two phase-cell cycle models. The cell cycle is di-
vided into the resting and proliferating phase. The
duration of the resting phase is random variable
tA which depends on the maturity of a cell. The
duration tB of the proliferating phase is almost
constant. Therefore, we assume that tB = τ,
where τ is a positive constant. A cell can move
from the resting phase to the proliferating phase
with rate ϕ(m). We assume that cells age with
unitary velocity and mature with a velocity g1(m)
in the resting phase and with a velocity g2(m) in
the proliferating phase. The age variable a in the
proliferating phase is assumed to range from a = 0
at the moment of entering the proliferating phase
to a = τ at the point of cytokinesis. The maturity
of the daughter cell m is a function of the maturity
of the mother cell m, i.e. m = h(m) (see Fig. 3).

We consider a version of the model studied in
[38]. Now we collect the assumptions concerning
the model:

1
a

m

aτ

m

a

m m

h

(1) (2) (3) (4)

Fig. 3. Evolution of maturity of a mother cell: (1) – resting
phase; (2) – proliferating phase and a daughter cell: (3) –
resting phase; (4) – proliferating phase.

(M1) ϕ is a continuous function such that ϕ(m) =
0 for m ≤ mP and ϕ(m) > 0 for m > mP , where
mP > 0 is the minimum cell size of which it can
enter the proliferating phase,
(M2) h: [mP ,∞) → [0,∞) is a C1 function such
that h′(m) > 0,
(M3) g1 : [0,∞) → (0,∞) and g2 : [mP ,∞) →
(0,∞) are C1 functions which increase sublin-
early,

(M4) lim
m→∞

m∫
0

ϕ(r)

g1(r)
dr = ∞.

Denote by πi(t,m0) the solution of the equation

m′(t) = gi(m(t)), i = 1, 2, (11)

with the initial condition m(0) = m0 ≥ 0.
Now, we introduce two auxiliary functions. Let

λ(m) =π2(−τ,h−1(m)) and Q(m) =
m∫
0

ϕ(r)

g1(r)
dr.

According to (M4) lim
m→∞

Q(m) = ∞, which
guaranties that each cell enters the proliferating
phase with probability one. Under this notation the
initial maturity of the daughter cells has density
Pf, if f is the analogous density of the mother
cells, where P is the stochastic operator given by
(10). Now we present some results concerning the
operator P .

Theorem 5.1: The operator P satisfies the
Foguel alternative, i.e. P is asymptotically stable
or sweeping from compact sets.

Theorem 5.2: Let α(m) = Q(λ(m)) − Q(m).
The following conditions hold:

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(a) if lim inf
m→∞

α(m) > 1, then P is asymptotically
stable,
(b) if α(m) ≤ 1 for sufficiently large m, then P
is sweeping from each bounded interval,
(c) if inf α(m) > −∞, then the operator P is
completely mixing, i.e.

lim
n→∞

‖Pnf −Png‖ = 0 for f,g ∈ D.

Theorem 5.1 was proved in [38]. The results from
Theorem 5.2 were proved, respectively, (a) in [14],
(b) in [25], and (c) in [45]. The sweeping property
in this case means that the maturity of cells statis-
tically tends to infinity, which also means that the
length of the cell cycle tends to infinity.

Now we consider a continuous version of the
model. The cell cycle can be described as a piece-
wise deterministic Markov process. We consider
a sequence of consecutive descendants of a single
cell. Let sn be the time, when a cell from the nth-
generation enters a resting phase and tn = sn + τ
be the time of its division. If tn−1 ≤ t < tn then
the state ξ(t) = (a(t),m(t), i(t)) of the n-th cell
is described by the age a(t), maturity m(t) and
the index i(t), where i = 1 if the cell is in the
resting phase and i = 2 if it is in the proliferating
phase. Random moments t0,s1, t1,s2, t2, . . . are
called jump times. Between jump times the param-
eters change according to the following system of
equations: 


a′(t) = 1,

m′(t) = gi(t)(m(t)),

i′(t) = 0.

(12)

The process ξ(t) changes at the jump points ac-
cording to the following rules:

a(sn) = 0, m(sn) = m(s
−
n ), i(sn) = 2,

a(tn) = 0, m(tn) = h(m(t
−
n )), i(tn) = 1.

If m(tn−1) = m0, then the cumulative distribution
function F of sn − tn−1 is given by

F(t) = 1 −eQ(m0)−Q(π1(t,m0)). (13)

Then ξ(t) is a homogeneous Markov process. If
the distribution of ξ(0) is given by a density

2
a

m

m = π1(a,0)

aτ

m

mP m = π2(a,mP )

1-phase 2-phase

Fig. 4. The set X

function f(0,a,m,i), i.e. a measurable function
of (a,m,i) such that

P(ξ(0) ∈ A× i) =
∫∫
A

f(0,a,m,i) dadm

for any Borel set A and i = 1, 2, then ξ(t) has a
density f(t,a,m,i).

Having a homogeneous Markov process ξ(t)
with the property that if the random variable ξ(0)
has a density f0, then ξ(t) has a density ft,
we can define a stochastic semigroup {P(t)}t≥0
corresponding to ξ(t) by P(t)f0 = ft. The proper
choice of the space X of the values of the process
ξ(t) plays an important role in investigations of the
process and the semigroup {P(t)}t≥0. We define
X = X1 ∪X2, where

X1 = {(a,m, 1) : m ≥ π1(a, 0), a ≥ 0},

X2 = {(a,m, 2) : m ≥ π2(a,mp), a ∈ [0,τ]},

Σ = B(X) and µ is the product of the two-
dimensional Lebesgue measure and the counting
measure on the set {1, 2} (see Fig. 4).

We need two additional assumptions:

ψ(m) = h(π2(τ,m)) < m for m ≥ mP (14)

and

h′(π2(τ,m̄))g2(π2(τ,m̄))g1(m̄)

6= g1(h(π2(τ,m̄)))g2(m̄) (15)

for some m̄ > mP .
Condition (14) guarantees that with a positive

probability each cell will have a descendant with

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a sufficiently small maturity in some generation
and thanks to that property each two states from
the interior of the set X communicate. Condition
(15) seems to be technical but if

h′(π2(τ,m))g2(π2(τ,m))g1(m)

= g1(h(π2(τ,m)))g2(m)

for all m ≥ mP , then all descendants of a single
cell in the same generation have the same maturity
at a given time t. It means that the cells have
synchronous growth and we cannot expect the
model to be asymptotically stable. In particular,
if g1 ≡ g2 and h(m) = m/2, then (15) reduces
to 2g2(m) 6= g2(2m) for some m > π2(τ,mP ).
A similar condition appears in many papers con-
cerning size-structured models [5], [11], [15], [52],
[54].

The following results are proved in [38].
Theorem 5.3: The semigroup {P(t)}t≥0 sat-

isfies the Foguel alternative, i.e. {P(t)}t≥0 is
asymptotically stable or sweeping from compact
sets.
The proof of this result is based on Theorem 2.1
and Corollary 1.

Theorem 5.4: If the operator P given by (10)
has an invariant density and ϕ(m) ≥ ε > 0 for
sufficiently large m, then the semigroup {P(t)}t≥0
is asymptotically stable. If P has no invariant
density and ϕ is a bounded function, then the
semigroup {P(t)}t≥0 is sweeping from compact
sets.

According to Theorem 5.2 and Theorem 5.4 we
have the following alternative.

Corollary 2: If lim inf
m→∞

(Q(λ(m)) −Q(m)) > 1
and there is ε > 0 such that ϕ(m) ≥ ε for
sufficiently large m, then the semigroup {P(t)}t≥0
is asymptotically stable. If Q(λ(m)) −Q(m) ≤ 1
for sufficiently large m and ϕ is bounded, then the
semigroup {P(t)}t≥0 is sweeping.

Remark 1: One can give an example of the
operator P which is asymptotically stable but the
semigroup {P(t)}t≥0 is sweeping. Such a case can
happen when limm→∞ϕ(m) = 0. The explanation
of this phenomenon is that in this example the rate
of entering the proliferating phase is very small

for large m. Then the mean length of the resting
phase can be large and more and more mature cells
dominate the population as t → ∞. In [48], [60]
one can find the comparison of the discrete time
model presented here with a two-phase model of
maturity structured population considered in the
paper [27] and briefly presented in the next section.

VI. TWO-PHASE POPULATION MODELS

Now we recall a two-phase maturity structured
model of a cellular population from the paper [27].
The model is based on the same biological as-
sumption as that of Section V, but we include
also the mortality rates µr(m) and µp(m) in both
phases. Denote by r(t,m,a) and p(t,m,a) the
maturity-age distribution of resting and prolifer-
ating cells, respectively. We also assume that the
rate of entering the proliferating phase ϕ depends
on m and the total number of cells in the resting
phase R̄(t) =

∫
R(t,m) dm, where R(t,m) =∫

r(t,m,a) da. The time evolution of p and r is
described by the following system of equations:

∂r

∂t
+
∂r

∂a
+
∂(g1(m)r)

∂m
= −(µr(m) + ϕ(R̄,m))r,

∂p

∂t
+
∂p

∂a
+
∂(g2(m)p)

∂m
= −µp(m)p

with the boundary conditions

r(t, 0,m) = 2(h−1(m))′p(t,τ,h−1(m)),

p(t, 0,m) = ϕ(m,R̄(t))R(t,m).

Additionally, we assume that µp, µr and ϕ do
not depend on m. Integrating the above equations
over the age variable a and using the boundary
conditions we obtain
∂R

∂t
+
∂(g1R)

∂m
= −(µr + ϕ(R̄))R +

2e−µpτϕ(R̄(t− τ))λ′(m)R(t− τ,λ(m)).
(16)

We recall that λ(m) = π2(−τ,h−1(m)) is the ma-
turity of the mother cell at the moment of entering
proliferating phase, if the new born daughter cell
has maturity m. Integrating both sides of (16) over
m we obtain

R̄′(t) =−(µr+ϕ(R̄))R̄+2e−µpτϕ(R̄(t−τ))R̄(t−τ).
(17)

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The following result is proved in [27].
Theorem 6.1: Assume that equation (17) has

a constant solution R̄0 > 0 and R̄0 is globally
asymptotically stable. If

(µr + ϕ(R̄0)) log(h
−1)′(0) < g′1(0) (18)

then there exists a stationary solution R0(m) of
equation (16) and for every solution R(t,m) of it
we have

lim
t→∞

∫
|R(t,m) −R0(m)|dm = 0. (19)

Condition (18) has an interesting biological inter-
pretation. It shows that the stability of the popula-
tion depends on the dynamics of immature (small)
cells. The term (h−1)′(0) describes the relation
between the maturity of the mother and daughter
cells. If m is the maturity of a small mother cell
at the moment of entering the proliferating phase,
then the maturity of a new-born daughter cell is
m/(h−1)′(0). The term c = µr +ϕ(R̄0) is the rate
of leaving of the resting phase (by being lost or
by entering the proliferating phase). Since g′1(0)
is the rate at which small cells mature, condition
(18) means that the maturity of a large part of
small cells will increase in the next generation.

Now we present a model from the paper [28].
In this model we assume that the rate of entering
proliferating phase ϕ for cell with maturity m
depends on the total number of cells with this
maturity R(t,m). We consider a simplified version
of this model with g1(m) = g1m, g2(m) = g2m,
h(m) = hm, where g1,g2,h are positive con-
stants. We have also λ(m) = λm, λ > 0.

Then equation (16) becomes a special case of
the following nonlinear equation:

∂u

∂t
+g(x)

∂u

∂x
= f(t,u(t,x),u(t−τ,λ(x))), (20)

where u = R, x = m. Equations of the form (20)
were used in description of cellular models in the
papers [9], [12], [40].

We assume that the fuctions g : [0, 1] → R,
λ: [0, 1] → [0, 1] and f : [0,∞) × R × R → R
have continuous derivatives and
(a) g(0) = 0, g(x) > 0 for x ∈ ( 0, 1 ],

(b) λ(0) = 0, λ(x) < x for x ∈ ( 0, 1 ],
(c) there exist continuous functions α1 and α2

such that

|f(t,u,v)| ≤ α1(t,v)|u| + α2(t,v).

We consider the solution of (20) with the initial
condition

u(t,x) =ψ(t,x) for (t,x)∈[−τ, 0]×[0, 1]. (21)

Now we consider the following delay differen-
tial equation associated with (20):

z′(t) = f(t,z(t),z(t− τ)). (22)

The following theorem plays a central role in
investigations of equation (20).

Theorem 6.2: Let u(t,x) be a solution of (20).
Let z(t) be the solution of (22) with the initial
condition z(t) = u(t, 0) for t ∈ [−τ, 0]. Then for
every t0 ≥ 0 and ε > 0 there exist t1 > 0 and
another solution ū(t,x) of (20) such that
(i) sup{|ū(t,x)−z(t)|: (t,x)∈[−τ,t0]×[0, 1]}<ε,
(ii) ū(t,x) = u(t,x) for (t,x) ∈ [t1,∞)×[0, 1].

The proof of this result is given in [28]. From
Theorem 6.2 the entire strategy of studying of
equation (20) becomes clear. Namely, if z0(t) is
a globally asymptotically stable solution of (22)
and u0(x,t) = z0(t) is a locally asymptotically
stable solution of (20), then u0(x,t) is globally
asymptotically stable solution of (20). Thus, rather
surprisingly, the question of determining the global
stability of a solution of (20) can be reduced to the
problem of examining the global stability of the
corresponding differential delay equation (22) and
the local stability of (20). Therefore, in the general
case it is sufficient to focus on the global stability
of the associated differential delay equation (22),
which is itself usually quite difficult, and the local
stability of (20), which is often easier.

We now turn to considerations of the local
stability of the full partial differential equation
(20). We assume that the function f does not
depend on t. Then equation (20) takes the form

∂u

∂t
+ g(x)

∂u

∂x
= f(u,uτ ), (23)

where uτ = u(t− τ,λ(x)).

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Let ū(x,t) be a given solution of (23) and let A
be a subset of C([0, 1]× [−τ, 0]). We say that the
solution ū of (23) is exponentially stable on the set
A if there exists µ > 0 such that for every ψ ∈ A
the solution of the problem (23), (21) satisfies the
inequality

max{|u(t,x)−ū(t,x)|: x ∈ [0, 1]}≤Ce−µt, (24)
where C is a constant which depends only on ψ.
Let

Aε={ψ : |ψ(t,x)−ū(t,x)|<ε, (t,x)∈[−τ, 0]×[0, 1]}.
We say that ū is locally exponentially stable if
there exist an ε > 0, µ and C such that condition
(24) holds for every solution of the problem (23),
(21) with ψ ∈ Aε.

Theorem 6.3: Let w be a constant such that
f(w,w) = 0 and

∂f

∂u
(w,w) < −

∣∣∣∣ ∂f∂uτ (w,w)
∣∣∣∣ . (25)

Then the solution ū(x,t) ≡ w of (23) is locally
exponentially stable.

Let us summarize the results. Consider the as-
sociated delay differential equation corresponding
to (23):

z′(t) = f(z(t),z(t− τ)). (26)
Let ϕ ∈ C[−τ, 0] and denote by zϕ the solution of
(26) satisfying the initial condition zϕ(t) = ϕ(t)
for t ∈ [−τ, 0]. Let w ∈ R be a constant such that
f(w,w) = 0. Then w is a stationary solution of
(26). The set

B(w) = {ϕ ∈ C[−τ, 0] : lim
t→∞

zϕ(t) = w}

is called the basin of attraction of w. Denote by
P the projection operator

P : C([0, 1] × [−τ, 0]) → C[−τ, 0]
given by (Pψ)(t) = ψ(0, t) for t ∈ [−τ, 0].

Corollary 3: Let w ∈ R satisfies (25) and
f(w,w) = 0. Then equation (23) is globally
exponentially stable on the set

A = {ψ ∈ C([0, 1] × [−τ, 0]) : Pψ ∈ B(w)}.
In paper [28] one can find applications of this

theory to maturation structured models and to
blood production systems.

VII. CHAOS

Thus far, we have restricted our mathematical
results to study such asymptotic properties of the
models as the asymptotic stability and sweep-
ing. But some of our models can have more
complicated behavior which can be studied using
theoretical methods of dynamical systems. Now
we present some results concerning chaotic and
ergodic properties.

It is not widely known that solutions of simple
linear partial differential equations may behave in
a chaotic way. The following equation with the
initial condition:

∂u

∂t
+ x

∂u

∂x
= cu, u(0,x) = v(x), c > 0, (27)

defines a semiflow on the space

X = {v ∈ C[0, 1] : v(0) = 0}

given by

Stv(x) = u(t,x) = ectv(e−tx),

which is chaotic, practically in each sense of the
meaning of this word. For example, for λ > 0
there exists a Gaussian measure with the support
X invariant under {St}t≥0 and the system is
mixing [44] (see also review papers [46], [51]).
This implies the topological chaos: the existence
of dense trajectories (topological transitivity) and
instability of trajectories. The invariant measure
µ for the semiflow {St}t≥0 can be given by the
formula µ(A) = P(ξx ∈ A), where ξx = wx2c
and wx is the standard Wiener process. Since
equation (27) can describe the evolution of the
distribution of maturity in a cellular population,
one can prefer to consider the semiflow {St}t≥0
restricted to the set X+ = {v ∈ X : v ≥ 0}.
In this case the invariant measure on X+ can be
induced by the process ξx = |wx2c| and we still
have very strong ergodic and chaotic properties of
this semiflow. Similar results can be obtained for
semiflows generated by equations of the form

∂u

∂t
+ c(x)

∂u

∂x
= f(x,u). (28)

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In structured population models we mainly in-
vestigate the long time behavior of densities. Al-
though in some models we expect chaotic behavior
on the set of densities, it is rather difficult to prove
such a result. One of the reasons is that there
are no good mathematical tools to investigate such
problems. Analytic methods of studying chaos in
infinite dimensional spaces, based on the paper
[10], do not work in this case because these meth-
ods are strictly connected with semiflows acting on
the whole linear space. Now we present one model
from the paper [49], which is well biologically
motivated, where, by using methods of ergodic
theory, we are able to prove chaotic behaviour of
a semiflow acting on the set of densities.

We consider a population of stem cells. These
cells live in the bone marrow and they are pre-
cursors of any blood cells. They are subjects of
two biological processes: maturation and division.
Stem cells can be at different levels of morphologi-
cal development called maturity. The maturity of a
cell differs from its age, because we assume that a
newly born cell is in the same morphological state
as its mother at the point of division. We assume
that maturity is a real number x ∈ [0, 1]. The
function u(t,x) describes the density distribution
function of cells with respect to their maturity.
The maturity grows according to the equation
x′ = g(x). When one cell reaches the maturity
1 it leaves the bone marrow, then one of cells
from the bone marrow splits. This cell is chosen
randomly according to the distribution given by the
density u(t,x). It follows from the assumptions
that a newly born cell has the same maturity as
its mother cell and each cell can divide with the
same probability (see Fig. 5).

Although our model is rather simple in compar-
ison with other models for erythroid production
(e.g. [26], [23]), it is based on the same continuous
maturation-proliferation scheme. We neglect here
the fact that with the growth of maturity cells pass
through consecutive morphological compartments
from pluripotential stem cell to erythrocytes. In
our model we do not have any exterior regulatory
system in which the production of erythrocytes is

0 1x
Fig. 5. Scheme of maturation and division of cells in the
bone marrow.

stimulated by the hormone erythropoietin and the
system tries to keep the number of erythrocytes on
a constant level. One can say that chaos appears
if the exterior regulatory system does not work (a
pathological case).

The model is described by a nonlinear semiflow
induced by the equation

∂u

∂t
+

∂

∂x
(g(x)u) = g(1)u(t, 1)u(t,x) (29)

with the initial condition

u(0,x) = u0(x), x ∈ [0, 1]. (30)

The semiflow is defined on the space of densities.
In the paper [49] it was shown that the semiflow
generated by the initial problem (29)–(30) posses
an invariant measure which is mixing and sup-
ported on the whole set of all densities. From this
result there follows instability of all trajectories
and topological transitivity. The main idea of the
proof is to show that the semiflow generated by
(29)–(30) is isomorphic to a semiflow generated
by (28). Then we construct an invariant measure
for the second semiflow and transfer it to the initial
semiflow. We skip the precise proof here.

Fig. 6 presents the spatial temporal plot of the
solution of (29) with the initial condition

u0(x) = x sin
2
(1
x

)
.

If we change a little the initial condition replacing
u0 by

ū0(x) = x sin
2
( 1
x + 0.05

)
we obtain the solution with the plot shown in
Fig. 7.

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

0

1

2

3

4

u
(t

,x
)

0.0

0.2

0.4

0.6

0.8

1.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t

Fig. 6. The plot of the solution of (29) with the initial condition u0(x) = x sin2
(
1
x

)
in the time interval [0, 3].

0

1

2

3

4

u
(t

,x
)

0.0

0.2

0.4

0.6

0.8

1.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t

Fig. 7. The plot of the solution of (29) with the initial condition ū0(x) = x sin2
(

1
x+0.05

)
.

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Katarzyna Pichór, Ryszard Rudnicki, One and two-phase cell cycle models

The second example is the Bell and Anderson
model of size structured cellular population given
by the equation

∂u

∂t
+
∂

∂x
(g(x)u) =−(µ+b)u(t,x)+4bu(t,2x), (31)

where x ∈ [0, 1] and we put u(t, 2x) = 0 if 2x >
1. It is well known that if g(2x) 6= 2g(x) at least
for one x ∈ [0, 1], then the solutions of (31) have
asynchronous exponential growth, i.e., there exist
λ ∈ R, positive functions f∗, and a constant c
which depends on the initial condition u(0,x) such
that

e−λtu(t, ·) → cf∗ in L1[0, 1].

Having in mind this result, it is difficult to imagine
that some versions of this model can be chaotic. It
is interesting that if g(x) = ax, then the semiflow
generated by the equation (31) is chaotic. The
chaotic behaviour of the semiflow generated by
this equation was studied using analytic methods
by Howard [17] and El Mourchid et al. [13]. In
[50] it was shown that for this semiflow there
exists a mixing invariant measure supported on the
whole space. From this property we can deduce
chaotic properties of the semiflow.

ACKNOWLEDGMENT

This research was partially supported by the
National Science Centre (Poland) Grant No.
2017/27/B/ST1/00100.

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http://dx.doi.org/10.11145/j.biomath.2019.05.261

	Introduction
	Asymptotic properties of stochastic operators and semigroups
	Rubinow-type models
	Bell-Anderson-type models
	Two-phase model
	Two-phase population models
	Chaos
	References