On the age-dependent population dynamics with delayed dependence of the structure

Nonlinear Analysis 71 (2009) e2657–e2664

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On the age-dependent population dynamics with delayed dependence of the structure Antoni Leon Dawidowicz a , Anna Poskrobko b,∗ a

Institute of Mathematics, Jagiellonian University, Ul. Łojasiewicza 6, 30-348 Kraków, Poland

b

Faculty of Computer Science, Bialystok Technical University, Ul. Wiejska 45A, 15-351 Białystok, Poland

article

abstract

info

Keywords: Delayed partial differential equations von Foerster model

The paper deals with the description of the new model of the population with delayed dependence of the structure. We present the proof of the existence and the uniqueness of the solution of this problem. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction McKendrick [1] was the first to introduce the age-structure for considering the dynamics of a population. He described a type of single-species population in which each individual is capable of reproduction. Age and time are independent variables. In the continuous model the size of a population is not considered, only its density, that is a number of individuals for a unit of a surface. Let u(a, t ) denote the density of a decomposition of the individuals of a population of age a at time t. This description is more exact as the unit of time is shorter. In our text we use interchangeably the expressions ‘‘density’’ and ‘‘number’’. We denote the total population at time t by z (t ) =

∞

Z

u(a, t )da. 0

In practice it is sensible to assume that for the variable a large enough the density u(a, t ) equals zero, that is z (t ) < ∞. The group of individuals of age a is of age a + h after time h. The rate Du(a, t ) = lim

u(a + h, t + h) − u(a, t )

h→0

h

(1.1)

denotes the intensity of changing of the population of this group in time. The sum of this rate and the number of individuals who died at time t at age a equal zero Du(a, t ) = −λ(a)u(a, t ). The above equation is called McKendrick equation or, more often, von Foerster equation in the literature. So McKendrick’s model considers mortality and the rate λ(a) is called the death-modulus. The birth process is described by the renewal equation u(0, t ) =

∞

Z

β(a)u(a, t )da. 0

∗

Corresponding author. E-mail addresses: [email protected] (A.L. Dawidowicz), [email protected] (A. Poskrobko).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.019

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A.L. Dawidowicz, A. Poskrobko / Nonlinear Analysis 71 (2009) e2657–e2664

The quantity β(a) is called the birth-modulus. It is the average number of offsprings produced (per unit time) by an individual of age a. To sum up, we get the classical von Foerster model

 Du(a, t ) = −λ(a)u(a, t )   u(a, 0) = Z ϕ(a) ∞  u(0, t ) = β(a)u(a, t )da. 0

The model proposed by Gurtin and MacCamy [2] was based on the assumption that the progress of the population depends on its number. Therefore the mortality as well as the reproductive abilities depend on this number.

Du(a, t ) = −λ(a, z (t ))u(a, t )    a) u(a, 0) = Zϕ(∞   β(a, z (t ))u(a, t )da u(0, t ) =  0  Z  ∞   z ( t ) = u(a, t )da. 0

However, it is common knowledge that other factors can have an influence on the progress of the population. Natural generalization of the population dynamics is taken into consideration, for example: two genders, limitation of sources of the natural environment, period of gestation or period of a response of a system to a stimulus (see Busoni and Palczewski [3]). The last two examples suggest the necessity to consider the descriptions with delayed parameter. This approach has deep biological justification. All natural processes occur with some delay with respect to the moment of their initiation. In the case of living organisms for a process of reproductiveness it is fertilization. In the case of cells it is the chemical signal being sent to the system. There is a problem with the process of mortality. It is difficult to define this subtle moment of its initiation. From the moment of sending the signal to the moment of the process’ initiation, for example birth of a new individual or proliferation of new cells, any period of time goes by. We call it delay. We should draw our attention to the fact that one of the most important equations with the delayed parameter was proposed by Polish scientists Ważewska-Czyżewska and Lasota [4]. They used it in the description of red blood cells’ population growth. The delay appears also in the papers concerning mathematical modeling in epidemiology and immunology, particularly see Marchuk [5], Foryś [6], Leszczyński and Zwierkowski [7]. The analysis of the equations with the delayed parameter can be found also in Łoskot [8], Haribash [9], Arino [12], Forystek [10] and Dawidowicz and Poskrobko [13]. The model considered in this paper assumes that the dependence of the mortality as well as the procreation on the structure of the population is functional. It means that both these rates depend on the structure not only in the moment t, but also in any preceding period of time. In particular the right-hand side of the equation of our model is not in the form λu, but Λu. Here Λ is the operator which does not apply, as in the classical model, on the value of the function u, but to its restriction to any suitable space. Similarly it is in the renewal equation, where we integrate the function u not only according the age, but also along any time interval. From necessity the initial condition should be determined on any time interval. In our paper we present the proof of the existence and the uniqueness of the solution of the new problem which is based on the Banach fixed point theorem. But to avoid the necessity of the extension of the solution we use Bielecki’s idea. 2. Construction of the model Let us consider the differential equation Du(a, t ) = −Λ(a, zt )u(a, t + ·) u(0, t ) =

∞

Z 0

z (t ) =

Z

Z

(2.1)

0

β(a, z (t + s))u(a, t + s)dsda

(2.2)

−r

∞

u(a, t )da for t ∈ [−r , T ]

(2.3)

0

with the initial condition u(a, s) = ϕ(a, s) for s ∈ [−r , 0]

(2.4)

where zt : [−r , 0] → R+ ,

r >0

is given by the formula zt = z (t + s). Definition 2.1. By the solution of problem (2.1)–(2.3) with the initial condition (2.4) up to time T > 0 we consider a nonnegative function u defined on R+ × [−r , T ], fulfilling the following properties: Du exists on R+ × [0, T ], u(., t ) ∈ L1 (R+ ),

A.L. Dawidowicz, A. Poskrobko / Nonlinear Analysis 71 (2009) e2657–e2664

R∞

the function z (t ) =

0

u(a, t )da is continuous for t ∈ [−r , T ] and

Du(a, t ) = −Λ(a, zt )u(a, t + ·) u(0, t ) =

∞

Z

e2659

(a > 0, − r ≤ t < T )

0

Z

β(a, z (t + s))u(a, t + s)dsda (0 < t 6 T ) −r

0

u(a, s) = ϕ(a, s)

(a > 0, s ∈ [−r , 0]).

3. Equivalent formulation of the problem In this section we shall mainly formulate problem (2.1)–(2.4) in terms of operator equations. Thanks to this we shall prove local and global existence of the solution of the presented population problem. Let us make the following assumptions:

(H1 ) ϕ R ∞∈ C (R+ × [−r , 0]); (H2 ) 0 sups∈[−r ,0] ϕ(a, s)da < ∞; (H3 ) Λ : R+ × C ([−r , 0]) → C (R+ )∗ , β : R2+ → R; (H4 ) The norm of Λ and the function β are bounded, i.e. kΛ(a, ψ)k ≤ Λ0 and |β(a, ψ)| ≤ β0 , where Λ0 and β0 are finite quantities;

(H5 ) ϕ > 0, Λ > 0, β > 0. (H6 ) There exist such Λ1 and β1 , that for every a, ψ1 , ψ2 kΛ(a, ψ1 ) − Λ(a, ψ2 )k 6 Λ1 sup |ψ1 (s) − ψ2 (s)| s∈[−r ,0]

and for every a, z

∂β(a, z ) ∂ z 6 β1 . Theorem 3.1. If u is the solution of problem (2.1)–(2.4) up to time T > 0 then the population zt and the birth-rate B(t ) = u(0, t ) satisfy on [0, T ] the operator equations z (t ) =

t

Z

B(a)da − 0

0

ϕ(a, 0)da −

+

 Λ(η, za+η )u(η, a + η + ·)dη da

0

∞

Z

t −a

Z t Z

0

Z 0

∞

t

Z



Λ(a + η, zη )u(a + η, η + ·)dη da

(3.1)

0

and B(t ) =

0

Z

t +s

Z

−r

Z

β(t + s − a, z (t + s))B(a)dads 0

0

t +s

Z

− −r

Z

0

0

0

β(a + t + s, z (t + s))ϕ(a, 0)dads

+ Z

0

0

∞

Z

− −r

 Λ(η, za+η )u(η, a + η + ·)dη dads

∞

Z

−r

t +s−a

 Z β(t + s − a, z (t + s))



β(a + t + s, z (t + s))

0

Z

t +s



Λ(a + η, zη )u(a + η, η + ·)dη dads.

(3.2)

0

If z and B are non-negative continuous functions satisfying (3.1) and (3.2) on [0, T ], and if u is defined on R+ × [0, T ] by the formula

 Z t   ϕ(a − t , 0) − Λ(a − t + η, zη )u(a − t + η, η + ·)dη for a > t Z a0 u( a, t ) =   Λ(η, zt −a+η )u(η, t − a + η + ·)dη for t > a B(t − a) −

(3.3)

0

then u is a solution of problem (2.1)–(2.4) up to time T . Proof. Let u be a solution of problem (2.1)–(2.4) up to time T and (a0 , t0 ) ∈ R+ × [0, T ]. Let us introduce the notations u(h) = u(a0 + h, t0 + h)

Λ = Λ(a0 + h, zt0 +h ) uh = u(a0 + h, t0 + h + ·).

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Then from (2.1) we have the equation du dh

= −Λ(h)uh

and its unique solution h

Z

u(h) = u(0) −

Λ(η)uη dη.

0

From the above notation we have u(a0 + h, t0 + h) = u(a0 , t0 ) −

Z

h

Λ(a0 + η, zt0 +η )u(a0 + η, t0 + η + ·)dη.

0

In particular, if we take (a0 , t0 ) = (a − t , 0) and h = t we get t

Z

u(a, t ) = ϕ(a − t , 0) −

Λ(a − t + η, zη )u(a − t + η, η + ·)dη for a > t .

(3.4)

0

And for (a0 , t0 ) = (0, t − a) and h = a we obtain u(a, t ) = B(t − a) −

a

Z

Λ(η, zt −a+η )u(η, t − a + η + ·)dη for t > a

(3.5)

0

where B(t ) = u(0, t ). Using the above relations we can express the total population z and the birth-rate B in the form of the operator equations z (t ) =

∞

Z

u(a, t )da 0

Z t

a



Λ(η, zt −a+η )u(η, t − a + η + ·)dη da 0   Z t ∞ + ϕ(a − t , 0) − Λ(a − t + η, zη )u(a − t + η, η + ·)dη da t 0  Z t Z t −a = B(a) − Λ(η, za+η )u(η, a + η + ·)dη da 0 0  Z ∞ Z t + ϕ(a, 0) − Λ(a + η, zη )u(a + η, η + ·)dη da 0 0  Z t Z t Z t −a = B(a)da − Λ(η, za+η )u(η, a + η + ·)dη da 0 0 0  Z ∞ Z ∞ Z t + ϕ(a, 0)da − Λ(a + η, zη )u(a + η, η + ·)dη da

=

B(t − a) −

Z

0

Z

0

0

0

and B(t ) = u(0, t ) =

∞

Z 0

Z

0

0

β(a, z (t + s))u(a, t + s)dsda −r

t +s

Z

β(t + s − a, z (t + s))B(a)dads

= −r

Z

0

Z

0

t +s

Z

− −r

Z

0

0

0

β(a + t + s, z (t + s))ϕ(a, 0)dads

+ Z

0

0

∞

Z

− −r

 Λ(η, za+η )u(η, a + η + ·)dη dads

∞

Z

−r

t +s−a

 Z β(t + s − a, z (t + s))

0

t +s

 Z β(a + t + s, z (t + s))

 Λ(a + η, zη )u(a + η, η + ·)dη dads.

0

To prove the second part of Theorem let us assume that z and B are continuous non-negative functions, defined respectively on the intervals [−r , T ] and [0, T ], fulfilling conditions (3.1) and (3.2). Let the function u be define on R+ × [0, T ] by the

A.L. Dawidowicz, A. Poskrobko / Nonlinear Analysis 71 (2009) e2657–e2664

e2661

formula (3.3). Since ϕ > 0 and β > 0 the function u is also non-negative. It is easily seen that (2.4) holds and u(., t ) = B(t ) for t > 0. u(., t ) ∈ L1 (R+ ) because the functions λ, β , ϕ and z are continuous. It follows from (3.1)–(3.3) that (2.2) and (2.3) are satisfied. To complete the proof let us notice that (3.3) implies Du existing on R+ × [0, T ] and that the equality (2.1) holds. We get the desired conclusion that u is a solution of problem (2.1)–(2.4) up to time T .  4. A priori estimates In this section let us get back to the operator Eqs. (3.1) and (3.2) and determine the estimations for each of their summands. These a priori estimations help us in the proof of the local and global existence of the solution of the population problem quoted in the next section. Recall the above mentioned equations and introduce: z (t ) =

B(a)da − ∞

ϕ(a, 0)da −

+

 Λ(η − a, zη )u(η − a, η + ·)dη da

a

0

0

Z

t

Z t Z

t

Z

Z

∞



Λ(a + η, zη )u(a + η, η + ·)dη da 0

0

0

t

Z

= I11 − I12 + I13 − I14 and B(t ) =

0

Z

t +s

Z

−r

β(t + s − a, z (t + s))B(a)dads 0

0

Z

t +s

Z

− −r

0

0

Z

0

β(a + t + s, z (t + s))ϕ(a, 0)dads

+ 0

0

Z

 Λ(η, za+η )u(η, a + η + ·)dη dads

∞

Z

−r

t +s−a

 Z β(t + s − a, z (t + s))

∞

Z



− −r

β(a + t + s, z (t + s))

0

Z

t +s

 Λ(a + η, zη )u(a + η, η + ·)dη dads

0

= I21 − I22 + I23 − I24 . For s ∈ [−r , 0] we can write u(η, a + η + s) ≤ u(0, a + s) ≤ B(a + s) u(a + η, η + s) ≤ u(a, s) ≤ ϕ(a, s)

|Λ(a, zt )u(a, τ + ·)| ≤ kΛ(a, zt )k sup |u(a, τ + s)|. s∈[−r ,0]

We assume that kΛ(a, ψ)k ≤ Λ0 and |β(a, ψ)| ≤ β0 . We thus get t −a

Z t Z |I12 | ≤ 0

Z

 Z Λ0 sup |u(η, a + η + s)|dη da ≤ Λ0 T s∈[−r ,0]

0

∞



|I14 | ≤

sup |B(a + s)|da

0 s∈[−r ,0]



t Λ0 sup ϕ(a, s) da s∈[−r ,0]

0

|I21 (t )| ≤ β0 r

t

Z

sup B(a + s)da

0 s∈[−r ,0]

|I22 (t )| ≤ β0 r Λ0 |I23 (t )| ≤ β0

t

0

Z

−r

|I24 (t )| ≤ r β0 Λ0

t

Z

sup B(a + s)da

0 s∈[−r ,0]

∞

Z

ϕ(a, 0)dads 0

∞

Z

sup ϕ(a, s)da.

0

s∈[−r ,0]

Therefore by the applied conditions (Φ = B(t ) ≤ M1 (T ) + M2 (T ) Denoting

B (t ) = sup B(τ ) τ ∈[−r ,t ]

Z

R∞ 0

sups∈[−r ,0] ϕ(a, s)da < ∞) we get the estimation

t

sup B(a + s)da.

0 s∈[−r ,0]

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A.L. Dawidowicz, A. Poskrobko / Nonlinear Analysis 71 (2009) e2657–e2664

we obtain t

Z

B (t ) ≤ M1 (T ) + M2 (T )

B (a)da,

0

it allows us to finally estimate B (t ) from Gronwall’s inequality

B (t ) ≤ M1 (T )etM2 (T ) .

(4.1)

5. Local and global existence According to Theorem 3.1 we see that to solve the population problem (2.1)–(2.4) it is enough to find functions z ∈ C + [−r , T ] and B ∈ C + [0, T ] satisfying the above operator equations (3.1) and (3.2). Here C + [a, b] denotes the space of non-negative continuous functions on [a, b]. Let us introduce a new function

Θ (ξ , a) =



β(ξ − a, z (ξ )) 0

for a 6 ξ . for a > ξ

Thanks to this we can rewrite the first summand of Eq. (3.2) in the following form

Z

0

−r

t +s

Z

β(t + s − a, z (t + s))B(a)dads =

0

Z

−r

0

t

Z

Θ (t + s, a)B(a)dads 0

Z t Z

0

= 0

 Θ (t + s, a)ds B(a)da.

−r

We see that (3.2) is a Volterra equation of B with a unique solution BT (z )(t ). Let us define on C + [−r , T ] the new operator ZT using the right-hand side of Eq. (3.1) with B replaced by BT (z )(t ).

ZT (z )(t ) =

Z

t

BT (z )(a)da −

0

Z t Z 0

∞

Z

t

Z

− 0

t

∞

 Z Λ(η − a, zη )u(η − a, η + ·)dη da +

a

ϕ(a, 0)da 0

 Λ(a + η, zη )u(a + η, η + ·)dη da,

(5.1)

0

where u(a, t ) is defined by (3.4) and (3.5) for t > 0 and u(a, t ) = ϕ(a, t ) for t 6 0. So finally, to solve the population problem (2.1)–(2.4) up to time T it is sufficient to show that the operator ZT has a unique fixed point. Thanks to the analogical reasoning as in the previous section we can conclude that the values of the operators BT and ZT fulfill the estimates a priori from Section 4. Lemma 5.1. For any T > 0 the operator ZT : C + [−r , T ] → C + [−r , T ] defined by (5.1) has a unique fixed point. Proof. Let us consider the Banach space C [−r , T ] with Bielecki norm kf kT ,K = supt ∈[−r ,T ] e−Kt |f (t )| with any K > 0 (see [11]). The norm k.kT ,K is equivalent to the classical supremum norm k · k in C [−r , T ] because it is obvious that e−KT kf k 6 kf kT ,K 6 eKr kf k. It is sufficient to show that ZT is contracting, then the proof of the Lemma will be an easy consequence of the Banach fixed point theorem. Choose z, zˆ ∈ C [−r , T ]. Applying (5.1) we obtain

kZT (z ) − ZT (ˆz )kT ,K Z t 6

sup e−Kt

t ∈[−r ,T ]

BT (z )(a) − BT (ˆz )(a) da

(5.2)

0

+ sup e−Kt

Z tZ

t ∈[−r ,T ]

0

+ sup e−Kt t ∈[−r ,T ]

0


Λ(η − a, zη ) − Λ(η − a, zˆη ) u(η − a, η + ·) dηda

(5.3)

a

∞

Z

t

t

Z


Λ(a + η, zη ) − Λ(a + η, zˆη ) u(a + η, η + ·) dηda.

(5.4)

0

Using assumption (H6 ) it is convenient to estimate each elements (5.2)–(5.4) separately. Let us start from the relations (5.3) and (5.4). sup e−Kt

t ∈[−r ,T ]

Z tZ 0

t


Λ(η − a, zη ) − Λ(η − a, zˆη ) u(η − a, η + ·) dηda a

A.L. Dawidowicz, A. Poskrobko / Nonlinear Analysis 71 (2009) e2657–e2664

6

Z tZ

sup e−Kt

t ∈[−r ,T ]

0

6 Λ1 kz − zˆ kT ,K

6

Λ1 kz − zˆ kT ,K eK η u(η − a, η + ·)dηda

a

sup e−Kt

eK η M1 (T )eaM2 (T ) dηda

a

0

M1 (T )

t

Z tZ

t ∈[−r ,T ]

eK η B (a)dηda

a

0

sup e−Kt

t

Z tZ

t ∈[−r ,T ]

6 Λ1 kz − zˆ kT ,K 6 Λ1 kz − zˆ kT ,K

t

sup etM2 (T ) e−Kt

t

Z

M2 (T ) t ∈[−r ,T ]

eK η d η

0

Λ1 M1 (T ) TM2 (T ) e kz − zˆ kT ,K KM2 (T )

and sup e−Kt

t ∈[−r ,T ]

6

∞

Z 0

0

sup e−Kt

t ∈[−r ,T ]

K

∞

t

Z

0

6 Λ1 kz − zˆ kT ,K

Λ1 Φ


Λ(a + η, zη ) − Λ(a + η, zˆη ) u(a + η, η + ·) dηda

Z

6 Λ1 kz − zˆ kT ,K

6

t

Z

Λ1 kz − zˆ kT ,K eK η u(a + η, η + ·)dηda

0

sup e−Kt

∞

Z

t ∈[−r ,T ]

0

sup e

eK η sup ϕ(a, s)dηda s∈[−r ,0]

0

−Kt Φ

eKt − 1

K

t ∈[−r ,T ]

t

Z




kz − zˆ kT ,K .

Condition (5.2) remains to be estimated. From formula (3.2)

BT (z )(t ) − BT (b z )(t )

Z

0

−r 0

Z

−r 0

BT (ˆz )(a) β(t + s − a, z (t + s)) − β(t + s − a, zˆ (t + s)) dads ∞

  ϕ(a + s, 0) β(a + t + s, z (t + s)) − β(a + t + s, zˆ (t + s)) dads 0

Z

t +s

 Z β(t + s − a, z (t + s))

t +s

 Z z (t + s)) − β(t + s − a, z (t + s))) (β(t + s − a,b

+ −r 0

Z

0

Z

−r 0

−r 0



β(a + t + s, z (t + s))

0

−r

 Λ(η,b za+η )u(η, a + η + ·)dη dads

t +s

Z



  Λ(a + η, zˆη ) − Λ(a + η, zη ) u(η, a + η + ·)dη dads

0

∞

Z

t +s −a 0

∞

" β(a + t + s, zˆ (t + s)) − β(a + t + s, z (t + s))

+




0

#

t +s

Z

   Λ(η, zˆa+η ) − Λ(η, za+η ) u(η, a + η + ·)dη dads

0

Z

+ Z

t +s−a

0

+ Z



0

Z

−r 0

0



+ Z

  β(t + s − a, z (t + s)) BT (z )(a) − BT (ˆz )(a) dads

t +s

Z

+ Z

t +s

Z

=

Λ(a + η, zˆη )u(η, a + η + ·)dη dads.

× 0

We introduce an inequality which let us rewrite the above condition in simpler form

BT (z )(t ) − BT (b z )(t ) 6 |F (t , r )| + r β0

Z

t

BT (z )(a) − BT (ˆz )(a) da.




0

From Gronwall’s inequality

BT (z )(t ) − BT (b z )(t ) 6 |F (t , r )| + r β0

Z 0

t

|F (a, r )|er β0 (t −a) da.

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A.L. Dawidowicz, A. Poskrobko / Nonlinear Analysis 71 (2009) e2657–e2664

By an analogous reasoning we can deduce that

kF (t , r )kT ,K 6 C0 kz − zˆ kT ,K where C0 is constant depending on quantities Λ0 , Λ1 , β0 , β1 , K , Φ and T . Summarizing, because in all the considered summands there is continuous dependence on the difference z − zˆ we can assert that for sufficiently large K

kZT (z ) − ZT (ˆz )kT ,K 6 C kz − zˆ kT ,K where C ∈ (0, 1).



According to the above Lemma we can draw a conclusion about the existence and the uniqueness of the solution of the population problem. So we can formulate the following theorem: Theorem 5.2. There exists T > 0 such that the population problem (2.1)–(2.3) has a unique solution up to time T . From above we have also the following corollary. Corollary 5.3. The population problem (2.1)–(2.3) has a unique solution for all time.

Acknowledgements The first author was supported by Jagiellonian University and the second author was supported by Bialystok Technical University Grant No. W/WI/8/07. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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