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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 908768 11 pageshttpdxdoiorg1011552013908768
Research ArticleOn the Multispecies Delayed Gurtin-MacCamy Model
Anna Poskrobko1 and Antoni Leon Dawidowicz2
1 Faculty of Computer Science Bialystok University of Technology Ulica Wiejska 45A 15-351 Białystok Poland2 Faculty of Mathematics and Computer Science Jagiellonian University Ulica Łojasiewicza 6 30-348 Krakow Poland
Correspondence should be addressed to Anna Poskrobko aposkrobkopbedupl
Received 8 November 2012 Revised 9 March 2013 Accepted 29 March 2013
Academic Editor Carlos Vazquez
Copyright copy 2013 A Poskrobko and A L Dawidowicz This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The paper deals with the description of multispecies model with delayed dependence on the size of population It is based on theGurtin and MacCamy model The existence and uniqueness of the solution for the new problem of n populations dynamics areproved as well as the asymptotical stability of the equilibrium age distribution
1 Introduction
In this paper we consider mutual influence of 119899 populationsWe assume that each population develops differently affect-ing each other Populations do not destroy each other How-ever they share the same natural resources and the space Inour model 119906 denotes the density of the ecosystem consistingof 119899 different populations Therefore 119906 = (119906
1 119906
119899) is the
vector function Each component 119906119894 for 119894 = 1 119899 denotesthe 119894th population densityThe development of 119894th populationcan be expressed by the following system of the equations
denotes the intensity of changing of the 119894th population intime In particular if 119906119894 is differentiable then119863119906119894 = 120597119906119894120597119886 +120597119906
119894120597119905 The quantity 119911119894(119905) is the total 119894th population at time
119905 We can express total population of the whole ecosystem attime 119905 by
119911 (119905) = (1199111(119905) 119911
119899(119905)) (7)
Thus
119911119905= (119911
1
119905 119911
119899
119905) (8)
2 Abstract and Applied Analysis
We assume that
119911119894
0(0) = int
infin
0
120593119894(119886 0) 119889119886 (9)
The birth and death processes of the 119894th population aredescribed by coefficients 120582119894 and 120573119894 We assume that both pro-cesses depend on the population size not only at the moment119905 but also at any preceding period of time (so-called historysegment) Introducing the delay parameter to the model hasdeep biological approach (see for instance [1]) All naturalprocesses occur with some delay with respect to the momentof their initiation We can take into consideration for exam-ple the period of pregnancy (constant delay) or morbidity(variable delay) In our model considered processes occurwith 119899 various delays typical for 119899 populations It is a novelapproach to the population dynamics research Moreoverthe system (1)ndash(4) consists of the general description of 119899populations dynamics If populations develop independentlythen the functions 120582119894 and 120573119894 depend on 119894th coordinate of thefunction 119911
119905only and the system consists of 119899 independent
von Foerster equations However if there are populationsrelationships we can consider three cases
(1) Competition for food in this case 120573119894 is decreas-
ing function of variables describing the numbers ofspecies competing for food with the 119894th one Forexample 120573119894
denotes total amount of food and 119868 is the set of speciescompeting with the 119894th one
(2) Schema predator prey if 119894th species feeds on 119895thspecies individuals then 120582
119895 and 120573119894 are increasing
functions of variables 119911119895 and 119911119894 respectively The
classical models show that balance state between twospecies competing for food is not possible In thiscase the model does not have nonzero equilibriumage distribution However if we additionally considerpredator feeding on one competing species then weget balance between three considered species (see forinstance [2ndash5])
(3) Symbiosis if 119894th and 119895th species live in symbiosis witheach other then 120573119894 can be an increasing function of119911119895 (when the presence of 119895th species individuals is thereproduction of 119894th species favour) or 120582119895 is a decreas-ing function of 119911119894 (when the presence of 119894th speciesprotects 119895th species individuals from death)
In this paper we develop the idea of Gurtin and MacCamymodel [6] for one age-dependent population
This classical model has many generalizations [7ndash16] Con-siderations of the dynamics of age-structured populations
have received substantial treatment on various fields [17ndash22] In particular our latest extension of this theme waspresented in the article [10] where we considered an age-dependent population dynamics with a delayed dependenceon the structure The right-hand side of the equation in ourmodel is not in the form 120582119906 butΛ119906 HereΛ is an operator thatdoes not apply to the value of the function 119906 as in the classicalmodel but to its restriction to some particular space
In our paper [10] we present the proof of the existence anduniqueness of the new problem solution Conditions of theexponential asymptotical stability for this model still remainto be formulated On account of the significant difficultieswith formulating conditions of the stability we considerslightly simpler version of the model than the one describedin [10] We resign from the delay in the structure during adeliberation on mortality However the delayed dependenceof the structure still remains the element of the birth processdescription In this paper we aim to state stability conditionsfor the ecosystem of 119899 populations The plan of the paperis as follows In Section 2 we formulate the problem in theterms of operator equations Section 3 contains the proof ofthe existence and uniqueness of the solution We also studyequilibriumage distributions that is solutions to the problemwhich are independent of time In Section 4 exponentialasymptotic stability of the equilibrium age distributions isanalyzed We use generalized Laplace transform to study thestability
We consider our model (1)ndash(4) under the followingassumptions
(1198671) 120593 = (120593
1 120593
119899) where 120593119894
isin 119862(R+times [minus119903
119894 0]) for 119894 =
1 119899(119867
2) Φ
119894= int
infin
0sup
119904isin[minus1199031198940]120593119894(119886 119904)119889119886 lt infin for 119894 = 1 119899
(1198673) The function [minus119903119894 0] ni 119905 997891rarr int
infin
0120593119894(119886 119905) 119889119886 is contin-
uous(119867
4) 120582 = (120582
1 120582
119899) 120573 = (120573
1 120573
119899) where 120582119894 isin
119862(R+times ⨉
119899
119894=1119862([minus119903
119894 0])) 120573119894
isin 119862(R119899+1
+) the Frechet
derivatives 119863120582119894 of 120582119894(119886 120595) with respect to 120595 exist forall 119886 ⩾ 0 and 120595 ⩾ 0
(1198675) The components of the function 120582(sdot 120595) = (120582
1(sdot 120595)
120582119899(sdot 120595)) belong to 119862(119862([minus119903
119894 0]) 119871
infin(R
+)) re-
spectively for 119894 = 1 119899(119867
6) The components of Frechet derivative 119863
1205950120582 =
(11986312059501205821 119863
1205950120582119899) in the point 120595
0as functions of
1205950belong to 119862(119862([minus119903119894 0])L(119862([minus119903
119894 0]) 119871
infin(R
+)))
119894 = 1 119899 Here L(119883 119884) denotes the Banach spaceof all bounded linear operators from119883 to 119884
Abstract and Applied Analysis 3
(1198677) 120593
119894⩾ 0 120582
119894⩾ 0 120573
119894⩾ 0 for 119894 = 1 119899
(1198678) The functions 120582 and 120573 are bounded that is120582(119886 120595) ⩽ 120582
In this section we will formulate problem (1)ndash(4) in termsof operator equations Thanks to this we will prove localand global existences of the presented problem solution Thenext theorem is analogous to these well-known results forexample von Foerster [23] or Gurtin and MacCamy models
for 119894 = 1 119899 are the solutions of the problems (2) and (3) upto time 119879 gt 0 if and only if the function 119906 = (1199061 119906119899) is thesolution of the age-dependent 119899 populations problem (1)ndash(4) on[0 119879] and 119906 is defined by the formula
119906119894(119886 119905) =
120593119894(119886 minus 119905 0) 119890
minusint119905
0120582119894(119886minus119905+120591119911120591)119889120591 for 119886 ⩾ 119905
119861119894(119905 minus 119886) 119890
minusint119886
0120582119894(120572119911119905minus119886+120572)119889120572 for 119905 gt 119886
(16)
where 119861119894(119905) = 119906
119894(0 119905) for each 119894 = 1 119899
Proof Theidea of the proof is analogous to the result includedin the paper [9] treating the theme of the age-dependent
population problem for one species Let 119906 = (1199061 119906
119899) be
a solution of the problem up to time 119879 Let 119906119894(ℎ) = 119906119894(119886
0+
ℎ 1199050+ ℎ) 120582
119894
(ℎ) = 120582119894(119886
0+ ℎ 119911
1199050+ℎ) then we can rewrite (1) as
the equation 119889119906119894119889ℎ + 120582119894
(ℎ)119906119894= 0 with the unique solution
119906119894(119886
0+ ℎ 119905
0+ ℎ) = 119906
119894(119886
0 119905
0) 119890
minusintℎ
0120582119894
(120578)119889120578 (17)
Substituting (1198860 119905
0) = (119886 minus 119905 0) ℎ = 119905 and (119886
0 119905
0) = (0 119905 minus 119886)
ℎ = 119886 into (17) yields the formula (16) Applying (16) to (2)and (3) we obtain the operator equations (14) and (15)
To prove the second part of the theorem we shouldassume that 119911119894
119905⩾ 0 and 119861119894
⩾ 0 for 119894 = 1 119899 are continuousfunctions on the interval [0 119879] fulfilling conditions (14) and(15) Let 119906 for each component 119906119894 be defined on R
+times [0 119879]
by the formula (16)The function 119906 is nonnegative because of(119867
7) An easy computation shows that (4) holds and 119906(0 119905) =
119861(119905) for 119905 gt 0 119906 isin 1198711(R+) because 120582 120573 and 119911
119905are continuous
and 120593 isin 1198711(R) It follows from (14)ndash(16) that (2) and (3) are
satisfied To complete the proof let us notice that (6) and(16) imply existing 119863119906 on R
+times [0 119879] and the equality (1)
holds
Let us make some estimation for the necessity of the nextsection We turn back to the operator equation (14) By theassumptions (119867
We define the previous operator using the system of (15)for 119894 = 1 119899 with 119861 = (119861
1 119861
119899) replaced by B
119879=
(B1
119879 B119899
119879)
4 Abstract and Applied Analysis
Theorem 2 The operator Z119879 ⨉
119899
119894=1119862+[minus119903
119894 119879] rarr
⨉119899
119894=1119862+
[minus119903119894 119879] defined by (22) has a unique fixed point
for any 119879 gt 0
Proof We prove that the operator Z119879is contracting and
then the assertion will be a consequence of the Banach fixedpoint theorem Let us consider the Banach space 119862[minus119903119894 119879]with the Bielecki norm 119891
Therefore we can choose sufficiently large constant 119870 forfixed 119879 that 119868
1 119868
2 and 119868
3are less than 119862119911 minus
119879with the
constant119862 isin [0 (13119899)) (independent of 119911 and )This showsthe contraction ofZ
119879and completes the proof
According to the previous mentioned there exists theexact one solution for any interval [0 119879] so the solutionsdefined on two different intervals coincide on their intersec-tion By the extension property we have the existence anduniqueness of the 119899 populations problem solution for alltimes
4 Stability of Equilibrium Age Distribution
A stationary solution 119906(119886) = (1199061(119886) 119906
119899(119886)) of the model
(1)ndash(4) satisfies the following system of the equations
(119906119894
0)
1015840
(119886) + 120582119894(119886 119911
0) 119906
119894
0= 0
119911119894
0= int
infin
0
119906119894
0(119886) 119889119886
119906119894
0(0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 119906
119894
0(119886) 119889119886
(31)
for 119894 = 1 119899
The population of the whole ecosystem 1199110= (119911
1
0 119911
119899
0)
and its birthrate 1198610= 119906
0(0) = (119906
1
0(0) 119906
119899
0(0)) are con-
stantsThe quantity 1199060isin 119862
1(R
+) is the solution of the system
(31) and it will be referred to the equilibrium age distributionThe probability that an individual of 119894th population survivesto age 119886 if the population of the ecosystem is on the constantsize level 119911
0can be expressed by
120587119894
0(119886) = 119890
minusint119886
0120582119894(1205721199110)119889120572
(32)
The quantity
119877119894(119911
0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 120587
119894
0(119886) 119889119886 (33)
is the number of offsprings expected to be born to anindividual of 119894th populationwhen the population of thewholeecosystem equals 119911
0We can formulate the following theorem
describing the connection between these three quantities
Theorem 3 Let 1199110= (119911
1
0 119911
119899
0) and 119911119894
0gt 0 for 119894 = 1 119899
and assume that 120587119894
0(sdot) 120573119894
(sdot 1199110)120587
119894
0(sdot) isin 119871
1(R
+) for each 119894 =
1 119899 Then
119877 (1199110) = (119877
1(119911
0) 119877
119899(119911
0)) (34)
with
119877119894(119911
0) = 1 119894 = 1 119899 (35)
is a necessary and sufficient condition that an equilibrium agedistribution exists The unique equilibrium age distribution1199060= (119906
1
0 119906
119899
0) corresponding to 119911
0= (119911
1
0 119911
119899
0) is given
by
119906119894
0(119886) = 119861
119894
0120587119894
0(119886) 119894 = 1 119899 (36)
where
119861119894
0=
119911119894
0
int
infin
0120587119894
0(119886) 119889119886
(37)
Proof The function (36) is the unique solution of (31)1with
the initial condition 119906119894
0(0) = 119861
119894
0 By (31)
2we obtain the
formula (37) for 119861119894
0 An easy computation shows that (35) is
equivalent to (31)3for each 119894 = 1 119899
We now turn to the problem of the equilibrium agedistribution stabilityWe consider ldquoperturbationsrdquo 120585119894(119886 119905) and119901119894(119905) Let us write
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
The birth and death processes of the 119894th population aredescribed by coefficients 120582119894 and 120573119894 We assume that both pro-cesses depend on the population size not only at the moment119905 but also at any preceding period of time (so-called historysegment) Introducing the delay parameter to the model hasdeep biological approach (see for instance [1]) All naturalprocesses occur with some delay with respect to the momentof their initiation We can take into consideration for exam-ple the period of pregnancy (constant delay) or morbidity(variable delay) In our model considered processes occurwith 119899 various delays typical for 119899 populations It is a novelapproach to the population dynamics research Moreoverthe system (1)ndash(4) consists of the general description of 119899populations dynamics If populations develop independentlythen the functions 120582119894 and 120573119894 depend on 119894th coordinate of thefunction 119911
119905only and the system consists of 119899 independent
von Foerster equations However if there are populationsrelationships we can consider three cases
(1) Competition for food in this case 120573119894 is decreas-
ing function of variables describing the numbers ofspecies competing for food with the 119894th one Forexample 120573119894
denotes total amount of food and 119868 is the set of speciescompeting with the 119894th one
(2) Schema predator prey if 119894th species feeds on 119895thspecies individuals then 120582
119895 and 120573119894 are increasing
functions of variables 119911119895 and 119911119894 respectively The
classical models show that balance state between twospecies competing for food is not possible In thiscase the model does not have nonzero equilibriumage distribution However if we additionally considerpredator feeding on one competing species then weget balance between three considered species (see forinstance [2ndash5])
(3) Symbiosis if 119894th and 119895th species live in symbiosis witheach other then 120573119894 can be an increasing function of119911119895 (when the presence of 119895th species individuals is thereproduction of 119894th species favour) or 120582119895 is a decreas-ing function of 119911119894 (when the presence of 119894th speciesprotects 119895th species individuals from death)
In this paper we develop the idea of Gurtin and MacCamymodel [6] for one age-dependent population
This classical model has many generalizations [7ndash16] Con-siderations of the dynamics of age-structured populations
have received substantial treatment on various fields [17ndash22] In particular our latest extension of this theme waspresented in the article [10] where we considered an age-dependent population dynamics with a delayed dependenceon the structure The right-hand side of the equation in ourmodel is not in the form 120582119906 butΛ119906 HereΛ is an operator thatdoes not apply to the value of the function 119906 as in the classicalmodel but to its restriction to some particular space
In our paper [10] we present the proof of the existence anduniqueness of the new problem solution Conditions of theexponential asymptotical stability for this model still remainto be formulated On account of the significant difficultieswith formulating conditions of the stability we considerslightly simpler version of the model than the one describedin [10] We resign from the delay in the structure during adeliberation on mortality However the delayed dependenceof the structure still remains the element of the birth processdescription In this paper we aim to state stability conditionsfor the ecosystem of 119899 populations The plan of the paperis as follows In Section 2 we formulate the problem in theterms of operator equations Section 3 contains the proof ofthe existence and uniqueness of the solution We also studyequilibriumage distributions that is solutions to the problemwhich are independent of time In Section 4 exponentialasymptotic stability of the equilibrium age distributions isanalyzed We use generalized Laplace transform to study thestability
We consider our model (1)ndash(4) under the followingassumptions
(1198671) 120593 = (120593
1 120593
119899) where 120593119894
isin 119862(R+times [minus119903
119894 0]) for 119894 =
1 119899(119867
2) Φ
119894= int
infin
0sup
119904isin[minus1199031198940]120593119894(119886 119904)119889119886 lt infin for 119894 = 1 119899
(1198673) The function [minus119903119894 0] ni 119905 997891rarr int
infin
0120593119894(119886 119905) 119889119886 is contin-
uous(119867
4) 120582 = (120582
1 120582
119899) 120573 = (120573
1 120573
119899) where 120582119894 isin
119862(R+times ⨉
119899
119894=1119862([minus119903
119894 0])) 120573119894
isin 119862(R119899+1
+) the Frechet
derivatives 119863120582119894 of 120582119894(119886 120595) with respect to 120595 exist forall 119886 ⩾ 0 and 120595 ⩾ 0
(1198675) The components of the function 120582(sdot 120595) = (120582
1(sdot 120595)
120582119899(sdot 120595)) belong to 119862(119862([minus119903
119894 0]) 119871
infin(R
+)) re-
spectively for 119894 = 1 119899(119867
6) The components of Frechet derivative 119863
1205950120582 =
(11986312059501205821 119863
1205950120582119899) in the point 120595
0as functions of
1205950belong to 119862(119862([minus119903119894 0])L(119862([minus119903
119894 0]) 119871
infin(R
+)))
119894 = 1 119899 Here L(119883 119884) denotes the Banach spaceof all bounded linear operators from119883 to 119884
Abstract and Applied Analysis 3
(1198677) 120593
119894⩾ 0 120582
119894⩾ 0 120573
119894⩾ 0 for 119894 = 1 119899
(1198678) The functions 120582 and 120573 are bounded that is120582(119886 120595) ⩽ 120582
In this section we will formulate problem (1)ndash(4) in termsof operator equations Thanks to this we will prove localand global existences of the presented problem solution Thenext theorem is analogous to these well-known results forexample von Foerster [23] or Gurtin and MacCamy models
for 119894 = 1 119899 are the solutions of the problems (2) and (3) upto time 119879 gt 0 if and only if the function 119906 = (1199061 119906119899) is thesolution of the age-dependent 119899 populations problem (1)ndash(4) on[0 119879] and 119906 is defined by the formula
119906119894(119886 119905) =
120593119894(119886 minus 119905 0) 119890
minusint119905
0120582119894(119886minus119905+120591119911120591)119889120591 for 119886 ⩾ 119905
119861119894(119905 minus 119886) 119890
minusint119886
0120582119894(120572119911119905minus119886+120572)119889120572 for 119905 gt 119886
(16)
where 119861119894(119905) = 119906
119894(0 119905) for each 119894 = 1 119899
Proof Theidea of the proof is analogous to the result includedin the paper [9] treating the theme of the age-dependent
population problem for one species Let 119906 = (1199061 119906
119899) be
a solution of the problem up to time 119879 Let 119906119894(ℎ) = 119906119894(119886
0+
ℎ 1199050+ ℎ) 120582
119894
(ℎ) = 120582119894(119886
0+ ℎ 119911
1199050+ℎ) then we can rewrite (1) as
the equation 119889119906119894119889ℎ + 120582119894
(ℎ)119906119894= 0 with the unique solution
119906119894(119886
0+ ℎ 119905
0+ ℎ) = 119906
119894(119886
0 119905
0) 119890
minusintℎ
0120582119894
(120578)119889120578 (17)
Substituting (1198860 119905
0) = (119886 minus 119905 0) ℎ = 119905 and (119886
0 119905
0) = (0 119905 minus 119886)
ℎ = 119886 into (17) yields the formula (16) Applying (16) to (2)and (3) we obtain the operator equations (14) and (15)
To prove the second part of the theorem we shouldassume that 119911119894
119905⩾ 0 and 119861119894
⩾ 0 for 119894 = 1 119899 are continuousfunctions on the interval [0 119879] fulfilling conditions (14) and(15) Let 119906 for each component 119906119894 be defined on R
+times [0 119879]
by the formula (16)The function 119906 is nonnegative because of(119867
7) An easy computation shows that (4) holds and 119906(0 119905) =
119861(119905) for 119905 gt 0 119906 isin 1198711(R+) because 120582 120573 and 119911
119905are continuous
and 120593 isin 1198711(R) It follows from (14)ndash(16) that (2) and (3) are
satisfied To complete the proof let us notice that (6) and(16) imply existing 119863119906 on R
+times [0 119879] and the equality (1)
holds
Let us make some estimation for the necessity of the nextsection We turn back to the operator equation (14) By theassumptions (119867
We define the previous operator using the system of (15)for 119894 = 1 119899 with 119861 = (119861
1 119861
119899) replaced by B
119879=
(B1
119879 B119899
119879)
4 Abstract and Applied Analysis
Theorem 2 The operator Z119879 ⨉
119899
119894=1119862+[minus119903
119894 119879] rarr
⨉119899
119894=1119862+
[minus119903119894 119879] defined by (22) has a unique fixed point
for any 119879 gt 0
Proof We prove that the operator Z119879is contracting and
then the assertion will be a consequence of the Banach fixedpoint theorem Let us consider the Banach space 119862[minus119903119894 119879]with the Bielecki norm 119891
Therefore we can choose sufficiently large constant 119870 forfixed 119879 that 119868
1 119868
2 and 119868
3are less than 119862119911 minus
119879with the
constant119862 isin [0 (13119899)) (independent of 119911 and )This showsthe contraction ofZ
119879and completes the proof
According to the previous mentioned there exists theexact one solution for any interval [0 119879] so the solutionsdefined on two different intervals coincide on their intersec-tion By the extension property we have the existence anduniqueness of the 119899 populations problem solution for alltimes
4 Stability of Equilibrium Age Distribution
A stationary solution 119906(119886) = (1199061(119886) 119906
119899(119886)) of the model
(1)ndash(4) satisfies the following system of the equations
(119906119894
0)
1015840
(119886) + 120582119894(119886 119911
0) 119906
119894
0= 0
119911119894
0= int
infin
0
119906119894
0(119886) 119889119886
119906119894
0(0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 119906
119894
0(119886) 119889119886
(31)
for 119894 = 1 119899
The population of the whole ecosystem 1199110= (119911
1
0 119911
119899
0)
and its birthrate 1198610= 119906
0(0) = (119906
1
0(0) 119906
119899
0(0)) are con-
stantsThe quantity 1199060isin 119862
1(R
+) is the solution of the system
(31) and it will be referred to the equilibrium age distributionThe probability that an individual of 119894th population survivesto age 119886 if the population of the ecosystem is on the constantsize level 119911
0can be expressed by
120587119894
0(119886) = 119890
minusint119886
0120582119894(1205721199110)119889120572
(32)
The quantity
119877119894(119911
0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 120587
119894
0(119886) 119889119886 (33)
is the number of offsprings expected to be born to anindividual of 119894th populationwhen the population of thewholeecosystem equals 119911
0We can formulate the following theorem
describing the connection between these three quantities
Theorem 3 Let 1199110= (119911
1
0 119911
119899
0) and 119911119894
0gt 0 for 119894 = 1 119899
and assume that 120587119894
0(sdot) 120573119894
(sdot 1199110)120587
119894
0(sdot) isin 119871
1(R
+) for each 119894 =
1 119899 Then
119877 (1199110) = (119877
1(119911
0) 119877
119899(119911
0)) (34)
with
119877119894(119911
0) = 1 119894 = 1 119899 (35)
is a necessary and sufficient condition that an equilibrium agedistribution exists The unique equilibrium age distribution1199060= (119906
1
0 119906
119899
0) corresponding to 119911
0= (119911
1
0 119911
119899
0) is given
by
119906119894
0(119886) = 119861
119894
0120587119894
0(119886) 119894 = 1 119899 (36)
where
119861119894
0=
119911119894
0
int
infin
0120587119894
0(119886) 119889119886
(37)
Proof The function (36) is the unique solution of (31)1with
the initial condition 119906119894
0(0) = 119861
119894
0 By (31)
2we obtain the
formula (37) for 119861119894
0 An easy computation shows that (35) is
equivalent to (31)3for each 119894 = 1 119899
We now turn to the problem of the equilibrium agedistribution stabilityWe consider ldquoperturbationsrdquo 120585119894(119886 119905) and119901119894(119905) Let us write
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
In this section we will formulate problem (1)ndash(4) in termsof operator equations Thanks to this we will prove localand global existences of the presented problem solution Thenext theorem is analogous to these well-known results forexample von Foerster [23] or Gurtin and MacCamy models
for 119894 = 1 119899 are the solutions of the problems (2) and (3) upto time 119879 gt 0 if and only if the function 119906 = (1199061 119906119899) is thesolution of the age-dependent 119899 populations problem (1)ndash(4) on[0 119879] and 119906 is defined by the formula
119906119894(119886 119905) =
120593119894(119886 minus 119905 0) 119890
minusint119905
0120582119894(119886minus119905+120591119911120591)119889120591 for 119886 ⩾ 119905
119861119894(119905 minus 119886) 119890
minusint119886
0120582119894(120572119911119905minus119886+120572)119889120572 for 119905 gt 119886
(16)
where 119861119894(119905) = 119906
119894(0 119905) for each 119894 = 1 119899
Proof Theidea of the proof is analogous to the result includedin the paper [9] treating the theme of the age-dependent
population problem for one species Let 119906 = (1199061 119906
119899) be
a solution of the problem up to time 119879 Let 119906119894(ℎ) = 119906119894(119886
0+
ℎ 1199050+ ℎ) 120582
119894
(ℎ) = 120582119894(119886
0+ ℎ 119911
1199050+ℎ) then we can rewrite (1) as
the equation 119889119906119894119889ℎ + 120582119894
(ℎ)119906119894= 0 with the unique solution
119906119894(119886
0+ ℎ 119905
0+ ℎ) = 119906
119894(119886
0 119905
0) 119890
minusintℎ
0120582119894
(120578)119889120578 (17)
Substituting (1198860 119905
0) = (119886 minus 119905 0) ℎ = 119905 and (119886
0 119905
0) = (0 119905 minus 119886)
ℎ = 119886 into (17) yields the formula (16) Applying (16) to (2)and (3) we obtain the operator equations (14) and (15)
To prove the second part of the theorem we shouldassume that 119911119894
119905⩾ 0 and 119861119894
⩾ 0 for 119894 = 1 119899 are continuousfunctions on the interval [0 119879] fulfilling conditions (14) and(15) Let 119906 for each component 119906119894 be defined on R
+times [0 119879]
by the formula (16)The function 119906 is nonnegative because of(119867
7) An easy computation shows that (4) holds and 119906(0 119905) =
119861(119905) for 119905 gt 0 119906 isin 1198711(R+) because 120582 120573 and 119911
119905are continuous
and 120593 isin 1198711(R) It follows from (14)ndash(16) that (2) and (3) are
satisfied To complete the proof let us notice that (6) and(16) imply existing 119863119906 on R
+times [0 119879] and the equality (1)
holds
Let us make some estimation for the necessity of the nextsection We turn back to the operator equation (14) By theassumptions (119867
We define the previous operator using the system of (15)for 119894 = 1 119899 with 119861 = (119861
1 119861
119899) replaced by B
119879=
(B1
119879 B119899
119879)
4 Abstract and Applied Analysis
Theorem 2 The operator Z119879 ⨉
119899
119894=1119862+[minus119903
119894 119879] rarr
⨉119899
119894=1119862+
[minus119903119894 119879] defined by (22) has a unique fixed point
for any 119879 gt 0
Proof We prove that the operator Z119879is contracting and
then the assertion will be a consequence of the Banach fixedpoint theorem Let us consider the Banach space 119862[minus119903119894 119879]with the Bielecki norm 119891
Therefore we can choose sufficiently large constant 119870 forfixed 119879 that 119868
1 119868
2 and 119868
3are less than 119862119911 minus
119879with the
constant119862 isin [0 (13119899)) (independent of 119911 and )This showsthe contraction ofZ
119879and completes the proof
According to the previous mentioned there exists theexact one solution for any interval [0 119879] so the solutionsdefined on two different intervals coincide on their intersec-tion By the extension property we have the existence anduniqueness of the 119899 populations problem solution for alltimes
4 Stability of Equilibrium Age Distribution
A stationary solution 119906(119886) = (1199061(119886) 119906
119899(119886)) of the model
(1)ndash(4) satisfies the following system of the equations
(119906119894
0)
1015840
(119886) + 120582119894(119886 119911
0) 119906
119894
0= 0
119911119894
0= int
infin
0
119906119894
0(119886) 119889119886
119906119894
0(0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 119906
119894
0(119886) 119889119886
(31)
for 119894 = 1 119899
The population of the whole ecosystem 1199110= (119911
1
0 119911
119899
0)
and its birthrate 1198610= 119906
0(0) = (119906
1
0(0) 119906
119899
0(0)) are con-
stantsThe quantity 1199060isin 119862
1(R
+) is the solution of the system
(31) and it will be referred to the equilibrium age distributionThe probability that an individual of 119894th population survivesto age 119886 if the population of the ecosystem is on the constantsize level 119911
0can be expressed by
120587119894
0(119886) = 119890
minusint119886
0120582119894(1205721199110)119889120572
(32)
The quantity
119877119894(119911
0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 120587
119894
0(119886) 119889119886 (33)
is the number of offsprings expected to be born to anindividual of 119894th populationwhen the population of thewholeecosystem equals 119911
0We can formulate the following theorem
describing the connection between these three quantities
Theorem 3 Let 1199110= (119911
1
0 119911
119899
0) and 119911119894
0gt 0 for 119894 = 1 119899
and assume that 120587119894
0(sdot) 120573119894
(sdot 1199110)120587
119894
0(sdot) isin 119871
1(R
+) for each 119894 =
1 119899 Then
119877 (1199110) = (119877
1(119911
0) 119877
119899(119911
0)) (34)
with
119877119894(119911
0) = 1 119894 = 1 119899 (35)
is a necessary and sufficient condition that an equilibrium agedistribution exists The unique equilibrium age distribution1199060= (119906
1
0 119906
119899
0) corresponding to 119911
0= (119911
1
0 119911
119899
0) is given
by
119906119894
0(119886) = 119861
119894
0120587119894
0(119886) 119894 = 1 119899 (36)
where
119861119894
0=
119911119894
0
int
infin
0120587119894
0(119886) 119889119886
(37)
Proof The function (36) is the unique solution of (31)1with
the initial condition 119906119894
0(0) = 119861
119894
0 By (31)
2we obtain the
formula (37) for 119861119894
0 An easy computation shows that (35) is
equivalent to (31)3for each 119894 = 1 119899
We now turn to the problem of the equilibrium agedistribution stabilityWe consider ldquoperturbationsrdquo 120585119894(119886 119905) and119901119894(119905) Let us write
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
[minus119903119894 119879] defined by (22) has a unique fixed point
for any 119879 gt 0
Proof We prove that the operator Z119879is contracting and
then the assertion will be a consequence of the Banach fixedpoint theorem Let us consider the Banach space 119862[minus119903119894 119879]with the Bielecki norm 119891
Therefore we can choose sufficiently large constant 119870 forfixed 119879 that 119868
1 119868
2 and 119868
3are less than 119862119911 minus
119879with the
constant119862 isin [0 (13119899)) (independent of 119911 and )This showsthe contraction ofZ
119879and completes the proof
According to the previous mentioned there exists theexact one solution for any interval [0 119879] so the solutionsdefined on two different intervals coincide on their intersec-tion By the extension property we have the existence anduniqueness of the 119899 populations problem solution for alltimes
4 Stability of Equilibrium Age Distribution
A stationary solution 119906(119886) = (1199061(119886) 119906
119899(119886)) of the model
(1)ndash(4) satisfies the following system of the equations
(119906119894
0)
1015840
(119886) + 120582119894(119886 119911
0) 119906
119894
0= 0
119911119894
0= int
infin
0
119906119894
0(119886) 119889119886
119906119894
0(0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 119906
119894
0(119886) 119889119886
(31)
for 119894 = 1 119899
The population of the whole ecosystem 1199110= (119911
1
0 119911
119899
0)
and its birthrate 1198610= 119906
0(0) = (119906
1
0(0) 119906
119899
0(0)) are con-
stantsThe quantity 1199060isin 119862
1(R
+) is the solution of the system
(31) and it will be referred to the equilibrium age distributionThe probability that an individual of 119894th population survivesto age 119886 if the population of the ecosystem is on the constantsize level 119911
0can be expressed by
120587119894
0(119886) = 119890
minusint119886
0120582119894(1205721199110)119889120572
(32)
The quantity
119877119894(119911
0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 120587
119894
0(119886) 119889119886 (33)
is the number of offsprings expected to be born to anindividual of 119894th populationwhen the population of thewholeecosystem equals 119911
0We can formulate the following theorem
describing the connection between these three quantities
Theorem 3 Let 1199110= (119911
1
0 119911
119899
0) and 119911119894
0gt 0 for 119894 = 1 119899
and assume that 120587119894
0(sdot) 120573119894
(sdot 1199110)120587
119894
0(sdot) isin 119871
1(R
+) for each 119894 =
1 119899 Then
119877 (1199110) = (119877
1(119911
0) 119877
119899(119911
0)) (34)
with
119877119894(119911
0) = 1 119894 = 1 119899 (35)
is a necessary and sufficient condition that an equilibrium agedistribution exists The unique equilibrium age distribution1199060= (119906
1
0 119906
119899
0) corresponding to 119911
0= (119911
1
0 119911
119899
0) is given
by
119906119894
0(119886) = 119861
119894
0120587119894
0(119886) 119894 = 1 119899 (36)
where
119861119894
0=
119911119894
0
int
infin
0120587119894
0(119886) 119889119886
(37)
Proof The function (36) is the unique solution of (31)1with
the initial condition 119906119894
0(0) = 119861
119894
0 By (31)
2we obtain the
formula (37) for 119861119894
0 An easy computation shows that (35) is
equivalent to (31)3for each 119894 = 1 119899
We now turn to the problem of the equilibrium agedistribution stabilityWe consider ldquoperturbationsrdquo 120585119894(119886 119905) and119901119894(119905) Let us write
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
Therefore we can choose sufficiently large constant 119870 forfixed 119879 that 119868
1 119868
2 and 119868
3are less than 119862119911 minus
119879with the
constant119862 isin [0 (13119899)) (independent of 119911 and )This showsthe contraction ofZ
119879and completes the proof
According to the previous mentioned there exists theexact one solution for any interval [0 119879] so the solutionsdefined on two different intervals coincide on their intersec-tion By the extension property we have the existence anduniqueness of the 119899 populations problem solution for alltimes
4 Stability of Equilibrium Age Distribution
A stationary solution 119906(119886) = (1199061(119886) 119906
119899(119886)) of the model
(1)ndash(4) satisfies the following system of the equations
(119906119894
0)
1015840
(119886) + 120582119894(119886 119911
0) 119906
119894
0= 0
119911119894
0= int
infin
0
119906119894
0(119886) 119889119886
119906119894
0(0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 119906
119894
0(119886) 119889119886
(31)
for 119894 = 1 119899
The population of the whole ecosystem 1199110= (119911
1
0 119911
119899
0)
and its birthrate 1198610= 119906
0(0) = (119906
1
0(0) 119906
119899
0(0)) are con-
stantsThe quantity 1199060isin 119862
1(R
+) is the solution of the system
(31) and it will be referred to the equilibrium age distributionThe probability that an individual of 119894th population survivesto age 119886 if the population of the ecosystem is on the constantsize level 119911
0can be expressed by
120587119894
0(119886) = 119890
minusint119886
0120582119894(1205721199110)119889120572
(32)
The quantity
119877119894(119911
0) = 119903
1sdot sdot sdot 119903
119899int
infin
0
120573119894(119886 119911
0) 120587
119894
0(119886) 119889119886 (33)
is the number of offsprings expected to be born to anindividual of 119894th populationwhen the population of thewholeecosystem equals 119911
0We can formulate the following theorem
describing the connection between these three quantities
Theorem 3 Let 1199110= (119911
1
0 119911
119899
0) and 119911119894
0gt 0 for 119894 = 1 119899
and assume that 120587119894
0(sdot) 120573119894
(sdot 1199110)120587
119894
0(sdot) isin 119871
1(R
+) for each 119894 =
1 119899 Then
119877 (1199110) = (119877
1(119911
0) 119877
119899(119911
0)) (34)
with
119877119894(119911
0) = 1 119894 = 1 119899 (35)
is a necessary and sufficient condition that an equilibrium agedistribution exists The unique equilibrium age distribution1199060= (119906
1
0 119906
119899
0) corresponding to 119911
0= (119911
1
0 119911
119899
0) is given
by
119906119894
0(119886) = 119861
119894
0120587119894
0(119886) 119894 = 1 119899 (36)
where
119861119894
0=
119911119894
0
int
infin
0120587119894
0(119886) 119889119886
(37)
Proof The function (36) is the unique solution of (31)1with
the initial condition 119906119894
0(0) = 119861
119894
0 By (31)
2we obtain the
formula (37) for 119861119894
0 An easy computation shows that (35) is
equivalent to (31)3for each 119894 = 1 119899
We now turn to the problem of the equilibrium agedistribution stabilityWe consider ldquoperturbationsrdquo 120585119894(119886 119905) and119901119894(119905) Let us write
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
K (119905 minus 120591) y120591119889120591 = f (119905) (63)
where A and K are block diagonal matrices We have
A =
[
[
[
[
[
A1 O OO A2
O
d
O O A119899
]
]
]
]
]
K =
[
[
[
[
[
K1 O OO K2
O
d
O O K119899
]
]
]
]
]
(64)
for zero matrix O A119894= [
1 0
minus1205811198941] A119894y
119905= [
119901119894
119905(0) 0
minus120581119894119901119894
119905119887119894
119905(0)] =
[119901119894(119905) 0
minus120581119894119901119894
119905119887119894(119905)] and
K119894(119905) =
[
[
[
[
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) 119889119886 minus120587
119894
0(119905)
int
0
minus119903119894
int
infin
0
120573119894
0(119886 + 119905 + 119904)
119894
0(119886 119886 + 119905 + 119904) 120596
119894(119886) 119889119886 119889119904 minusint
0
minus119903119894
120573119894
0(119905 + 119904) 120587
119894
0(119905 + 119904) 119889119904
]
]
]
]
(65)
Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover
f (119905) = [[[
f1 (119905)
f119899 (119905)
]
]
]
(66)
for
f 119894 (119905) = [119891119894
1(119905)
119891119894
2(119905)
] minus A119894y1198941119905minus int
119905
0
K119894(119905 minus 120591) y119894
1120591119889120591
119891119894
1(119905) = int
119905
0
int
infin
119905minus120591
119909119894(120591 + 119886 minus 119905 120591)
119894
0(120591 + 119886 minus 119905 119886) 119889119886 119889120591
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
1(119905 + 119904) for 119904 isin [minus119903119894 0]
We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903
We return to the previous deliberation in the nexttheorem We take some additional assumptions
(11986710) The Frechet derivatives 119863
119911119894120582
119894(119886 119911
0) and the deriva-
tion 119863119911119894120573
119894(119886 119911
0) for 119894 = 1 2 119899 as functions of the
variable 119886 belong to 119871infin(R+)
(11986711) (Λ
119894(119886 120588)120588
0) and (Ω
119894(119886 120588)120588
0) tend to zero as
1205880rarr 0 uniformly for 119886 rarr 0 here sdot
0denotes
the supremum norm in the Banach space 119862([minus119903119894 0])(119867
12) 120582
lowast= inf
119886⩾0
120582119894(119886 119911
0) 119894 = 1 2 119899 gt 0
Denote 119890120579 (minusinfin 0] rarr R by the formula 119890
120579(119904) = 119890
120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem
Theorem 4 Let 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 +
119904119894)119889119904 and let 120587119894
1(119905) = int
[minus1199030]120573119894
0(119905+119904
119894)120587
119894
0(119905+119904
119894)119889119904 Let us assume
that there exists some 120583 gt 0 that the equation
1 = int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905
minus int
infin
0
119890minus119905120574
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905
sdot (1 minus int
infin
0
119890minus119905120574120587119894
1(119905) 119889119905)
minus int
infin
0
119890minus119905120574120587119894
0(119905) 119889119905
sdot (int
infin
0
119890minus119905120574
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120574) 119889119886 119889119905 minus 120581
119894(119890
120574))
(68)
has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906
01198711 lt 120575 the corresponding solution of the population
problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911
0
1003817100381710038171003817= 119874 (119890
minus120583119905) (69)
1003817100381710038171003817119906 (119886 119905) minus 119906
Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903
119894 0])
lowast
be the function with measure value Define
984858 (120579) = int
infin
0
984858 (119905) (119890120579) 119890
minus120579119905119889119905 (71)
If we define the convolution by the formula
(984858 lowast V120591) (119905) = int
119905
0
984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)
where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is
984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)
Using generalized Laplace transform to (63) we get
A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)
where
A (120579) =[
[
[
[
[
A1(120579) O sdot sdot sdot OO A2
(120579) sdot sdot sdot O
d
O O sdot sdot sdot A119899(120579)
]
]
]
]
]
A119894(120579) = [
1 0
minus120581119894(119890
120579) 1
]
K (120579) =[
[
[
[
[
K1(120579) O sdot sdot sdot OO K2
(120579) sdot sdot sdot O
d
O O sdot sdot sdotK119899
(120579)
]
]
]
]
]
K119894(120579)
=[
[
[
int
infin
0
119890minus119905120579
int
infin
0
119894
0(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
0(120579)
int
infin
0
119890minus119905120579
int
infin
0
119894
1(119886 119886 + 119905) 120596
119894(119886) (119890
120579) 119889119886 119889119905 minus
119894
1(120579)
]
]
]
(75)
for 119894
1(119886 119886 + 119905) = int
[minus1199030]120573119894
0(119886 + 119905 + 119904
119894)
119894
0(119886 119886 + 119905 + 119904
119894)119889119904 and
120587119894
1(119905) = int
[minus1199030]120573119894
0(119905 + 119904
119894)120587
119894
0(119905 + 119904
119894)119889119904 From the equality (68)
we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have
Aminus1(120579) =
[
[
[
[
[
[
[
(A1)
minus1
(120579) O sdot sdot sdot OO (
A2)
minus1
(120579) sdot sdot sdot O
d
O O sdot sdot sdot (A119899)
minus1
(120579)
]
]
]
]
]
]
]
(A119894)
minus1
(120579) = [
1 0
120581119894(119890
120579) 1
]
(76)
so A is analytic for120579 isin C
Abstract and Applied Analysis 9
The solution of the matrix equation (63) exists and isgiven by
y (119905) = Aminus1f119905+ int
119905
0
J (119905 minus 120591) f120591119889120591 (77)
where f120591(119904) =
[
[
f1(120591+119904)
f119899(120591+119904)
]
]
f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906
01198711 = 120578
1198711 lt 120575 In
that case from what has already been proved it follows that 119909(sdot 119905)
1198711 120595(119905)
1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore
(41) (86)2imply (69) and (53)
2 (80) and (39) imply (70)
Example 5 Let 119899 = 2 Let us consider
120582119894(119886 119911) = 120582
119894(119886 119911
1
1 119911
1
2 119911
2
1 119911
2
2)
120573119894(119886 119911) = 120573
119894(119886 119911
1 119911
2)
(89)
where
120582119894(119886 119911) = 120582
119894(119911
1(0) 119911
1(minus119903
1) 119911
2(0) 119911
2(minus119903
2))
120573119894(119886 119911) = 120573
119894(119911
1 119911
2) 119890
minus120572119894119886
(90)
with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define
120582119894
0= 120582
119894(119911
1
0 119911
1
0 119911
2
0 119911
2
0)
120573119894
0= 120573
119894(119911
1
0 119911
2
0) 119890
minus120572119894119886
(91)
for an arbitrary 1199110= (119911
1
0 119911
2
0) isin R2 From (32) and (33) we
conclude that
120587119894
0(119886) = 119890
minus119886120582119894
0
119877119894(119911
0) =
11990311199032120573119894
0
120572119894+ 120582
119894
0
(92)
By Theorem 3 there exists an equilibrium age distribution119906119894
0(119886) = 119911
119894
0120582119894
0119890minus120582119894
0119886 where 119861
0= 119911
119894
0120582119894
0if and only if 11990311199032120573119894
0(120572
119894+
120582119894
0) = 1 To investigate the stability of the equilibrium age
10 Abstract and Applied Analysis
distributionwe consider (68) Let 1199110= (119911
1
0 119911
2
0) be the solution
of the system of the equations
119877119894(119911
0) = 1 119894 = 1 2 (93)
First let us notice that
120596119894(119890
120574) = (
120597120582119894
120597119911119894
1
+
120597120582119894
120597119911119894
2
119890minus120574119903119894
)120582119894
0119911119894
0119890minus119886120582119894
0 (94)
with the derivatives in the point 1199110 It is easy to notice that
|120597120582119894120597119911
119894
1+(120597120582
119894120597119911
119894
2)119890
minus120574119903119894
| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously
120581119894(119890
120574) =
120582119894
0
120573119894
0
sdot
119911119894
0
119903119894sdot
120597120573119894
120597119911119894(119911
0) int
0
minus119903119894
119890120574119904119889119904 (95)
It is also possible to prove that for every real 120583 the function|120581
119894(119890
120574)| is bounded on half-plane Re(120574) gt 120583 Consider the
right-hand side of (68) All components can be presented inthe form
119860
119861 + 120574
(96)
where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
In consequence for 120572119894 and 1205821198940sufficiently large we can find
that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583
Acknowledgment
The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)
References
[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986
[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006
[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009
[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983
[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980
[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974
[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008
[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994
[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010
[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009
[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979
[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003
[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004
[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977
[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980
[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989
[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997
[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997
Abstract and Applied Analysis 11
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959
[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012
[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992
[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009
[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994
[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959