# On the Multispecies Delayed Gurtin-MacCamy Model

Authors: Anna Poskrobko; Antoni Leon Dawidowicz
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The paper deals with the description of multispecies model
with delayed dependence on the size of population. It is based on the
Gurtin and MacCamy model. The existence and uniqueness of the solution
for the new problem ofn populations dynamics are proved, as well
as the asymptotical stability of the equilibrium age distribution.


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## Body

## 1. Introduction

In this paper we consider mutual influence ofn populations. We assume that each population develops differently, affecting each other. Populations do not destroy each other. However, they share the same natural resources and the space. In our model u denotes the density of the ecosystem consisting of n different populations. Therefore u=(u1,…,un) is the vector function. Each component ui for i=1,…,n denotes the ith population density. The development of ith population can be expressed by the following system of the equations:
(1)Dui(a,t)=-λi(a,zt)ui(a,t),(2)ui(0,t)=∫0∞‍∫[-r,0]‍βi(a,z(t+s))ui(a,t+si)dsda,(3)zi(t)=∫0∞‍ui(a,t)da
with the initial condition
(4)ui(a,s)=φi(a,s)fors∈[-ri,0],
where
(5)[-r,0]=[-r1,0]×[-r2,0]×···×[-rn,0],z(t+s)=(z1(t+s1),z2(t+s2),…,zn(t+sn)),zti:[-ri,0]→ℝ+,ri⩾0,ℝ+=[0,∞),zti(s)=zi(t+s).
The rate
(6)Dui(a,t)=limh→0ui(a+h,t+h)-ui(a,t)h
denotes the intensity of changing of the ith population in time. In particular, if ui is differentiable then Dui=∂ui/∂a+∂ui/∂t. The quantity zi(t) is the total ith population at time t. We can express total population of the whole ecosystem at time t by
(7)z(t)=(z1(t),…,zn(t)).
Thus
(8)zt=(zt1,…,ztn).
We assume that
(9)z0i(0)=∫0∞‍φi(a,0)da.
The birth and death processes of the ith population are described by coefficients λi and βi. We assume that both processes depend on the population size not only at the moment t, but also at any preceding period of time (so-called history segment). Introducing the delay parameter to the model has deep biological approach (see, for instance, [1]). All natural processes occur with some delay with respect to the moment of their initiation. We can take into consideration, for example, the period of pregnancy (constant delay) or morbidity (variable delay). In our model, considered processes occur with n various delays, typical for n populations. It is a novel approach to the population dynamics research. Moreover the system (1)–(4) consists of the general description of n populations dynamics. If populations develop independently, then the functions λi and βi depend on ith coordinate of the function zt only and the system consists of n independent von Foerster equations. However, if there are populations relationships, we can consider three cases. (1)
Competition for food: in this caseβi is decreasing function of variables describing the numbers of species competing for food with the ith one. For example, βi(a,z)=max{a(F-∑j∈I‍kjzj),0}, where F denotes total amount of food and I is the set of species competing with the ith one.(2)
Schema predator prey: ifith species feeds on jth species individuals, then λj and βi are increasing functions of variables zj and zi, respectively. The classical models show that balance state between two species competing for food is not possible. In this case the model does not have nonzero equilibrium age distribution. However, if we additionally consider predator feeding on one competing species, then we get balance between three considered species (see, for instance, [2–5]).(3)
Symbiosis: ifith and jth species live in symbiosis with each other, then βi can be an increasing function of zj (when the presence of jth species individuals is the reproduction of ith species favour) or λj is a decreasing function of zi (when the presence of ith species protects jth species individuals from death).In this paper we develop the idea of Gurtin and MacCamy model [6] for one age-dependent population
(10)Du(a,t)=-λ(a,z(t))u(a,t),u(0,t)=∫0∞‍β(a,z(t))u(a,t)da,z(t)=∫0∞‍u(a,t)da,u(a,0)=φ(a).
This classical model has many generalizations [7–16]. Considerations of the dynamics of age-structured populations have received substantial treatment on various fields [17–22]. In particular our latest extension of this theme was presented in the article [10], where we considered an age-dependent population dynamics with a delayed dependence on the structure. The right-hand side of the equation in our model is not in the form λu but Λu. Here Λ is an operator that does not apply to the value of the function u, as in the classical model, but to its restriction to some particular space:
(11)Du(a,t)=-Λ(a,zt)u(a,t+·),u(0,t)=∫0∞‍∫-r0‍β(a,z(t+s))u(a,t+s)dsda,z(t)=∫0∞‍u(a,t)dafort∈[-r,T],u(a,s)=φ(a,s)fors∈[-r,0].
In our paper [10] we present the proof of the existence and uniqueness of the new problem solution. Conditions of the exponential asymptotical stability for this model still remain to be formulated. On account of the significant difficulties with formulating conditions of the stability we consider slightly simpler version of the model than the one described in [10]. We resign from the delay in the structure during a deliberation on mortality. However, the delayed dependence of the structure still remains the element of the birth process description. In this paper we aim to state stability conditions for the ecosystem of n populations. The plan of the paper is as follows. In Section 2 we formulate the problem in the terms of operator equations. Section 3 contains the proof of the existence and uniqueness of the solution. We also study equilibrium age distributions, that is, solutions to the problem which are independent of time. In Section 4 exponential asymptotic stability of the equilibrium age distributions is analyzed. We use generalized Laplace transform to study the stability.We consider our model (1)–(4) under the following assumptions. (H1)
φ=(φ1,…,φn) where φi∈C(ℝ+×[-ri,0]) for i=1,…,n.(H2)
Φi=∫0∞‍sups∈[-ri,0]φi(a,s)da<∞ for i=1,…,n.(H3)
The function[-ri,0]∋t↦∫0∞‍φi(a,t)da is continuous.(H4)
λ=(λ1,…,λn), β=(β1,…,βn) where λi∈C(ℝ+×⨉i=1nC([-ri,0])), βi∈C(ℝ+n+1); the Fréchet derivatives Dλi of λi(a,ψ) with respect to ψ exist for all a⩾0 and ψ⩾0.(H5)
The components of the functionλ(·,ψ) =(λ1(·,ψ),…,λn(·,ψ)) belong to C(C([-ri,0]);  L∞(ℝ+)), respectively, for i=1,…,n.(H6)
The components of Fréchet derivativeDψ0λ=(Dψ0λ1,…,Dψ0λn) in the point ψ0 as functions of ψ0 belong to C(C([-ri,0]);ℒ(C([-ri,0]),L∞(ℝ+))), i=1,…,n. Here ℒ(X,Y) denotes the Banach space of all bounded linear operators from X to Y.(H7)
φi⩾0,λi⩾0,βi⩾0 for i=1,…,n.(H8)
The functionsλ and β are bounded; that is, ∥λ(a,ψ)∥⩽λ0 and ∥β(a,z)∥⩽β0 where λ0 and β0 are finite quantities.(H9)
There existλ1 and β1 such that
(12)∥λ(a,ψ1)-λ(a,ψ2)∥⩽λ1∥ψ1-ψ2∥      for every a,ψ1,ψ2 and
(13)∑i=1n‍∑j=1n‍|∂βi(a,z)∂zj|⩽β1
for every a,z.


## 2. Equivalent Formulation of the Problem

In this section we will formulate problem (1)–(4) in terms of operator equations. Thanks to this we will prove local and global existences of the presented problem solution. The next theorem is analogous to these well-known results, for example, von Foerster [23] or Gurtin and MacCamy models.Theorem 1.
Nonnegative continuous functionszt=(zt1,…,ztn) and B=(B1,…,Bn), where
(14)Bi(t)=∫[-r,0]‍∫0t+si‍βi(t+si-a,z(t+s))∫[-r,0]‍∫0t+si‍×Bi(a)e-∫at+si‍λi(τ-a,zτ)dτdads+∫[-r,0]‍∫0∞‍βi(a+t+si,z(t+s))∫[-r,0]‍∫0t+si‍×φi(a)e-∫0t+si‍λi(a+τ,zτ)dτdads,(15)zti(si)=∫0t+si‍Bi(a)e-∫at+si‍λi(τ-a,zτ)dτda00+∫0∞‍φi(a,0)e-∫0t+si‍λi(a+τ,zτ)dτdafort+si⩾0,zti(si)=z0i(t+si)fort+si<0,
for i=1,…,n, are the solutions of the problems (2) and (3) up to time T>0 if and only if the function u=(u1,…,un) is the solution of the age-dependent n populations problem (1)–(4) on [0,T] and u is defined by the formula
(16)ui(a,t)={φi(a-t,0)e-∫0t‍λi(a-t+τ,zτ)dτfora⩾t,Bi(t-a)e-∫0a‍λi(α,zt-a+α)dαfort>a,
where Bi(t)=ui(0,t) for each i=1,…,n.Proof.
The idea of the proof is analogous to the result included in the paper [9], treating the theme of the age-dependent population problem for one species. Let u=(u1,…,un) be a solution of the problem up to time T. Let u¯i(h)=ui(a0+h,t0+h), λ¯i(h)=λi(a0+h,zt0+h) then we can rewrite (1) as the equation du¯i/dh+λ¯i(h)u¯i=0 with the unique solution
(17)ui(a0+h,t0+h)=ui(a0,t0)e-∫0h‍λ¯i(η)dη.
Substituting (a0,t0)=(a-t,0), h=t and (a0,t0)=(0,t-a), h=a 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 should assume thatzti⩾0 and Bi⩾0 for i=1,…,n are continuous functions on the interval [0,T] fulfilling conditions (14) and (15). Let u for each component ui be defined on ℝ+×[0,T] by the formula (16). The function u is nonnegative because of (H7). An easy computation shows that (4) holds, and u(0,t)=B(t) for t>0. u∈L1(ℝ+) because λ, β, and zt are continuous and φ∈L1(ℝ). It follows from (14)–(16) that (2) and (3) are satisfied. To complete the proof, let us notice that (6) and (16) imply existing Du on ℝ+×[0,T] and the equality (1) holds.Let us make some estimation for the necessity of the next section. We turn back to the operator equation (14). By the assumptions (H2) and (H8) we get
(18)Bi(t)⩽β0r1⋯rn∫0t‍sups∈[-ri,0]Bi(a+s)da+β0r1⋯rnΦi.
Denoting
(19)ℬi(t)=supτ∈[-ri,t]Bi(τ),
we obtain
(20)ℬi(t)⩽β0r1⋯rn∫0t‍ℬi(a)da+β0r1⋯rnΦi,
and by Gronwall's inequality we get
(21)ℬi(t)⩽β0r1⋯rnΦieβ0r1⋯rnt.


## 3. Existence and Uniqueness

According to the theorem in Section2 to solve the population problem up to time T it is sufficient to find functions z=(z1,…,zn)∈⨉i=1nC+([-ri,T]) and B=(B1,…,Bn)∈⨉i=1nC+([-ri,T]) satisfying the system of operator equations (14) and (15). We can notice that (14) is a Volterra equation of Bi with a unique solution ℬTi(z). Let us define on ⨉i=1nC+([-ri,T]) the new operator 𝒵T=(𝒵T1,…,𝒵Tn) where
(22)𝒵Ti(z)(t)=∫0t‍ℬTi(z)(a)e-∫at‍λi(τ-a,zτ)dτda00+∫0∞‍φi(a,0)e-∫0t‍λi(a+τ,zτ)dτda.
We define the previous operator using the system of (15) for i=1,…,n with B=(B1,…,Bn) replaced by ℬT=(ℬT1,…,ℬTn).Theorem 2.
The operator𝒵T:⨉i=1nC+[-ri,T]→⨉i=1nC+[-ri,T] defined by (22) has a unique fixed point for any T>0.Proof.
We prove that the operator𝒵T is contracting, and then the assertion will be a consequence of the Banach fixed point theorem. Let us consider the Banach space C[-ri,T] with the Bielecki norm ∥fi∥T=supt∈[-ri,T]e-Kt|fi(t)| for any K>0. Such norm is equivalent to classical supremum norm ∥·∥ in C[-ri,T]. Choose z,z^∈⨉i=1nC[-ri,T]. Applying the definition (22) we obtain
(23)∥𝒵Ti(z)-𝒵Ti(z^)∥T⩽supt∈[-ri,T]e-Kt×∫0t‍|e-∫at‍λi(τ-a,zτ)dτ-e-∫at‍λi(τ-a,z^τ)dτ|ℬTi(z)(a)da+supt∈[-ri,T]e-Kt×∫0t‍e-∫at‍λi(τ-a,z^τ)dτ|ℬTi(z)(a)-ℬTi(z^)(a)|da+supt∈[-ri,T]e-Kt×∫0∞‍|e-∫0t‍λi(a+τ,zτ)dτ-e-∫0t‍λi(a+τ,z^τ)dτ|φi(a,0)da=I1+I2+I3.
Estimating the quantities I1,  I2, and I3 we use the assumptions (H2), (H7), and (H8), inequalities (21) and |ek-1|⩽|k|e|k|. Thus
(24)I1⩽supt∈[-ri,T]e-Kt×∫0t‍|1-e-∫at‍[λi(τ-a,zτ)-λi(τ-a,z^τ)]dτ|ℬTi(z)(a)da⩽λ1∥z-z^∥Tsupt∈[-ri,T]e-Kt×∫0t‍∫at‍e2λ0(t-a)eKτℬTi(z)(a)dτda⩽λ1Kβ0r1⋯rnΦiβ0r1⋯rn-2λ0eβ0r1⋯rnT∥z-z^∥T,I3⩽supt∈[-ri,T]e-Kt×∫0∞‍∫at‍e2λ0(t-a)eKτλ1∥z-z^∥Tφi(a,0)dτda⩽λ1KΦie2λ0T∥z-z^∥T.
From (14) and the definition of ℬT we have
(25)ℬTi(z)(t)-ℬTi(z^)(t)=∫[-r,0]‍∫0t+si‍βi(t+si-a,z(t+s))·e-∫at+si‍λi(τ-a,zτ)dτ∫[-r,0]‍∫0t+si‍×(ℬTi(z)(a)-ℬTi(z^)(a))dads(26)+∫[-r,0]‍∫0t+si‍ℬTi(z^)(a)∫[-r,0]‍∫0t+si‍∫0t+si‍×(e-∫at+si‍λi(τ-a,z^τ)dτβi(t+si-a,z(t+s))∫[-r,0]‍∫0t+si‍∫0t+si‍0000×e-∫at+si‍λi(τ-a,zτ)dτ-βi(t+si-a,z^(t+s))∫[-r,0]‍∫0t+si‍∫0t+si‍0000×e-∫at+si‍λi(τ-a,z^τ)dτ)dads+∫[-r,0]‍∫0∞‍φi(a,0)∫[-r,0]‍∫0t+si‍∫0t+si‍×(e-∫at+si‍λi(τ-a,z^τ)dτβi(a+t+si,z(t+s))∫[-r,0]‍∫0t+si‍∫0t+si‍0000×e-∫0t+si‍λi(a+τ,zτ)dτ-βi(a+t+si,z^(t+s))∫[-r,0]‍∫0t+si‍∫0t+si0000‍×e-∫0t+si‍λi(a+τ,z^τ)dτ)dads.
Let us denote (26) by fi(t); then
(27)ℬTi(z)(t)-ℬTi(z^)(t)⩽β0r1⋯rn∫0t‍(ℬTi(z)(a)-ℬTi(z^)(a))da+|fi(t)|,
and hence, by Gronwall's inequality we have
(28)ℬTi(z)(t)-ℬTi(z^)(t)⩽|fi(t)|+β0r1⋯rn∫0t‍|fi(a)|eβ0r1⋯rn(t-a)da.
Let us estimate |fi(t)|(29)|fi(t)|⩽∫[-r,0]‍∫0t+si‍C1ℬTi(z^)(a)∫[-r,0]‍∫0t+si‍000×(βi(t+si-a,z(t+s))000000000000000-βi(t+si-a,z^(t+s)))dads+∫[-r,0]‍∫0∞‍C2φi(a,0)00000000000×(βi(a+t+si,z(t+s))000000000000000-βi(a+t+si,z^(t+s)))dads⩽C~β1r1⋯rneKt∥z-z^∥T×(Φi∫0t‍β0r1⋯rneβ0r1⋯rnada+∫0∞‍φi(a,0)da)⩽C~β1r1⋯rnΦieKteβ0r1⋯rnt∥z-z^∥T,
where C1 and C2 are positive constants and C~=max{C1,C2}. Finally, we have
(30)I2⩽supt∈[-ri,T]e-Kt×∫0t‍(|fi(a)|+β0r1⋯rn∫0a‍|fi(ξ)|eβ0r1⋯rn(a-ξ)dξ)da⩽C~β1r1⋯rnΦieKt∥z-z^∥T×supt∈[-ri,T]e-Kt(∫0t‍e(K+β0r1⋯rn)ada0000000000000000+β0r1⋯rn∫0t‍∫0a‍eKξeβ0r1⋯rnadξda∫0t‍)⩽C~β1r1⋯rnΦiK+β0r1⋯rn∥z-z^∥Teβ0r1⋯rnT(1+β0r1⋯rnK).
Therefore, we can choose sufficiently large constant K for fixed T that I1, I2, and I3 are less than C∥z-z^∥T with the constant C∈[0,(1/3n)) (independent of z and z^). This shows the contraction of 𝒵T and completes the proof.According to the previous mentioned there exists the exact one solution for any interval[0,T] so the solutions defined on two different intervals coincide on their intersection. By the extension property we have the existence and uniqueness of the n populations problem solution for all times.


## 4. Stability of Equilibrium Age Distribution

A stationary solutionu(a)=(u1(a),…,un(a)) of the model (1)–(4) satisfies the following system of the equations:
(31)(u0i)′(a)+λi(a,z0)u0i=0,z0i=∫0∞‍u0i(a)da,u0i(0)=r1⋯rn∫0∞‍βi(a,z0)u0i(a)da
fori=1,…,n.The population of the whole ecosystemz0=(z01,…,z0n) and its birthrate B0=u0(0)=(u01(0),…,u0n(0)) are constants. The quantity u0∈C1(ℝ+) is the solution of the system (31), and it will be referred to the equilibrium age distribution. The probability that an individual of ith population survives to age a if the population of the ecosystem is on the constant size level z0 can be expressed by
(32)π0i(a)=e-∫0a‍λi(α,z0)dα.
The quantity
(33)Ri(z0)=r1⋯rn∫0∞‍βi(a,z0)π0i(a)da
is the number of offsprings expected to be born to an individual of ith population when the population of the whole ecosystem equals z0. We can formulate the following theorem describing the connection between these three quantities.Theorem 3.
Letz0=(z01,…,z0n) and z0i>0 for i=1,…,n, and assume that π0i(·),  βi(·,z0)π0i(·)∈L1(ℝ+) for each i=1,…,n. Then
(34)R(z0)=(R1(z0),…,Rn(z0))
with
(35)Ri(z0)=1,i=1,…,n
is a necessary and sufficient condition that an equilibrium age distribution exists. The unique equilibrium age distribution u0=(u01,…,u0n) corresponding to z0=(z01,…,z0n) is given by
(36)u0i(a)=B0iπ0i(a),i=1,…,n,
where
(37)B0i=z0i∫0∞‍π0i(a)da.Proof.
The function (36) is the unique solution of (31)1 with the initial condition u0i(0)=B0i. By (31)2 we obtain the formula (37) for B0i. An easy computation shows that (35) is equivalent to (31)3 for each i=1,…,n.We now turn to the problem of the equilibrium age distribution stability. We consider “perturbations”ξi(a,t) and pi(t). Let us write
(38)u(a,t)=u0(a)+ξ(a,t)=(u01(a)+ξ1(a,t),u02(a)+ξ2(a,t),…,u0n(a)+ξn(a,t))(39)ui(a,t)=u0i(a)+ξi(a,t),
where ξi(a,t)=φi(a,t)-u0i(a) for t∈[-ri,0]. And
(40)z(t)=z0+p(t)=(z01+p1(t),z02+p2(t),…,z0n+pn(t)),zt=z0+pt=(z01+pt1,z02+pt2,…,z0n+ptn),(41)zi(t)=z0i+pi(t),zti=z0i+pti,
where pti(s)=pi(t+s) for s∈[-ri,0]. Our goal is to formulate relations for ξi and pi which guarantee that ui and z0i obey the basic equations (1)–(4). From (1) we obtain
(42)Dξi(a,t)+λ0i(a)ξi(a,t)+ωi(a)pti=xi(a,t)
for
(43)xi(a,t)=-Dziλi(a,z0)ptiξi(a,t)-Λi(a,pt)[B0iπ0i(a)+ξi(a,t)],λ0i(a)=λi(a,z0),
where ωi:ℝ+×C([-ri,0])→ℝ is the functional
(44)ωi(a)pti=Dziλi(a,z0)pti(B0iπ0i(a)),Λi(a,pt)=λi(a,z0+pt)-λ0i(a)-Dziλi(a,z0)pti.
We conclude from (2) that
(45)ξi(0,t)=∫0∞‍∫[-r,0]‍β0i(a)ξi(a,t+si)dsda+κipti+ψi(t),
where
(46)βi(a,z0)=β0i(a),ψi(t)=∫0∞‍∫[-r,0]‍Dziβi(a,z0)pi(t+si)ξi(a,t+si)dsda+∫0∞‍∫[-r,0]‍Ωi(a,p(t+s))+∫0∞‍∫[-r,0]‍×[B0iπ0i(a)+ξi(a,t+si)]dsdaκipti=B0i∫0∞‍∫[-r,0]‍Dziβi(a,z0)ptiπ0i(a)dsda.
Here κi:C([-ri,0])→ℝ is the functional. Moreover, we have
(47)Ωi(a,p(t+s))=βi(a,z0+p(t+s))-β0i(a)-Dziβi(a,z0)pi(t+si).
A trivial verification shows that (3) gives the condition for p in the form
(48)pi(t)=∫0∞‍ξi(a,t)da.
We consider the system (42)–(48) with the initial condition
(49)ξi(a,0)=ηi(a),
where
(50)ηi(a)=φi(a,0)-B0i.
We first express the solution of (42)–(48) in the form of the matrix equation. Let ξi be the solution of (42) up to time T. Let (a0,t0)∈ℝ+×[0,T], ξi¯(h)=ξi(a0+h,t0+h), λ0i¯(h)=λ0i(a0+h), ωi¯(h)=ωi(a0+h), phi¯=pt+hi, and xi¯(h)=xi(a0+h,t0+h). Then we can rewrite (42) as the equation
(51)ddhξi¯(h)+λ0i¯(h)ξi¯(h)=xi¯(h)-ωi¯(h)phi¯
with the unique solution
(52)ξi(a0+h,t0+h)=ξi(a0,t0)e-∫0h‍λ0i(a0+τ)dτ+e-∫0h‍λ0i(a0+τ)dτ×∫0h‍[xi(a0+τ,t0+τ)-ωi(a0+τ)pt0+τi]+∫0∞‍∫[-r,0]00×e∫0τ‍λ0i(a0+α)dαdτ.
Substituting (a0,t0)=(a-t,0), h=t for a⩾t and (a0,t0)=(0,t-a), h=a for a<t yields the formulas
(53)ξi(a,t)=ηi(a-t)π~i(a-t,a)+∫0t‍[xi(a-t+τ,τ)-ωi(a-t+τ)pτi]+∫0∞‍00×π~0i(a-t+τ,a)dτfora⩾t,ξi(a,t)=bi(t-a)π0i(a)+∫t-at‍[xi(τ+a-t,τ)-ωi(τ+a-t)pτi]+∫0∞‍00×π~0i(τ+a-t,a)dτfora<t,
where
(54)π~0i(a-t,a)=π0i(a)π0i(a-t),(55)bi(t-a)=ξi(0,t-a).
Furthermore, (45) implies that
(56)pi(t)=∫0t‍ξi(a,t)da+∫t∞‍ξi(a,t)da.
It follows that
(57)pi(t)+∫0t‍∫0∞‍ωi(a)pτiπ~0i(a,a+t-τ)dadτ-∫0t‍bi(τ)π0i(t-τ)dτ=∫0t‍∫t-τ∞‍xi(τ+a-t,τ)π~0i(τ+a-t,a)dadτ+∫0∞‍ηi(τ)π~0i(τ,τ+t)dτ.
We conclude from (48) that
(58)bi(t)=∫[-r,0]‍∫0t+si‍β0i(a)ξi(a,t+si)dads+∫[-r,0]‍∫t+si∞‍β0i(a)ξi(a,t+si)dads+κi(t)pti+ψi(t),
and hence
(59)-κipti+bi(t)+∫0t‍∫-[r,0]‍∫0∞‍β0i(a-τ+t+si)ωi(a)pτi=∫[-r,0]‍∫0t+si‍∫t+si-τ∞‍×π~0i(a,a-τ+t+si)dadsdτ-∫0t‍∫[-r,0]‍β0i(t+si-τ)bi(τ)π0i(t+si-τ)dsdτ=∫[-r,0]‍∫0t+si‍∫t+si-τ∞‍β0i(a)xi(τ+a-t-si,τ)=∫[-r,0]‍∫0t+si‍∫t+si-τ∞‍×π~0i(τ+a-t-si,a)dadτds+∫[-r,0]‍∫0∞‍β0i(τ+t+si)ηi(τ)=∫[-r,0]‍∫0t+si‍∫t+si-τ∞‍×π~0i(τ,τ+t+si)dτds+ψi(t).
Conditions (57) and (59) imply that
(60)y(t)=[y1(t)⋮yn(t)],
where
(61)yi(t)=[pi(t)bi(t)],(62)yti(s)=yi(t+s)fors∈[-ri,0],i=1,2,…,n,
satisfy the matrix equation
(63)Ayt+∫0t‍K(t-τ)yτdτ=f(t),
where A and K are block diagonal matrices. We have
(64)A=[A1O…OOA2…O⋮⋮⋱⋮OO…An],K=[K1O…OOK2…O⋮⋮⋱⋮OO…Kn]
for zero matrix O, Ai=[10-κi1], Aiyt=[pti(0)0-κiptibti(0)]=[pi(t)0-κiptibi(t)] and(65)Ki(t)=[∫0∞‍π~0i(a,a+t)ωi(a)da-π0i(t)∫-ri0‍∫0∞‍β0i(a+t+s)π~0i(a,a+t+s)ωi(a)dads-∫-ri0‍β0i(t+s)π0i(t+s)ds].Sinceω(a) is the functional so in the formula of K(t) we consider ω as the function with functional values. Furthermore, in the notation (63) looks like the ordinary matrix equation, but in fact it is the operator one. Moreover,
(66)f(t)=[f1(t)⋮fn(t)]
for
(67)fi(t)=[f1i(t)f2i(t)]-Aiy1ti-∫0t‍Ki(t-τ)y1τidτf1i(t)=∫0t‍∫t-τ∞‍xi(τ+a-t,τ)π~0i(τ+a-t,a)dadτ000000+∫0∞‍ηi(τ)π~0i(τ,τ+t)dτ,f2i(t)=∫[-r,0]‍∫0t+si‍∫t+si-τ∞‍β0i(a)xi(τ+a-t-si,τ)00000000000000000000×π~0i(τ+a-t-si,a)dadτds000000+∫[-r,0]‍∫0∞‍β0i(τ+t+si)ηi(τ)00000000000000000×π~0i(τ,τ+t+si)dτds+ψi(t)
and y1i(t)=[pi(t)0]|[-ri,0], y1ti(s)=y1i(t+s) for s∈[-ri,0].We can notice that the last two elements of the expression definingfi(t) equal zero for t>r.We return to the previous deliberation in the next theorem. We take some additional assumptions.(H10)
The Fréchet derivativesDziλi(a,z0) and the derivation Dziβi(a,z0) for i=1,2…,n as functions of the variable a belong to L∞(ℝ+).(H11)
(Λi(a,ρ)/∥ρ∥0) and (Ωi(a,ρ)/∥ρ∥0) tend to zero as ∥ρ∥0→0 uniformly for a→0; here ∥·∥0 denotes the supremum norm in the Banach space C([-ri,0]).(H12)
λ*=infa⩾0{λi(a,z0),i=1,2,…,n}>0.Denote eθ:(-∞,0]→ℝ by the formula eθ(s)=eθs. The exponential asymptotical stability of the model is established by the following theorem.Theorem 4.
Letπ~1i(a,a+t)=∫[-r,0]‍β0i(a+t+si)π~0i(a,a+t+si)ds, and let π1i(t)=∫[-r,0]‍β0i(t  +  si)  π0i(t+si)ds. Let us assume that there exists some μ¯>0 that the equation
(68)1=∫0∞‍e-tγπ1i(t)dt-∫0∞‍e-tγ∫0∞‍π~0i(a,a+t)ωi(a)(eγ)dadt·(1-∫0∞‍e-tγπ1i(t)dt)-∫0∞‍e-tγπ0i(t)dt·(∫0∞‍e-tγ∫0∞‍π~1i(a,a+t)ωi(a)(eγ)dadt-κi(eγ))
has no solution γ with Re(γ)⩾-μ¯. Then there exist real numbers δ>0 and μ>0 such that for any initial data φ with ∥φ-u0∥L1<δ, the corresponding solution of the population problem (1)–(4), if it exists for t>0, satisfies
(69)∥zt-z0∥=O(e-μt),(70)∥u(a,t)-u0(a)∥=O(e-μt)foreacha∥u(a,t)-u0(a)∥=O(e-μt)ast→∞.Proof.
Letu be the solution of the population problem (1)–(4) for t>0. In the proof we will use the properties of the generalized Laplace transform. Let ϱ:[0,∞)→C([-ri,0])* be the function with measure value. Define
(71)ϱ^(θ)=∫0∞‍ϱ(t)(eθ)e-θtdt.
If we define the convolution by the formula
(72)(ϱ*vτ)(t)=∫0t‍ϱ(t-τ)v(τ+·)dτ,
where v(t)=0 for t<0, then we have the equality analogous to the property of classical Laplace transform, that is,
(73)ϱ*vτ^(θ)=ϱ^(θ)v^(θ).
Using generalized Laplace transform to (63) we get
(74)A^(θ)y^(θ)+K^(θ)y^(θ)=f^(θ),
where
(75)A^(θ)=[A^1(θ)O⋯OOA^2(θ)⋯O⋮⋮⋱⋮OO⋯A^n(θ)],A^i(θ)=[10-κi(eθ)1],K^(θ)=[K^1(θ)O⋯OOK^2(θ)⋯O⋮⋮⋱⋮OO⋯K^n(θ)],K^i(θ)=[∫0∞‍e-tθ∫0∞‍π~0i(a,a+t)ωi(a)(eθ)dadt-π^0i(θ)∫0∞‍e-tθ∫0∞‍π~1i(a,a+t)ωi(a)(eθ)dadt-π^1i(θ)]
for π~1i(a,a+t)=∫[-r,0]‍β0i(a+t+si)π~0i(a,a+t+si)ds and π1i(t)=∫[-r,0]‍β0i(t+si)π0i(t+si)ds. From the equality (68) we conclude that the matrix A^+K^ has an analytic inverse for Re(θ)⩾-μ¯. Moreover, we have
(76)A^-1(θ)=[(A^1)-1(θ)O⋯OO(A^2)-1(θ)⋯O⋮⋮⋱⋮OO⋯(A^n)-1(θ)],(A^i)-1(θ)=[10κi(eθ)1],
so A^ is analytic for θ∈ℂ.
The solution of the matrix equation (63) exists and is given by
(77)y(t)=A-1ft+∫0t‍J(t-τ)fτdτ,
where fτ(s)=[f1(τ+s)⋮fn(τ+s)], fτi(s)=fi(τ+s) for s∈[-ri,0] and
(78)J(t)=12πe-μ¯t∫-∞∞‍eiζtJ^(-μ¯+iζ)dζ.J^=(A^(θ)+K^(θ))-1-A^-1(θ) is the Laplace transform of the function J and
(79)|J(t)|⩽C1e-μ¯t,
where |J| denotes the sum of moduli of the elements of the matrix J. Here and in the whole proof C1, C2,  C3,… denote positive constants.
By(H12), (32), and (54)
(80)π0i(a)⩽e-λ*a,π~0i(a′,a)⩽e-λ*(a-a′)(a′⩽a).
Forλ*>μ¯ and ∥β0∥L∞<∞ we can estimate that
(81)∥f(t)∥⩽C2{∫0t‍e-λ*(t-τ)∥x(·,τ)∥L1dτ∥f(t)∥⩽C2000+∥η∥L1e-λ*t+∥ψ(t)∥L1∫0t‍},(82)∥y(t)∥⩽C3{∫0t‍∥η∥L1e-μ¯t+∥ψ(t)∥L1∥f(t)∥⩽C2000+∫0t‍e-μ¯(t-τ)[∥ψ(τ)∥L1+∥x(·,τ)∥L1]dτ},
where ∥y(t)∥=(∑i=1n‍(pi)2+(bi)2)1/2,
(83)∥ξ(·,t)∥L1⩽∫0t‍e-λ*τ∥b(t-τ)∥dτ+∥η∥L1e-λ*t+∫0t‍e-λ*(t-τ)(∥x(·,τ)∥L1+∥ω∥L1∥pτ∥0)dτ⩽C4{∫0t‍∥η∥L1e-μ¯te-00+∫0t‍e-μ¯(t-τ)∫0t‍e-μ¯(t-τ)000×[∥ψ(τ)∥L1+∥x(·,τ)∥L1]dτ∫0t‍}.
Let ε>0. By (H11) there exists δ~=δ~(ε) such that
(84)∥Λ(a,pt)∥⩽ε∥pt∥0,∥Ω(a,pt)∥⩽ε∥pt∥0
for ∥pt∥0<δ~. By the previous inequalities and the fact that ∥Dzλ(a,z0)∥L∞, ∥Dzβ(a,z0)∥L∞, and ∥u0∥L1 are finite, we can estimate that
(85)∥x(·,t)∥L1,∥ψ(t)∥L1⩽C5{∥pt∥0·∥ξ(·,t)∥L1+ε∥pt∥0}.
We require that ε<1 and δ~<ε. Hence ∥pt∥0<ε. According to the previous remarks, we have
(86)∥x(·,t)∥L1,∥ψ(t)∥L1⩽C5εσ(t),σ(t)=∥ξ(·,t)∥L1+∥pt∥0.
Thus (82)–(86) imply that
(87)σ(t)⩽M{∥η∥L1e-μ¯t+2ε∫0t‍e-μ¯(t-τ)σ(τ)dτ},
where M>0. Gronwall's inequality yields
(88)σ(t)⩽M∥η∥L1e(-μ¯+2Mε)t.
Let us choose δ and ε>0 such that μ=μ¯-2Mε>0 and δ<min(δ~(ε),δ~(ε)/M). Assume that ∥φ-u0∥L1=∥η∥L1<δ. In that case, from what has already been proved, it follows that ∥x(·,t)∥L1, ∥ψ(t)∥L1, σ(t) and ∥b(t)∥ are O(e-μt). Therefore (41), (86)2 imply (69) and (53)2, (80), and (39) imply (70).Example 5.
Letn=2. Let us consider
(89)λi(a,z)=λi(a,z11,z21,z12,z22),βi(a,z)=βi(a,z1,z2),
where
(90)λi(a,z)=λi(z1(0),z1(-r1),z2(0),z2(-r2)),βi(a,z)=βi(z1,z2)e-αia
with λi>0,  βi>0,  ri>0,  αi>0 for i=1,2. Let us define
(91)λ0i=λi(z01,z01,z02,z02),β0i=βi(z01,z02)e-αia
for an arbitrary z0=(z01,z02)∈ℝ2. From (32) and (33) we conclude that
(92)π0i(a)=e-aλ0i,Ri(z0)=r1r2β0iαi+λ0i.
By Theorem 3 there exists an equilibrium age distribution u0i(a)=z0iλ0ie-λ0ia where B0=z0iλ0i if and only if r1r2β0i/(αi+λ0i)=1. To investigate the stability of the equilibrium age distribution we consider (68). Let z0=(z01,z02) be the solution of the system of the equations
(93)Ri(z0)=1,i=1,2.
First, let us notice that
(94)ωi(eγ)=(∂λi∂z1i+∂λi∂z2ie-γri)λ0iz0ie-aλ0i
with the derivatives in the point z0. It is easy to notice that |∂λi/∂z1i+(∂λi/∂z2i)e-γri| is bounded on half-plane Re(γ)>μ for every real μ (also negative). Analogously
(95)κi(eγ)=λ0iβ0i·z0iri·∂βi∂zi(z0)∫-ri0‍eγsds.
It is also possible to prove that for every real μ the function |κi(eγ)| is bounded on half-plane Re(γ)>μ. Consider the right-hand side of (68). All components can be presented in the form
(96)AB+γ,
where A is bounded and B⩾max{αi,λ0i}. Let μ<B, and let Re(γ)>-μ:
(97)|AB+γ|=|AB-μ+μ+γ|≤|AB-μ+μ+Re(γ)|≤|A2B-μμ+Re(γ)|.
Fix σ>0. For B sufficiently large there exist some μ and μ¯<μ that
(98)|A2B-μμ-μ¯|<σ.
For Re(γ)>-μ¯ we have
(99)|AB+γ|⩽|A2B-μμ+Re(γ)|⩽|A2B-μμ-μ¯|<σ.
In consequence for αi and λ0i sufficiently large, we can find that μ¯>0 such that the modulus of the right-hand side of (68) is less than 1 for every γ, for which Re(γ)>-μ¯.



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*Source: 908768-2013-04-29.xml*