CHAPTER 4. SINGULARLY PERTURBED MODEL 28
CHAPTER 4. SINGULARLY PERTURBED MODEL 29 defineL(b,a) asL(b)L−1(a). The (i, j)th elementli j(b,a) ofL(b,a) denotes the probability that a person in the jth state at ageawill survive to agebin theith state. We can prove the following.
Lemma 4.6 ForL(a,b), the following holds:
1. L(b,a)is nonnegative, 2. kL(b,a)k ≤e−µ(b−a),
where the operator norm is related to the l1-type norm inRn. Proof. 1. Clearly,L(b,a) satisfies
d
dbL(b,a)=Q(b)L(b,a), L(a,a)=I.
Letη:=sup
i,a
|qii(a)|. ThenQ(a)+ηIis a nonnegative matrix, and we have d
db
nL(b,a)eη(b−a)o
=
Q(b)+ηIL(b,a)eη(b−a). By Picard’s iteration method, we have the following representation
L(b,a)eη(b−a) =I+ Z b
a
Q(ρ)+ηI dρ +
Z b a
Q(ρ1)+ηI Z ρ1
a
Q(ρ2)+ηI dρ2dρ1+· · · .
Since the right-hand side is nonnegative, we can conclude thatL(b,a) is nonnegative.
2. Next, letli j(b,a) denote the (i, j)th element ofL(b,a). Then d
dbli j(b,a)= Xn
k=1
qik(b)lk j(b,a), where li j(a,a)=δi j, and summing over indexi, we have
d db
Xn
i=1
li j(b,a)= Xn
k=1
Xn
i=1
qik(b)lk j(b,a)= Xn
k=1
−µk(b)
lk j(b,a)
≤(−µ) Xn
i=1
li j(b,a).
Thus we obtain
Xn
i=1
li j(b,a)≤e−µ(b−a).
CHAPTER 4. SINGULARLY PERTURBED MODEL 30 This shows thatkL(b,a)k ≤e−µ(b−a). This completes the proof.
In the following series of lemmas, we are going to prove the second condition of Theorem 2.18, and that will subsequently prove our main Theorem 4.5.
Lemma 4.7 Ais a closed linear operator in L1(0,w; Rn).
Proof. Linearity ofA follows immediately from linear properties of differentiation and matrix multiplication.
Next, in order to show that A is a closed operator, we prove that if φn ∈ D(A),
nlim→∞φn =: φ ∈ L1([0,w],Rn) and lim
n→∞Aφn =: v ∈ L1([0,w],Rn), then φ ∈ D(A) and Aφ=v.
Since
Aφn
(a)=− d
daφn(a)+Q(a)φn(a), it follows that
Z a 0
Aφn
(σ)dσ=− Z a
0
d
dσφn(σ)dσ+ Z a
0
Q(σ)φn(σ)dσ
=−φn(a)+φn(0)L(a)+ Z a
0
Q(σ)φn(σ)dσ or, φn(0)=φn(a)+
Z a 0
Aφn
(σ)dσ+ Z a
0
Q(a)φn(σ)dσ.
Then we find that
φn(0)−φm(0) Rn
kLk
L1 ≤
φn−φm
L1 +w
Aφn−Aφm
L1 +w
Qφn−Qφm L1. Hence{φn(0)}n∈
Nis a Cauchy sequence. Let lim
n→∞φn(0)=α. If we define ω(a) :=αL(a)−
Z a
0
v(σ)L(a, σ)dσ,
then we can see thatωis absolutely continuous, because the right hand side is differ- entiable almost everywhere and its derivative is integrable on [0,w] (we assumed that Q(a) is bounded in [0,w]). Also we have
φn(a)−ω(a)=φn(0)L(a)− Z a
0
Aφn
(σ)L(a, σ)dσ−αL(a)+ Z a
0
v(σ)L(a, σ)dσ,
CHAPTER 4. SINGULARLY PERTURBED MODEL 31 which shows that limn→∞
φn−ω
L1 = 0, so φ = ωalmost everywhere. Moreover, from φn(0) = R w
0 B(s)φn(s)ds we haveα = Rw
0 B(s)φ(s)dswhen n → ∞. Since φ = ω almost everywhere, we obtain
α=ω(0)= Z w
0
B(s)ω(s)ds.
Since we can identifyφasω(as an element ofL1) by modifying the values on the null set, so we can say thatφ ∈D(A) andAφ =v. This completes the proof.
Lemma 4.8 ρ(A)⊃ {λ: Reλ >β¯−µ}.
Proof. [28] Define the characteristic matrix ˜Γ: C→B(Rn,Rn) by Γ(λ) :=˜
Z w 0
Γ(a)e−λada,
with Γ(a) :=B(a)L(a).
Let us first prove the following result which we are going to use in this lemma.
σ(A)=n
λ:λ ∈C, det
I−Γ(λ)˜
=0o
. (4.6)
To prove (4.6), let λ ∈ C be such that det
I−Γ(λ)˜
= 0. Then there must exist x∈Rn,x,0 such that
I−Γ(λ)˜
x=0. Define
φ(0) :=x, (4.7)
φ(a) :=e−λaL(a)φ(0), a≥0. (4.8) Thenφ,0and using
I−Γ(λ)˜
x=0we get φ(0)=
Z w 0
e−λaB(a)L(a)φ(0)da
= Z w
0
B(a)φ(a)da,
i.e.,φ∈D(A). Differentiating (4.8) with respect toa, we get φ0(a)=−λe−λaL(a)φ(0)+e−λa d
daL(a)φ(0)
=−λe−λaL(a)φ(0)+e−λaQ(a)L(a)φ(0) (by (4.4))
=−λφ(a)+Q(a)φ(a) (by (4.8)
or, λφ(a)=−φ0(a)+Q(a)φ(a)=Aφ(a),
CHAPTER 4. SINGULARLY PERTURBED MODEL 32 and this implies λ ∈ σ(A). Now to prove our lemma, let F(λ), λ ∈ R denote the Frobenius root of the characteristic matrix ˜Γ(λ), let ˜Γi j(λ) be the (i, j)th element of Γ(λ). Let˜ Sp(A) denote the spectral radius of the operator A. Then it can be shown that Sp( ˜Γ(λ))≤F(λ) [24], p.57
. From
Γ(Reλ)˜ ≤
Z w 0
kΓ(v)k e−vReλdv
≤ Z w
0
kB(v)k kL(v)k e−vReλdv
≤β¯ Z w
0
e−
µ+Reλ
vdv by Lemma 4.6 and assumption (A3)
≤ β¯ Reλ+µ
1−e−
Reλ+µ w
, we obtain
F(Reλ)≤max
j n
X
i=1
Γ˜ij(Reλ)=
Γ(Reλ)˜
(4.9)
≤ β¯ Reλ+µ
h1−e−(Reλ+µ)wi ,
where for the proof of the inequality in (4.9), we refer to [24, p.63]. Therefore if Reλ > β¯ − µ then Sp( ˜Γ(λ))≤F(Reλ)<1. Thus det(I−Γ(λ))˜ ,0 for Reλ >β¯−µ.
Hence using (4.6), we getρ(A)⊃ {λ:Reλ >β¯−µ}. Lemma 4.9 Ifλ∈ρ(A)withλ >β¯−µ, then R(λ,A)is given by
R(λ,A)ψ(a)=e−λaL(a) I− Z w
0
B(b)L(b)e−λbdb
!−1
× Z w
0
B(a)L(a)e−λa Z a
0
eλbL−1(b)ψ(b)db da+e−λaL(a) Z a
0
eλbL−1(b)ψ(b)db.
Proof. Following [65], define
∆(λ)x:=x− Z w
0
e−λaB(a)L(a)xda,
wherex∈Rnand Reλ > β−µ. Letψ∈L1. There existsφ∈L1satisfying (λI−A)φ=ψ if and only if
λφ(a)+φ0(a)−Q(a)φ(a)=ψ(a), (4.10)
φ(0)= Z w
0
B(a)φ(a)da. (4.11)
CHAPTER 4. SINGULARLY PERTURBED MODEL 33
The solution of (4.10) is given by (see [42, Proposition 5.2, p. 242]) φ(a)=e−λaL(a)φ(0)+
Z a 0
e−λ(a−b)L(a,b)ψ(b)db. (4.12) Substituting this formula forφinto the boundary equation (4.11), we obtain
φ(0)= Z w
0
e−λaB(a)L(a)φ(0)da+ Z w
0
B(a)
"Z a 0
e−λ(a−b)L(a,b)ψ(b)db
# da.
Sinceλ > β−µ,λ<σ(A), so that∆(λ)−1exists and thus we get φ(0)= I−
Z w 0
e−λaB(a)L(a)da
!−1Z w 0
B(a)
"Z a 0
e−λ(a−b)L(a)L−1(b)ψ(b)db
#
da. (4.13) Hence, putting the expression ofφ(0) from (4.13) into the equation (4.12), we get
φ(a)=R(λ,A)ψ(a)=e−λaL(a) I− Z w
0
B(b)L(b)e−λbdb
!−1
× Z w
0
B(a)L(a)e−λa Z a
0
eλbL−1(b)ψ(b)db da+ e−λaL(a) Z a
0
eλbL−1(b)ψ(b)db.
Lemma 4.10 D(A)=L1(0,w; Cn).
Proof. To prove the result, we follow [28] but we will go through the detailed calcu- lations. If λ >β¯−µ, we can defineφλ = λ(λI−A)−1ψfor allψ ∈ L1(0,w; Cn). Since φλ ∈ D(A), it is sufficient to show thatφλ → ψasλ → ∞inL1(0,w;Cn). Now using the expression of the resolvent from Lemma 4.9 we have
φλ(a)=λe−λaL(a) I− Z w
0
B(b)L(b)e−λbdb
!−1
× Z w
0
B(u)L(u)e−λu Z u
0
eλvL−1(v)ψ(v)dv du+ λe−λaL(a) Z a
0
eλvL−1(v)ψ(v)dv.
We can write
φλ−ψ L
1
≤J1+J2, where
J1=λ Z w
0
e−λaL(a) I− Z w
0
B(b)L(b)e−λbdb
!−1
× Z w
0
B(u)L(u)e−λu Z u
0
eλvL−1(v)ψ(v)dv du da,
CHAPTER 4. SINGULARLY PERTURBED MODEL 34 and
J2= Z w
0
λe−λaL(a) Z a
0
eλvL−1(v)ψ(v)dv−ψ(a)
da. (4.14)
Now
I− Z w
0
e−λaB(a)L(a)da
≥ 1−
Z w 0
e−λaL(a)B(a)da
≥1−β Z w
0
e−(λ+µ)ada
∵β Z w
0
e−(λ+µ)ada< β¯ λ+µ<1
or,
I− Z w
0
e−λaB(a)L(a)da
!−1
≤ λ+µ
λ−(β−µ)+βe−(λ+µ)w ≤ λ+µ λ−(β−µ). So
J1 ≤λ Z w
0
e−(λ+µ)a
I− Z w
0
e−λaB(a)L(a)da
!−1 Z w
0
B(u)L(u)e−λu ×
Z u 0
eλv
L−1(v)ψ(v) dv du
# da
≤λβ λ+µ λ−(β−µ)
Z w 0
"
e−(λ+µ)a Z w
0
e−(λ+µ)u Z u
0
e(λ+µ)v ψ(v)
dv du
# da
=λβ λ+µ λ−(β−µ)
Z w 0
"
e−(λ+µ)a Z w
0
e(λ+µ)v ψ(v)
Z w v
e−(λ+µ)udu dv
# da
≤λβ 1 λ−(β−µ)
Z w 0
"
e−(λ+µ)a Z w
0
e−(λ+µ)v.e(λ+µ)v ψ(v)
dv
# da
=λβ 1 λ−(β−µ).
1−e−(λ+µ)w
λ+µ .
ψ L
1
or J1 ≤λ1−e−(λ+µ)w λ−( ¯β−µ). β¯
λ+µ ψ
L
1 , (4.15)
and hence lim
λ→∞J1 =0.
CHAPTER 4. SINGULARLY PERTURBED MODEL 35 To show limλ→∞J2 =0, we notice that the expressionJ2in (4.14) can be written as J2 =
Z w 0
λL(a)
Z a 0
e−λ(a−v)L−1(v)ψ(v)dv−ψ(a) da,
≤ Z w
0
kL(a)k
Z a
0
λe−λ(a−v)L−1(v)ψ(v)dv−L−1(a)ψ(a)
da
≤ Z w
0
λ
Z a 0
λe−µae−λ(a−v)L−1(v)ψ(v)dv−e−µaL−1(a)ψ(a)
da
∵kL(a)k ≤e−µa
= Z w
0
Z a 0
λe−µte−λte−µ(a−t)L−1(a−t)ψ(a−t)dt−e−µaL−1(a)ψ(a)
da (wheret:=a−v)
= Z w
0
Z a 0
λe−λte−µtψ(a¯ −t)dt−ψ(a)¯
da, (4.16)
where ¯ψ(a) :=e−µaL−1(a)ψ(a) and since
ψ(a)¯
=e−µa
L−1(a) ψ(a)
≤e−µaeµa ψ(a)
=
ψ(a) ,
we have ¯ψ∈L1. Now, following [65], to prove that the expression (4.16) tends to zero asλ→ ∞, lett>0 and defineψt ∈L1by
ψt(a) :=
0, a<t;
e−µtψ(a¯ −t), a>t.
Since
ψt−ψ¯ L1 =
Z ∞
0
ψt(a)−ψ(a)¯ da
= Z t
0
ψ(a)¯
da+
Z ∞
t
e−µtψ(a¯ −t)−ψ(a)¯ da
≤ Z t
0
ψ(a)¯
da+
Z ∞
t
e−µt−1
ψ(a¯ −t) da+
Z ∞
t
ψ(a¯ −t)−ψ(a)¯ da
= Z t
0
ψ(a)¯
da+
Z ∞
0
e−µt−1 ψ(z)¯
dz+
Z ∞
0
ψ(z)¯ −ψ(t¯ +z) dz,
(4.17) we claim that
limt→0
ψt−ψ¯
L1 =0. (4.18)
To prove (4.18), the last term from (4.17) needs some attention and we show that limt→0
Z ∞
0
ψ(t¯ +z)−ψ(z)¯
dz=0
or, lim
t→0
ψ¯t−ψ¯
L1 =0, (4.19)
where ¯ψt(z) :=ψ(t¯ +z).
CHAPTER 4. SINGULARLY PERTURBED MODEL 36
Now as the continuous functions with compact support are dense inL1([59, Theorem 2.4]), the proof of the fact (4.19) is a simple consequence of the approximation of integrable functions by continuous functions with compact support. In fact for any >0, we can find such a functiongsuch thatkψ¯ −gk
L1 < . Now ψ¯t−ψ¯ = gt−g+
ψ¯t−gt
− ψ¯ −g,
wheregt(x) :=g(t+x). However,kψ¯t−gtkL1 =kψ¯ −gkL1 < , while sincegis continuous and has compact support, we have
gt−g L1 =
Z ∞
0
g(t+z)−g(z)
dz→0 as t→0.
Therefore, if |t| < δ, whereδ is sufficiently small, then kgt −gkL1 < , and as a result kψ¯t−ψ¯k
L1 <3, whenever|t|< δ. This proves (4.19) and subsequently we have (4.18).
Now from (4.16), we have J2 ≤
Z ∞ 0
Z a
0
λe−λte−µtψ(a¯ −t)dt−ψ(a)¯
da
= Z ∞
0
Z a 0
λe−λte−µtψ(a¯ −t)dt− Z ∞
0
λe−λtψ(a)¯ dt
da ∵ Z ∞
0
λe−λtdt=1
!
= Z ∞
0
Z ∞
0
λe−λth
ψt(a)−ψ(a)¯ i dt
da
≤ Z ∞
0
( λe−λt
Z ∞
0
ψt(a)−ψ(a)¯ da
) dt
≤ sup
0≤t≤
ψt−ψ¯ L1
Z
0
λe−λtdt+2 ψ¯
L1
Z ∞
λe−λtdt
≤ sup
0≤t≤
ψt−ψ¯
L1 +2 ψ¯
L1e−λ. Therefore, using (4.18), we find lim
λ→∞J2 =0.Hence we can conclude that
λlim→∞
φλ−ψ L1 =0,
and this completes the proof.
Lemma 4.11 k(λI−A)−1k ≤ 1 λ−( ¯β−µ).
Proof. From the proof of Lemma (4.10) we have
(λI−A)−1ψ ≤ 1
λJ1+ Z w
0
e−λa Z a
0
eλvkL(a,v)k ψ(v)
dv da,
CHAPTER 4. SINGULARLY PERTURBED MODEL 37
where, from (4.15), we have
J1 ≤ λβ¯
λ+µ. 1
λ−( ¯β−µ). ψ
L1 ,
and Z w
0
e−λa Z a
0
eλv kL(a,v)k ψ(v)
dv da≤ 1 λ+µ.
ψ L1. So, we have
(λI−A)−1 ≤ 1
λ+µ
β¯
λ−( ¯β−µ) +1
= 1
λ−( ¯β−µ).
Proof of Theorem 4.5. Considering above lemmas, we can see that (A,D(A)) satisfies condition (2) of Hille–Yosida Theorem 2.18. Hence, the proof of Theorem 4.5 follows immediately due to equivalence conditions (1) and (2) in theorem 2.18.
The specific spectral properties of the migration matrix what has been discussed in this chapter is going to be used in the asymptotic analysis of the model in next chapter. Also the existence of the C0-semigroup (Theorem 4.5) is one of the crucial result needed for all of our further analysis.
CHAPTER 5
Asymptotic Analysis of the Perturbed Model
5.1 Aggregated Model
Biological heuristics suggests that no geographical structure should persist for very large interstate migration rates; that is, for →0. Here we also note that both biological and mathematical analysis rely onλ=0 being the dominant simple eigenvalue ofC(a) for eacha∈R+with the corresponding positive eigenvector, denoted byk(a), and the left eigenvector 1 = (1,1, . . . ,1). Vector k(a) is normalized to satisfy 1 ·k = 1 and k(a) = (k1(a), . . . ,kn(a)) is the so-called stable patch structure; that is, the asymptotic (ast → ∞ and disregarding demographic processes) distribution of the population among the patches for a given agea. Thus, in population theory, the components ofk are approximated aski ≈ui/ufori=1, . . . ,n, where
u:=u·1= Xn
i=1
ui.
Adding together equations in (4.1) and using the above approximation we obtain
ut ≈ −ua−µ∗u, (5.1)
where µ∗ := 1 ·Mk is the ‘aggregated’ mortality. This model, supplemented with boundary condition
u(0,t)≈ Z ∞
0
β∗(a)u(a,t)da, (5.2)
38
CHAPTER 5. ASYMPTOTIC ANALYSIS OF THE PERTURBED MODEL 39 whereβ∗ :=1·Bkis the ‘aggregated fertility’, is expected to provide an approximate description of the averaged population. Thus, (5.1) is the macroscopic and (4.1) the mesoscopic description of the population.
The main goal of this chapter is a rigorous validation of the above heuristics;
that is, that the true total populationucan be approximated by the solution ¯uof the aggregated problem (5.1)–(5.2) (where ‘≈’ is replaced by ‘=’) with an -order error.
The analysis is involved due to the initial and boundary conditions which are not consistent with those of the aggregated model. This makes the problem singularly perturbed and thus necessitates a careful analysis of the boundary, corner and initial layer phenomena.