Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 56, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF POSITIVE ALMOST-PERIODIC SOLUTIONS
FOR A NICHOLSON’S BLOWFLIES MODEL
QING-LONG LIU, HUI-SHENG DING
Abstract. This article concerns the existence of solutions to a first-order
differential equation with time-varying delay, which is known as a Nicholson’s
blowflies model. By using fixed a point theorem and the Lyapunov functional
method, we establish the existence and locally exponential stability of almost
periodic solutions for the model. Also, we apply our results to two examples,
for which some earlier results can not be applied.
1. Introduction and preliminaries
Nicholson [10] and Gurney [5] proposed the following delay differential equation
as a population model
x0 (t) = −δx(t) + P x(t − τ )e−ax(t−τ ) ,
(1.1)
where x(t) is the size of the population at time t, P is the maximum per capita daily
egg production, 1/a is the size at which the population reproduces at its maximum
rate, δ is the per capita daily adult death rate, τ is the generation time.
Nicholson’s blowflies model and its analogous equations have attracted much
attention. Especially, equation (1.1) and its variants have been of great interest for
many mathematicians. As pointed out in [6], compared with periodic effects, almost
periodic effects are more frequent in many biological dynamic systems. Therefore,
recently, the existence and stability of almost periodic solutions for (1.1) and its
variants have attracted more and more attention. We refer the reader to [1, 2, 3,
4, 7, 8, 9] and references therein for some recent development on such topic.
Motivated by the above works, we study further this topic, by investigate the
Nicholson’s blowflies model
m
X
x0 (t) = −α(t)x(t) +
βj (t)x(t − τj (t))e−γj (t)x(t−τj (t)) ,
(1.2)
j=1
where m is a given positive integer, and α, βj , γj , τj : R → (0, +∞) are almost
periodic functions, j = 1, 2, . . . , m.
2000 Mathematics Subject Classification. 34C27, 34K14.
Key words and phrases. Nicholson’s blowflies model; almost periodic solution;
exponential stability.
c
2013
Texas State University - San Marcos.
Submitted October 15, 2012. Published February 21, 2013.
1
2
Q.-L. LIU, H.-S. DING
EJDE-2013/56
In this article, for a bounded continuous function g on R, we denote
g + = sup g(t),
g − = inf g(t).
t∈R
t∈R
In addition, for j = 1, 2, . . . , m, we assume that
βj− > 0,
α− > 0,
γj− > 0,
τj− > 0.
Next, we recall some notation and basic results about almost periodic functions.
For more details, we refer the reader to [6].
Definition 1.1. A continuous function f : R → R is called almost periodic if for
each ε > 0, there exists l(ε) > 0 such that every interval I of length l(ε) contains a
number τ with the property that
sup |f (t + τ ) − f (t)| < ε.
t∈R
We denote the set of all such functions by AP (R).
Definition 1.2. Let x(·) and Q(·) be n × n continuous matrix defined on R. The
linear system
x0 (t) = Q(t)x(t)
(1.3)
is said to admit an exponential dichotomy on R if there exist positive constants
k, α and a projection P such that
kX(t)P X −1 (s)k ≤ ke−α(t−s) ,
t ≥ s,
kX(t)(I − P )X −1 (s)k ≤ ke−α(s−t) ,
t ≤ s,
for a fundamental solution matrix X(t) of (1.3).
Lemma 1.3. If the linear system (1.3) admits an exponential dichotomy, then the
almost periodic system
x0 (t) = Q(t)x(t) + g(t)
has a unique almost periodic solution x(t) given by
Z t
Z +∞
−1
x(t) =
X(t)P X (s)g(s)ds −
X(t)(I − P )X −1 (s)g(s)ds.
−∞
t
2. Main results
2.1. Existence of almost periodic solutions. For the next theorem we use the
following two assumptions:
(H1) n2 ≥ n1 ≥ min 1 {γ + } , where
1≤j≤m
n2 =
(H2)
βj+
j=1 α−
Pm
j
m
X
βj+
j=1
α− γj− e
k
+
,
n1 =
m
X
n2 βj− e−γj
j=1
α+
n2
.
e
< min{ |1−k|
, e2 }, where k = n1 min1≤j≤m {γj− }. Moreover, if
k = 1, we denote
ek
min
, e2 = e2 .
|1 − k|
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EXISTENCE OF POSITIVE ALMOST-PERIODIC SOLUTIONS
3
Theorem 2.1. Under assumptions (H1)–(H2), Equation (1.2) has a unique almost
periodic solution in
Ω = {ϕ ∈ AP (R) : n1 ≤ ϕ(t) ≤ n2 , ∀t ∈ R}.
Proof. For each ϕ ∈ AP (R), we consider the almost periodic differential equation
x0 (t) = −α(t)x(t) +
m
X
βj (t)ϕ(t − τj (t))e−γj (t)ϕ(t−τj (t)) .
(2.1)
j=1
Since α− > 0, we know that
x0 (t) = −α(t)x(t)
admits an exponential dichotomy on R with P = I. Combining this with Lemma
1.3, we conclude that (2.1) has a unique almost periodic solution:
Z t
m
X
Rt
xϕ (t) =
e− s α(u)du
βj (s)ϕ(s − τj (s))e−γj (s)ϕ(s−τj (s)) ds.
(2.2)
−∞
j=1
Now, we define a mapping T on AP (R) by
(T ϕ)(t) = xϕ (t),
t ∈ R.
Next, we show that T (Ω) ⊂ Ω. It suffices to prove that
n1 ≤ (T ϕ)(t) ≤ n2
for all t ∈ R and ϕ ∈ Ω. Noticing that
−
sup xe−γj
x
1
1 ≤ j ≤ m,
,
γj− e
=
x≥0
(2.3)
for all t ∈ R and ϕ ∈ Ω, we have
Z t
m
X
Rt
(T ϕ)(t) = xϕ (t) =
e− s α(u)du
βj (s)ϕ(s − τj (s))e−γj (s)ϕ(s−τj (s)) ds
−∞
Z
j=1
t
e−
≤
Rt
α(u)du
s
m
X
−∞
Z
e−
Rt
α(u)du
s
m
X
βj+ −∞
≤
=
=
j=1
Z
m
X
βj+
γ−e
j=1 j
ds
γ−e
j=1 j
t
e−
Rt
s
γj− e
α− du
t
ds
eα
−
(s−t)
ds
−∞
m
X
βj+
j=1
α− γj− e
= n2 .
On the other hand, by (H1), we know that n1 ≥
n1 ≤x≤n2
ds
−∞
Z
m
X
βj+
inf
ϕ(s−τj (s))
j=1
t
≤
−
βj+ ϕ(s − τj (s))e−γj
+
xe−γj
x
+
= n2 e−γj
n2
,
1
.
min1≤j≤m {γj+ }
1 ≤ j ≤ m,
Thus, we have
(2.4)
4
Q.-L. LIU, H.-S. DING
EJDE-2013/56
which yields
(T ϕ)(t) = xϕ (t) =
Z
t
e−
Rt
s
α(u)du
m
X
α(u)du
m
X
α(u)du
j=1
m
X
−∞
Z
j=1
t
e−
≥
Rt
s
−∞
Z
t
e−
≥
Rt
s
−∞
≥
≥
m
X
j=1
m
X
+
βj− ϕ(s − τj (s))e−γj
+
βj− n2 e−γj
n2
ϕ(s−τj (s))
ds
ds
j=1
+
βj− n2 e−γj
+
βj− n2 e−γj
n2
Z
n2
e−
Rt
s
α+ du
ds
Z
t
+
eα
(s−t)
ds
−∞
+
m
X
n2 βj− e−γj
n2
= n1
α+
j=1
t
−∞
j=1
=
βj (s)ϕ(s − τj (s))e−γj (s)ϕ(s−τj (s)) ds
for all t ∈ R and ϕ ∈ Ω.
By a direct calculation, one can obtain
|xe−x − ye−y | ≤ max
|1 − k| 1 , 2 |x − y|,
ek
e
x, y ≥ k.
(2.5)
Denoting
Γj (s) = γj (s)ϕ(s − τj (s))e−γj (s)ϕ(s−τj (s)) − γj (s)ψ(s − τj (s))e−γj (s)ψ(s−τj (s)) ,
for ϕ, ψ ∈ Ω, we have
kT ϕ − T ψk
= sup |(T ϕ)(t) − (T ψ)(t)|
t∈R
Z
= sup t∈R
≤ max
≤ max
= max
t
e−
Rt
s
α(u)du
−∞
m
X
βj (s)
j=1
|1 − k| 1 , 2 sup
ek
e t∈R
Z
γj (s)
t
e−
Γj (s)ds
Rt
s
α− du
−∞
m
X
m
X
βj+ |ϕ(s − τj (s)) − ψ(s − τj (s))|ds
j=1
|1 − k| 1 β + kϕ − ψk · sup
, 2
ek
e j=1 j
t∈R
Z
t
eα
−
(s−t)
ds
−∞
m
|1 − k| 1 X
βj+
,
kϕ − ψk
ek
e2 j=1 α−
βj+
j=1 α−
kϕ
ek
min{ |1−k|
, e2 }
Pm
=
− ψk.
Now, by (H2), T has a unique fixed point in Ω, i.e., Equation (1.2) has a unique
almost periodic solution in Ω.
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EXISTENCE OF POSITIVE ALMOST-PERIODIC SOLUTIONS
5
Remark 2.2. Compared with some earlier results (cf. [4, 9]), in Theorem 2.1, we
removed the following two restrictive conditions:
1
γj− > 1, n1 ≥
.
min1≤j≤m {γj− }
For the next corollary we use the assumption
(H3) there exists λ > 0 such that
Pm
ek
j=1 kβj kS λ
<
min{
, e2 },
−
|1 − k|
1 − e−λα
where
k = n1 ·
Z
min {γj− },
1≤j≤m
t+λ
and kβj kS λ = sup
t∈R
βj (s)ds.
t
Corollary 2.3. Under assumptions (H1), (H3), Equation (1.2) has a unique almost
periodic solution in Ω.
Proof. Let T be the self mapping on Ω in Theorem 2.1, i.e.,
Z t
m
X
R
− st α(u)du
(T ϕ)(t) =
e
βj (s)ϕ(s − τj (s))e−γj (s)ϕ(s−τj (s)) ds
−∞
j=1
for ϕ ∈ Ω and t ∈ R. For all t ∈ R and ϕ, ψ ∈ Ω, by (2.5), we obtain
|(T ϕ)(t) − (T ψ)(t)|
≤ max
|1 − k| 1 , 2
ek
e
Z
t
e−α
−
(t−s)
−∞
m
X
βj (s)|ϕ(s − τj (s)) − ψ(s − τj (s))|ds
j=1
m Z t
X
|1 − k| 1 −
, 2 kϕ − ψk ·
e−α (t−s) βj (s)ds
≤ max
ek
e
j=1 −∞
m
X Z +∞
|1 − k| 1 −
= max
,
kϕ
−
ψk
e−α s βj (t − s)ds
k
2
e
e
j=1 0
= max
m X
+∞ Z (k+1)λ
X
|1 − k| 1 −
,
kϕ
−
ψk
·
e−α s βj (t − s)ds
k
2
e
e
kλ
j=1
k=0
≤ max
= max
m X
+∞
X
|1 − k| 1 , 2 kϕ − ψk
ek
e
j=1
k=0
−
e−α
kλ
Z
(k+1)λ
βj (t − s)ds
kλ
Z t−kλ
m X
+∞
X
|1 − k| 1 −α− kλ
,
kϕ
−
ψk
e
βj (s)ds
ek
e2
t−kλ−λ
j=1
k=0
m X
+∞
X
|1 − k| 1 −
, 2 kϕ − ψk
e−α kλ kβj kS λ
k
e
e
j=1 k=0
Pm
|1 − k| 1 j=1 kβj kS λ
= max
, 2 kϕ − ψk
.
ek
e
1 − e−λα−
Then, by (H3), T is a contraction, and thus (1.2) has a unique almost periodic
solution in Ω.
≤ max
6
Q.-L. LIU, H.-S. DING
EJDE-2013/56
Remark 2.4. Note that in some cases, (H3) is weaker than (H2) (see for example
2.8).
2.2. Locally exponential stability of the almost periodic solution.
Theorem 2.5. Suppose that (H1), (H2) are satisfied, x(t) is the unique almost
periodic solution of (1.2) in Ω, and y(t) is an arbitrary solution of (1.2) with
y(t) ≥ n1 ,
where r =
max1≤j≤m {τj+ }.
∀t ∈ [−r, +∞),
(2.6)
Then, there exists a constant λ > 0 such that
|x(t) − y(t)| ≤ M e−λt ,
∀t ∈ [−r, +∞),
where M = max−r≤t≤0 |x(t) − y(t)|.
Proof. Let e
k = max{ |1−k|
, e12 }. Then, by (H2), we have
ek
m
X
βj+
1
< .
−
e
α
k
j=1
Thus, there exists λ > 0 such that
λ − α− + e
k
m
X
βj+ eλr < 0.
(2.7)
j=1
Next, setting z(t) = x(t) − y(t), we consider Lyapunov functional
V (t) = |x(t) − y(t)|eλt = |z(t)|eλt .
It can see easily that
V (t) ≤ M,
t ∈ [−r, 0].
Now, we claim that
V (t) ≤ M,
t ∈ (0, +∞).
(2.8)
Otherwise, the set
S = {t > 0 : V (t) > M } =
6 ∅.
We denote t1 = inf S. It is not difficult to prove that
V (t1 ) = M,
V (t) ≤ M,
∀t ∈ (0, t1 ).
0
In addition, for each δ > 0, there exists t ∈ (t1 , t1 + δ) such that V (t0 ) ≥ M . It
follows that
V (t) − V (t1 )
≥ 0.
D+ V (t1 ) = inf
sup
δ>0 t∈(t1 ,t1 +δ)
t − t1
On the other hand, by using (2.5), we obtain
D+ V (t1 ) = D+ (|z(t1 )|eλt1 )
≤ eλt1 D+ |z(t1 )| + λ|z(t1 )|eλt1
≤ eλt1 sgn(z(t1 ))z 0 (t1 ) + λ|z(t1 )|eλt1
m
X
βj (t1 )
≤ eλt1 − α(t1 )z(t1 )sgn(z(t1 )) +
|Γj (t1 )| + λ|z(t1 )|eλt1
γ (t )
j=1 j 1
≤ −α(t1 )eλt1 |z(t1 )| + e
k
m
X
j=1
βj+ |z(t1 − τj (t1 ))|eλt1 + λ|z(t1 )|eλt1
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EXISTENCE OF POSITIVE ALMOST-PERIODIC SOLUTIONS
≤ −α− M + e
k
m
X
7
βj+ M eλτj (t1 ) + λM
j=1
m
X
≤ (λ − α− + e
k
βj+ eλr )M,
j=1
where
Γj (t1 ) = γj (t1 )x(t1 −τj (t1 ))e−γj (t1 )x(t1 −τj (t1 )) −γj (t1 )y(s−τj (t1 ))e−γj (t1 )y(t1 −τj (t1 )) .
Noting that D+ V (t1 ) ≥ 0, we obtain
λ − α− + e
k
m
X
βj+ eλr ≥ 0,
j=1
which contradicts (2.7). Hence, (2.8) holds; i.e.,
|x(t) − y(t)| ≤ M e−λt ,
t ∈ [−r, +∞).
This completes the proof.
2.3. Examples. In this section, we give two examples to illustrate our results.
Example 2.6. We consider the following Nicholson’s blowflies model with multiple
time-varying delays:
x0 (t) = −α(t)x(t) +
m
X
βj (t)x(t − τj (t))e−γj (t)x(t−τj (t)) ,
j=1
where
√
√
| sin 2t + sin 3t|
,
m = 2, α(t) = 18 +
2
√
√
β1 (t) = ee−1 (10 + 0.005| sin 3t + sin 2t|),
√
√
β2 (t) = ee−1 (10 + 0.005| sin 5t + sin 3t|),
τ1 (t) = e| cos
√
2t+cos t|
, τ2 (t) = e| cos
√
2t+cos
γ1 (t) = γ2 (t) = 0.25 + 0.025| sin t + sin
√
√
3t|
,
3t|.
By a direct calculation, we can obtain
α− = 18,
β1+ = β2+ = 10.01ee−1 ,
n2 =
β1− = β2− = 10ee−1 ,
α+ = 19,
2
X
γ1− = γ2− = 0.25,
βj+
α− γj− e
j=1
=
80.08ee−2
> 9,
18
+
3.4 < n1 =
2
X
n2 βj− e−γj
j=1
γ1+ = γ2+ = 0.3,
n2
α+
< 3.5.
It is easy to see that
n2 ≥ n1 ≥
1
min1≤j≤2 {γj+ }
=
10
.
3
(2.9)
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Q.-L. LIU, H.-S. DING
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So (H1) holds. In addition, we have
2
X
βj+
20.02ee−1
=
< e2 ,
−
α
18
j=1
exp(0.25n1 )
> e2 .
1 − 0.25n1
Thus, (H2) holds. By Theorem 2.1 and Theorem 2.5, Equation (2.9) has a unique
almost periodic solution x(t) in Ω, and every solution y(t) satisfying (2.6) converges
exponentially to x(t) as t → +∞.
Remark 2.7. In Example 2.9, γ1− = γ2− < 1. Thus, the results in [4] cannot be
applied to Example 2.9. In addition, here we have
1
= 4.
n1 <
min1≤j≤m {γj− }
So the results in [9] can not be applied to Example 2.6.
Example 2.8. Let m = 1, α(t) ≡ 1, γ1 (t) ≡ 1, and τ1 (t) = 1 + | sin t + sin πt| for
all t ∈ R. Moreover, β1 is a continuous 21 -periodic function, which is defined on
[0, 12 ] by

2

t ∈ [ 14 − ε, 14 + ε],
e ,
e−1
β1 (t) = e ,
t ∈ [0, 14 − 2ε] ∪ [ 41 + 2ε, 12 ],


line segments t ∈ [ 14 − 2ε, 14 − ε] ∪ [ 14 + ε, 41 + 2ε],
where ε ∈ (0, 18 ) is a fixed constant. It is easy to check that
α+ = α− = γ + = γ− = 1,
n2 = e,
n1 = 1,
and
Z
kβ1 kS 1/2 =
0
1/2
1
β1 (s)ds ≤ 4εe2 + ( − 4ε)ee−1 .
2
Then we conclude that (H1) holds and (H3) holds for sufficiently small ε and λ =
since
1
1
lim 4εe2 + ( − 4ε)ee−1 = ee−1 < e2 (1 − e−1/2 ).
ε→0
2
2
Then, by Corollary 2.3, we know that
x0 (t) = −x(t) + β1 (t)x(t − τ (t))e−x(t−τ (t))
has an almost periodic solution provided that ε is sufficiently small.
Remark 2.9. In Example 2.8, we have
m
X
βj+
= β1+ = e2 .
−
α
j=1
However, in [4, 9],
m
X
βj+
< e2
−
α
j=1
is a key assumption. So the results in [4, 9] can not be applied to Example 2.8.
1
2
EJDE-2013/56
EXISTENCE OF POSITIVE ALMOST-PERIODIC SOLUTIONS
9
Acknowledgements. This work was supported by the following grants: 11101192
from the NSF of China, 211090 from Key Project of Chinese Ministry ofEducation, 20114BAB211002 from the NSF of Jiangxi Province, and GJJ12173 from the
Jiangxi Provincial Education Department.
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Qing-Long Liu
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
E-mail address: 758155543@qq.com
Hui-Sheng Ding
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
E-mail address: dinghs@mail.ustc.edu.cn