Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 354016, 12 pages
doi:10.1155/2011/354016
Research Article
Positive Almost Periodic Solutions for
a Time-Varying Fishing Model with Delay
Xia Li, Yongkun Li, and Chunyan He
Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 19 May 2011; Revised 8 August 2011; Accepted 12 August 2011
Academic Editor: Dexing Kong
Copyright q 2011 Xia Li et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper is concerned with a time-varying fishing model with delay. By means of the
continuation theorem of coincidence degree theory, we prove that it has at least one positive almost
periodic solution.
1. Introduction
Consider the following differential equation which is widely used in fisheries 1–4:
Ṅ NLt, N − Mt, N − NFt,
1.1
where N Nt is the population biomass, Lt, N is the per capita fecundity rate, Mt, N
is the per capita mortality rate, and Ft is the harvesting rate per capita.
In 1.1, let Lt, N be a Hills’ type function 1, 2
Lt, N a
1 N/Kγ
1.2
and take into account the delay and the varying environments; Berezansky and Idels 5
proposed the following time-lag model based on 1.1 1–6
Ṅt Nt
where bt Mt, N Ft.
at
−
bt
,
1 Nθt/Ktγ
1.3
2
International Journal of Differential Equations
The model 1.3 has recently attracted the attention of many mathematicians and
biologists; see the differential equations which are widely used in fisheries 1, 2. However,
one can easily see that all equations considered in the above-mentioned papers are subject to
periodic assumptions, and the authors, in particular, studied the existence of their periodic
solutions. On the other hand, ecosystem effects and environmental variability are very
important factors and mathematical models cannot ignore, for example, reproduction rates,
resource regeneration, habitat destruction and exploitation, the expanding food surplus, and
other factors that affect the population growth. Therefore it is reasonable to consider the
various parameters of models to be changing almost periodically rather than periodically
with a common period. Thus, the investigation of almost periodic behavior is considered
to be more accordant with reality. Although it has widespread applications in real life, the
generalization to the notion of almost periodicity is not as developed as that of periodic
solutions; we refer the reader to 7–18.
Recently, the authors of 19 proved the persistence and almost periodic solutions
for a discrete fishing model with feedback control. In 20, 21, the contraction mapping
principle and the continuation theorem of coincidence degree have been employed to
prove the existence of positive almost periodic exponential stable solutions for logarithmic
population model, respectively. A primary purpose of this paper, nevertheless, is to utilize
the continuation theorem of coincidence degree for this purpose. To the best of the authors’
observation, there exists no paper dealing with the proof of the existence of positive almost
periodic solutions for 1.3 using the continuation theorem of coincidence degree. Therefore,
our result is completely different and presents a new approach.
2. Preliminaries
Our first observation is that under the invariant transformation Nt eyt , 1.3 reduces to
ẏt 1
at
γ
eyθt /Kt
− bt
2.1
for γ > 0, with the initial function and the initial value
yt φt,
y0 y0 , t ∈ −∞, 0.
2.2
For 2.1 and 2.2, we assume the following conditions:
A1 at, bt ∈ C0, ∞, 0, ∞ and Kt ∈ C0, ∞, 0, ∞;
A2 θt is a continuous function on 0, ∞ that satisfies θt ≤ t;
A3 φt : −∞, 0 → 0, ∞ is a continuous bounded function, φt ≥ 0, y0 > 0.
By a solution of 2.1 and 2.2 we mean an absolutely continuous function yt
defined on −∞, ∞ satisfying 2.1 almost everywhere for t ≥ 0 and 2.2. As we are
interested in solutions of biological significance, we restrict our attention to positive ones.
According to 22, the initial value problem 2.1 and 2.2 has a unique solution
defined on −∞, ∞.
International Journal of Differential Equations
3
Let X, Y be normed vector spaces, L : Dom L ⊂ X → Y be a linear mapping, and
N : X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping
of index zero if dim Ker L codim Im L < ∞ and Im L is closed in Y . If L is a Fredholm
mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y
such that Im P Ker L, Ker Q Im L ImI − Q, it follows that the mapping L|Dom L∩Ker P :
I − P X → Im L is invertible. We denote the inverse of that mapping by KP . If Ω is an
open bounded subset of X, then the mapping N will be called L-compact on Ω, if QNΩ
is bounded and KP I − QN : Ω → X is compact. Since Im Q is isomorphic to Ker L, there
exists an isomorphism J : Im Q → Ker L.
Theorem 2.1 see 19. Let Ω ⊂ X be an open bounded set and let N : X → Y be a continuous
operator which is L-compact on Ω. Assume that
1 Ly /
λNy for every y ∈ ∂Ω ∩ Dom L and λ ∈ 0, 1;
2 QNy /
0 for every y ∈ ∂Ω ∩ Ker L;
3 the Brouwer degree deg{JQN, Ω ∩ Ker L, 0} /
0.
Then Ly Ny has at least one solution in Dom L ∩ Ω.
3. Existence of Almost Periodic Solutions
Let AP R denote the set of all real valued almost periodic functions on R, for f ∈ AP R we
denote by
Λ f 1
λ ∈ R : lim
T →∞T
T
fse
0
−iλs
ds /
0 ,
⎧
⎫
m
⎬
⎨
mod f nj λj : nj ∈ Z, m ∈ N, λj ∈ Λ f , j 1, 2, . . . , m ,
⎩ j1
⎭
3.1
the set of Fourier exponents and the module of f, respectively. Let Kf, ε, S denote the set
of ε-almost periods for f with respect to S ⊂ C−∞, 0, R, lε denote the length of the
T
inclusion interval, and mf lim 1/T 0 fs ds denote the mean value of f.
T →∞
Definition 3.1. yt ∈ CR, R is said to be almost periodic on R if for any ε > 0 the set Ky, ε {δ : |yt δ − yt| < ε, ∀t ∈ R} is relatively dense; that is, for any ε > 0 it is possible to find
a real number lε > 0 for any interval with length lε; there exists a number δ δε in this
interval such that |yt δ − yt| < ε for any t ∈ R.
Throughout the rest of the paper we assume the following condition for 2.1:
H at, bt, Kt, t − θt ∈ AP R, mb/K γ /
0 and ma /
mb.
In our case, we set
X Y V1 ⊕ V2 ,
3.2
4
International Journal of Differential Equations
where
V1 y ∈ AP R : mod y ⊆ modF, ∀μ ∈ Λ y satisfies μ > α ,
V2 yt ≡ k, k ∈ R ,
3.3
where
F F t, φ at
γ − bt,
1 eφθ0 /Kt
φ ∈ C−∞, 0, R
3.4
and α is a given constant; define the norm
y supyt,
t∈R
y ∈ X or Y.
3.5
t
Remark 3.2. If f is ε-almost periodic function, then fsds is ε-almost periodic if and only
if mf 0. Whereas f ∈ AP R does not necessarily have an almost periodic primitive,
mf 0. That is why we can not make V1 {z ∈ AP R : mz 0} and let V1 {y ∈
AP R : mody ⊂ modF, ∀μ ∈ Λy satisfy |μ| > α}.
We start with the following lemmas.
Lemma 3.3. X and Y are Banach spaces endowed with the norm · .
Proof. If yn ∈ V1 and yn converge to y0 , then it is easy to show that y0 ∈ AP R with mod
≤ α we have
y0 ⊂ modF. Indeed, for all |λ|
1
lim
T →∞T
t
yn se−iλs ds 0.
3.6
3.7
0
Thus
1
T →∞T
T
lim
y0 se−iλs ds 0,
0
which implies that y0 ∈ V1 . One can easily see that V1 is a Banach space endowed with the
norm · . The same can be concluded for the spaces X and Y . The proof is complete.
Lemma 3.4. Let L : X → Y and
Ly 1
at
γ
eyθt /Kt
− bt,
where Ly y dy/dt. Then L is a Fredholmmapping of index zero.
3.8
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5
Proof. It is obvious that L is a linear operator and Ker L V2 . It remains to prove that Im L V1 . Suppose that φt ∈ Im L ⊂ Y . Then, there exist φ1 ∈ V1 and φ2 ∈ V2 such that
φ φ1 φ2 .
3.9
t
t
From the definitions of φt and φ1 t, one can deduce that φsds and φ1 sds are almost
periodic functions and thus φ2 t ≡ 0, which implies that φt ∈ V1 . This tells us that
Im L ⊂ V1 .
On the other hand, if ϕt ∈ V1 \ {0} then we have
obtain
1
lim
T →∞T
T t
0
ϕsds e
0
−iλt
t
0
3.10
ϕsds ∈ AP R. Indeed, if λ /
0 then we
1
1
dt lim
iλ T → ∞ T
T
ϕte−iλt dt.
3.11
0
It follows that
t
Λ
ϕsds − m
t
0
0
t
t
ϕsds
Λ ϕt .
3.12
Thus
0
ϕsds − m
ϕsds
∈ V1 ⊂ X.
3.13
0
t
t
Note that 0 ϕsds − m 0 ϕsds is the primitive of ϕ in X; therefore we have ϕ ∈ Im L.
Hence, we deduce that
V1 ⊂ Im L,
3.14
which completes the proof of our claim. Therefore,
Im L V1 .
3.15
Furthermore, one can easily show that Im L is closed in Y and
dim Ker L 1 codim Im L.
Therefore, L is a Fredholm mapping of index zero.
3.16
6
International Journal of Differential Equations
Lemma 3.5. Let N : X → Y , P : X → X, and Q : Y → Y such that
at
− bt, y ∈ X,
1
P y m y , y ∈ X, Qz mz, z ∈ Y.
Ny γ
eyθt /Kt
3.17
Then, N is L-compact on Ω (Ω is an open and bounded subset of X).
Proof. The projections P and Q are continuous such that
Im P Ker L,
Im L Ker Q.
3.18
It is clear that
I − QV2 {0},
3.19
I − QV1 V1 .
Therefore
ImI − Q V1 Im L.
3.20
In view of
Im P Ker L,
3.21
Im L Ker Q ImI − Q,
we can conclude that the generalized inverse of L KP : Im L → Ker P ∩ Dom L exists and
is given by
KP z t
zsds − m
t
0
zsds .
3.22
0
Thus
QNy m
at
γ − bt ,
1 eyθt /Kt
3.23
KP I − QNy f yt − Qf yt ,
where fyt is defined by
f yt t
0
Nys − QNys ds.
3.24
International Journal of Differential Equations
7
The integral form of the terms of both QN and I − QN implies that they are
continuous. We claim that KP is also continuous. By our hypothesis, for any ε < 1 and any
compact set S ⊂ C−∞, 0, R, let lε, S be the inclusion interval of KF, ε, S. Suppose that
t
{zn t} ⊂ Im L V1 and zn t uniformly converges to z0 t. Because 0 zn sds ∈ Y n t
0, 1, 2, 3, . . ., there exists ρ0 < ρ < ε such that KF, ρ, S ⊂ K o zn s ds, ε. Let lρ, S be the
inclusion interval of KF, ρ, S and
l max l ρ, S , lε, S .
3.25
It is easy to see that l is the inclusion interval of both KF, ε, S and KF, ρ, S. Hence, for all
t
t∈
/ 0, l, there exists μt ∈ KF, ρ, S ⊂ K 0 zn sds, ε such that t μt ∈ 0, l. Therefore, by
the definition of almost periodic functions we observe that
t
t
zn sds sup zn sds
0
0
t∈R
tμ
tμt
t
t
t
≤ sup zn sds sup zn sds −
zn sds zn sds
0
0
0
t∈0,l 0
t/
∈ 0,l
tμt
t
t
≤ 2 sup zn sds sup zn sds −
zn sds
0
0
0
t∈0,l
t/
∈ 0,l
≤2
l
0
|zn s|ds ε.
3.26
t
By applying 3.26, we conclude that 0 zsdsz ∈ Im L is continuous and consequently KP
and KP I − QNy are also continuous.
t
From 3.26, we also have that 0 zsds and KP I − QNy are uniformly bounded
in Ω. In addition, it is not difficult to verify that QNΩ is bounded and KP I − QNy is
equicontinuous in Ω. Hence by the Arzelà-Ascoli theorem, we can immediately conclude
that KP I − QNΩ is compact. Thus N is L-compact on Ω.
Theorem 3.6. Let condition (H) hold. Then 2.1 has at least one positive almost periodic solution.
Proof. It is easy to see that if 2.1 has one almost periodic solution y, then N ey is a positive
almost periodic solution of 1.3. Therefore, to complete the proof it suffices to show that 2.1
has one almost periodic solution.
In order to use the continuation theorem of coincidence degree theory, we set the
Banach spaces X and Y the same as those in Lemma 3.3 and the mappings L, N, P , Q the
same as those defined in Lemmas 3.4 and 3.5, respectively. Thus, we can obtain that L is a
Fredholm mapping of index zero and N is a continuous operator which is L-compact on Ω.
8
International Journal of Differential Equations
It remains to search for an appropriate open and bounded subset Ω. Corresponding to the
operator equation
Ly λNy,
λ ∈ 0, 1,
3.27
we may write
ẏt λ
at
γ − bt .
1 eyθt /Kt
3.28
Assume that y yt ∈ X is a solution of 3.28 for a certain λ ∈ 0, 1. Denote
y∗ sup yt,
y∗ inf yt.
3.29
t∈R
t∈R
In view of 3.28, we obtain
mat − bt m
bt yθt γ
e
K γ t
3.30
and consequently,
mat − bt ≥ m
bt
eγy∗ ,
K γ t
3.31
which implies from H that
mat − bt
.
ln
mbt/K γ t
3.32
mat − bt
y∗ ≥ γ −1 ln
.
mbt/K γ t
3.33
y∗ ≤ γ
−1
Similarly, we can get
By inequalities 3.32 and 3.33, we can find that there exists t1 ∈ R such that
yt1 ≤ M,
3.34
−1
mat − bt 1.
M γ ln
mbt/K γ t 3.35
where
International Journal of Differential Equations
9
Then from 3.26, we have
t
t
y sds ε
yt ≤ yt1 sup y sds ≤ M 2 sup
t∈R t2
t∈t2 ,t2 l t2
3.36
or
yt ≤ M 2
t2 l
t2
y sds 1.
3.37
Choose the point ν − t2 ∈ l, 2l ∩ KF, ρ, S, where ρ0 < ρ < ε satisfies KF, ρ ⊂ Kz, ε.
Integrating 3.28 from t2 to ν, we get
λ
ν
t2
at
ds λ
1 Ks−γ eγyθs
≤λ
ν
t2
ν
t2
|bs|ds ν
t2
y sds
3.38
|bs|ds ε.
However, from 3.28 and 3.38, we obtain
ν
t2
y sds ≤ λ
ν
t2
|bs|ds λ
≤2
≤2
ν
t2
ν
t2
ν
t2
at
ds,
1 Ks−γ eγyθs
|bs|ds ε
3.39
|bs|ds 1.
Substituting back in 3.37 and for ν ≥ t2 l, we have
yt ≤ M ,
3.40
where
M M 4
ν
|bs|ds 3.
3.41
0
Let M M M . Obviously, it is independent of λ. Take
"
!
Ω y ∈ X : y < M .
3.42
10
International Journal of Differential Equations
8
7
6
5
4
3
2
1
0
0
10
30
20
40
50
Time (t)
N(t)
Figure 1: Transient response of state Nt when γ 2.
It is clear that Ω satisfies assumption 1 of Theorem 2.1. If y ∈ ∂Ω ∩ Ker L, then y is a
constant with y M. It follows that
QNy m
at
γ − bt
1 eyθt /Kt
0,
/
3.43
which implies that assumption 2 of Theorem 2.1 is satisfied. The isomorphism J : Im Q →
Ker L is defined by Jz z for z ∈ R. Thus, JQNy /
0. In order to compute the Brouwer
degree, we consider the homotopy
H y, s −sy 1 − sJQNy,
0 ≤ s ≤ 1.
3.44
For any y ∈ ∂Ω ∩ Ker L, s ∈ 0, 1, we have Hy, s /
0. By the homotopic invariance of
topological degree, we get
deg{JQN, Ω ∩ Ker L, 0} deg −y, Ω ∩ Ker L, 0 / 0.
3.45
Therefore, assumption 3 of Theorem 2.1 holds. Hence, Ly Ny has at least one solution in
Dom L ∩ Ω. In other words, 2.1 has at least one positive almost periodic solution. Therefore,
1.3 has at least one positive almost periodic solution. The proof is complete.
International Journal of Differential Equations
11
4. An Example
√
√
√
Let √
at eπ 3 cos 2t, bt 1/2eπ 3 cos 2t, Kt 4 sin t, γ > 0, θt t − 2 −
sin 3t. Then 1.3 has the form
⎡
⎤
√ eπ 3 cos 2t
√
1
⎢
⎥
π
Ṅt Nt⎣
√ √ γ − 2 e 3 cos 2t ⎦.
1 N t − 2 − sin 3t / 4 sin t
4.1
One can easily realize that mbt/Ktγ > 0 and mat > mbt; thus condition H
holds. Therefore, by the consequence of Theorem 3.6, 4.1 has at least one positive almost
periodic solution Figure 1.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant 10971183.
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