R E S E A R C H
Open Access
Positive periodic solution for
Nicholson-type delay systems with impulsive
effects
Ruojun Zhang
*, Yanping Huang and Tengda Wei
*Correspondence:
[email protected] School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China
Abstract
In this paper, a class of Nicholson-type delay systems with impulsive effects is considered. First, an equivalence relation between the solution (or positive periodic solution) of a Nicholson-type delay system with impulsive effects and that of the corresponding Nicholson-type delay system without impulsive effects is established. Then, by applying the cone fixed point theorem, some criteria are established for the existence and uniqueness of positive periodic solutions of the given systems. Finally, an example and its simulation are provided to illustrate the main results. Our results extend and improve greatly some earlier works reported in the literature.
Keywords: Nicholson-type systems; positive periodic solutions; delay; impulsive effect; cone fixed point theorem
1 Introduction
To describe the population of the Australian sheep-blowfly and to agree with the
experi-mental data obtained in [], Gurneyet al.[] proposed the following Nicholson blowflies
model:
N(t) = –δN(t) +PN(t–τ)e–aN(t–τ), (.)
whereN(t) is the size of the population at timet,Pis the maximumper capitadaily egg production,
a is the size at which the population reproduces at its maximum rate,δis
theper capitadaily adult death rate, andτ is the generation time. Nicholson’s blowflies model and many generalized Nicholson’s blowflies models have attracted more attention because of their extensively realistic significance; see [–]. Recently, in order to describe the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics,
which are examples of Nicholson-type delay differential systems, Berezanskyet al.[],
Wanget al.[], and Liu [] studied the following Nicholson-type delay systems:
N(t) = –α(t)N(t) +β(t)N(t) +
m
j=cj(t)N(t–τj(t))e–γj(t)N(t–τj(t)), N(t) = –α(t)N(t) +β(t)N(t) +
m
j=cj(t)N(t–τj(t))e–γj(t)N(t–τj(t)),
(.)
whereαi(t),βi(t),cij(t),γij(t),τij(t)∈C(R, (,∞)),i= , ,j= , , . . . ,m. For constant
coeffi-cients and delays, Berezanskyet al.[] presented several results for the permanence and
globally asymptotic stability of system (.). Supposing thatαi(t),βi(t), cij(t),γij(t), and
τij(t) are almost periodic functions, Wanget al.[] obtained some criteria to ensure that
the solutions of system (.) converge locally exponentially to a positive almost periodic solution. Furthermore, Liu [] established some criteria for the existence and uniqueness of a positive periodic solution of system (.) by applying the method of the Lyapunov function.
However, species living in certain medium might undergo abrupt change of state at cer-tain moments, and this occurs due to some seasonal effects such as weather change, food supply, and mating habits. That is to say, besides delays, impulsive effects likewise exist widely in many evolution processes. In the last two decades, the theory of impulsive dif-ferential equations has been extensively investigated due to its widespread applications [–].
Therefore, it is more realistic to investigate Nicholson-type delay systems with impul-sive effects. However, to the best of our knowledge, few authors [] have considered the conditions for existence and uniqueness of positive periodic solution for system (.) with impulsive effects. Thus, techniques and methods on the existence and uniqueness of a positive periodic solution for system (.) with impulsive effects should be developed and explored.
In this paper, we consider the following class of Nicholson-type delay systems with im-pulsive effects:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
y(t) = –α(t)y(t) +β(t)y(t) +
m
j=cj(t)y(t–τj(t))e–γj(t)y(t–τj(t)), y(t) = –α(t)y(t) +β(t)y(t) +
m
j=cj(t)y(t–τj(t))e–γj(t)y(t–τj(t)), t≥t> ,t=tk,
yi(t+k) = ( +bk)yi(tk), t≥t,t=tki= , ,k= , , . . . ,
(.)
where αi(t),βi(t),cij(t),γij(t),τij(t)∈ C([,∞), (,∞)), i = , , j= , , . . . ,m. yi(tk) = yi(tk+) –yi(t–
k) are the impulses at momentstk.
In Equation (.), we shall use the following hypotheses:
(H) <t<t<t<· · ·,ti,i= , , . . .are fixed impulsive points withlimk→∞tk=∞;
(H) {bk}is a real sequence, andbk> –,k= , , . . .; (H) αi(t),βi(t),cij(t),γij(t),τij(t), and
<tk<t( +bk)are periodic functions with common
periodω> ,i= , ,j= , , . . . ,m,k= , , . . ..
Here and in the sequel, we assume that a product equals unit if the number of factors is equal to zero.
Letτ=max{τij+},τij+=max≤t≤ωτij(t),i= , ,j= , , . . . ,m. Ifyi(t) is defined on [t–τ,σ] witht,σ∈R, then we defineyt ∈C([–τ, ],R) asyt= (yt,yt) whereyit(θ) =yi(t+θ) for
θ∈[–τ, ] andi= , .
Due to the biological interpretation of system (.), only positive solutions are meaning-ful and admissible. Thus, we shall only consider the admissible initial conditions:
yit(s) =ϕi(s), s∈[–τ, ], (.)
The remaining parts of this paper is organized as follows. In Section , we introduce some notation, definitions, and lemmas. In Section , we first establish the equivalence between the solution (or positive periodic solution) of a Nicholson-type delay system with impulses and that of the corresponding Nicholson-type delay system without impulses. Then, we give some criteria ensuring the existence and uniqueness of positive periodic solutions of Nicholson-type delay systems with and without impulses. In Section , an example and its simulation are provided to illustrate our results obtained in the previous sections. Finally, some conclusions are drawn in Section .
2 Preliminaries
For convenience, in the following discussion, we always use the notation
g–= min
≤t≤ωg(t), g
+= max ≤t≤ωg(t),
wheregis a continuousω-periodic function defined onR.
Definition . A functiony(t) = (y(t),y(t))Tdefined on [t–τ,∞) is said to be a solution of Equation (.) with initial condition (.) if
(i) y(t)is absolutely continuous on the intervals(t,t]and(tk,tk+],k= , , . . .;
(ii) for alltk,k= , , . . .,y(tk+)andy(t–k)exist, andy(tk–) =y(tk);
(iii) y(t)satisfies the differential equation of (.) in[t,∞)\{tk}and the impulsive
conditions for allt=tk,k= , , . . .; (iv) yit(s) =ϕi(s),s∈[–τ, ].
Under hypotheses (H)-(H), we consider the following Nicholson-type delay systems
without impulsive effects:
⎧ ⎪ ⎨ ⎪ ⎩
x(t) = –α(t)x(t) +β(t)x(t) +
m
j=pj(t)x(t–τj(t))e–qj(t)x(t–τj(t)), x(t) = –α(t)x(t) +β(t)x(t) +
m
j=pj(t)x(t–τj(t))e–qj(t)x(t–τj(t)), t≥t> ,
(.)
with initial conditions
xit(s) =ϕi(s) fors∈[–τ, ],ϕ∈C [–τ, ], (,∞)
, (.)
where
pij(t) =
t–τij(t)≤tk<t
( +bk)–cij(t) and qij(t) = <tk<t–τij(t)
( +bk)γij(t),
i= , ,j= , , . . . ,m.
By a solutionx(t) of Equation (.) with initial condition (.) we mean an absolutely continuous functionx(t) = (x(t),x(t))T defined on [t,∞) satisfying Equation (.) for
t≥tand initial condition (.) on [–τ, ].
Similarly to the method of [], we have the following:
(i)if x(t) = (x(t),x(t))T is a solution(or positiveω-periodic solution)of Equation(.)
lution (or positiveω-periodic solution) of Equation(.)with initial condition (.) on
[–τ,∞);
solution(or positiveω-periodic solution)of Equation(.)with initial condition(.)on
Thus, for everyt=tk,k= , , . . . ,
yi tk+= ( +bk)yi(tk), i= , . (.)
Therefore, we arrive at the conclusion thaty(t) is the solution (or positiveω-periodic solution) of Equation (.) with initial condition (.). In fact, if x(t) is the solution (or positiveω-periodic solution) of Equation (.) with initial condition (.), thenyi(t) =
<tk<t( +bk)xi(t) =xi(t) =ϕi(t) on [–τ, ],i= , .
(ii) Sincey(t) = (y(t),y(t))Tis a solution (or positiveω-periodic solution) of Equation (.) with initial condition (.), it follows thaty(t) is absolutely continuous on all intervals (t,t] and (tk,tk+],k= , , . . . . Therefore,xi(t) =
<tk<t( +bk)
–yi(t) is absolutely
con-tinuous on all intervals (t,t] and (tk,tk+],k= , , . . . . Moreover, it follows that, for any
t=tk,k= , , . . . ,
xi t+k=lim
t→t+k
<tj<t
( +bj)–yi(t)
=
<tj≤tk
( +bj)–yi tk+
=
<tj<tk
( +bj)–yi(tk) =xi(tk) (.)
and
xi t–k
=lim
t→t–
k
<tj<t
( +bj)–yi(t)
=
<tj<tk
( +bj)–yi tk–
=
<tj<tk
( +bj)–yi(tk) =xi(tk), i= , , (.)
which implies thatx(t) is continuous and easy to prove absolutely continuous on [,∞). Now, similarly to the proof in case (i), we can easily check thatx(t) =<t
k<t( +bk)
–y(t)
is a solution (or positiveω-periodic solution) of Equation (.) with initial condition (.) on [–τ,∞].
From the above analysis we know that the conclusion of Lemma . is true. This
com-pletes the proof.
Lemma . Suppose that
(H)
β+β+ α–α– < .
Then every solution x(t)of Equation(.)with(.)and every solution y(t)of Equation
(.)with(.)are positive and bounded on[t,∞).
Proof Clearly, by Lemma ., we only need to prove that every solutionx(t) of Equation (.) with (.) is positive and bounded on [t,∞). In order to show that, we only need to see Lemma . in [].
with (.), whent>t,
be a completely continuous operator.If
(i) Tx ≤ xforx∈P∩∂andTx ≥ xforx∈P∩∂,or
(ii) Tx ≤ xforx∈P∩∂andTx ≥ xforx∈P∩∂,
then the operator T has at least one fixed point in P∩(\).
3 Existence and uniqueness of positive periodic solution
For ease of exposition, throughout this paper, we adopt the following notation:
|xi|∞= max
Define the operatorT by
(Tx)(t) =
Proof First, we proveT :P→P. From (H) we know thatαi(t),i= , , are continuous
ω-periodic functions. Further, we find
Moreover,
T:P→Pis completely continuous. The proof of Lemma . is complete.
Theorem . Assume that(H)-(H)hold.Then Equation(.)with(.)has at least one
positiveω-periodic solution.
Case.τ–≥ω. In view of (.), we have
(Tx)(t)≥N
ω
m
j=
pj(s)x s–τj(s)
e–qj(s)x(s–τj(s))
ds
≥Nω
m
j=
p–jϕ–e–q+jϕ+ A
> ,
whereϕ–=min–τ≤s≤ϕ(t),ϕ+ =max–τ≤s≤ϕ(t).
Case.τ–<ω. In view of (.), we have
(Tx)(t)≥N
τ–
m
j=
pj(s)x s–τj(s)
e–qj(s)x(s–τj(s))
ds
≥Nτ–
m
j=
p–jϕ–e–q+jϕ+ A
> .
Therefore,
(Tx)(t)≥min{A,A}A> .
Similarly, we have
(Tx)(t)≥min{A,A}A> ,
where A =Nω
m j=p–jϕ–e
–q+jϕ+
, A =Nτ–
m j=p–jϕ–e
–q+jϕ+
, ϕ– =min–τ≤s≤ϕ(t), ϕ+=max–τ≤s≤ϕ(t).
Then, for anyx∈Pandt>t,
Tx ≥min{A,A}A> . (.)
Let
=
x∈X:x<A
and
=
x∈X:x<B.
Clearly, andare open bounded subsets inX, andθ∈X,⊂. By Lemma .,
T:P∩(\)→Pis completely continuous.
Ifx∈P∩∂, which implies thatx=B, then from (.) we haveTx ≤B, and hence Tx ≤ xforx∈P∩∂.
Ifx∈P∩∂, which implies thatx=A, then from (.) we haveTx ≥A, and hence Tx ≥ xforx∈P∩∂.
with (.) has at least one positiveω-periodic solution. Therefore, Equation (.) with (.)
has at least one positiveω-periodic solution by Lemma .. This completes the proof of
Theorem ..
Theorem . Let(H)-(H)hold.Suppose further that the following condition holds:
(H) αi––βi+– m
j=p+ij> ,i= , .
Then Equation(.)with(.)has a unique positiveω-periodic solution.
Proof By Theorem . we know that Equation (.) with (.) has at least one positiveω -periodic solution. Thus, in order to prove Theorem ., we only need to prove the unique-ness of a positiveω-periodic solution for Equation (.) with (.).
The following proof is similar to that of Theorem . in [].
Assume thatx(t) andx(t) are two positiveω-periodic solutions of Equation (.). Set
zi(t) =xi(t) –xi(t), wheret∈[t–τ,∞),i= , . Then
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
z(t) = –α(t)z(t) +β(t)z(t) +
m
j=pj(t)[x(t–τj(t))e–qj(t)x(t–τj(t))
–x(t–τj(t))e–qj(t)x(t–τj(t))], z(t) = –α(t)z(t) +β(t)z(t) +
m
j=pj(t)[x(t–τj(t))e–qj(t)x(t–τj(t))
–x(t–τj(t))e–qj(t)x(t–τj(t))], t≥t> .
(.)
Set
i(u) = – αi––u
+βi++
m
j=
p+ijeuτi+, u∈[, ],τ+
i =max
≤j≤mτ
+
ij,i= , .
Clearly,i(u),i= , , are continuous functions on [, ]. From (H) we have
i() = –αi–+βi++ m
j=
p+ij< , i= , .
Hence, we can choose two constantsη> and <λ≤ such that
i(λ) = λ–α–i
+βi++
m
j=
p+ijeλτi+< –η< , i= , . (.)
Consider the Lyapunov functions
V(t) =z(t)eλt, V(t) =z(t)eλt.
Calculating the upper right derivative ofVi(t) (i= , ) along the solutionz(t) of (.), we
obtain
D+ V(t)
≤
λ–α(t)z(t)+β(t)z(t)+
m
j=
pj(t)x t–τj(t)
e–qj(t)x(t–τj(t))
–x t–τj(t)
e–qj(t)x(t–τj(t))
and
D+ V(t)
≤
λ–α(t)z(t)+β(t)z(t)+
m
j=
pj(t)x t–τj(t)
e–qj(t)x(t–τj(t))
–x t–τj(t)
e–qj(t)x(t–τj(t))
eλt for allt≥t. (.)
We claim that there isM> such that
Vi(t) =zi(t)eλt≤M for allt>t,i= , . (.)
Otherwise, one of the following cases must occur.
Case. There existsT>tsuch that
V(T) =M and Vi(t) <M for allt∈[t–τ,T],i= , . (.)
Case. There existsT>tsuch that
V(T) =M and Vi(t) <M for allt∈[t–τ,T],i= , . (.)
We will need the inequality
xe–x–ye–y≤ |x–y| forx,y∈[, +∞). (.)
Indeed, by the mean value theorem we have
xe–x–ye–y= –ξ
eξ
· |x–y|, whereξ is betweenxandy.
Forξ> , we have|–eξξ|=
ξ–
eξ ≤e < , and for ≤ξ≤, we have|
–ξ eξ |=
–ξ
eξ ≤. Therefore, inequality (.) holds.
In case , in view of (.) and inequality (.), (.) implies that
≤D+ V(T) –M
≤
λ–α(T)z(T)+β(T)z(T)
+
m
j=
pj(T)x T–τj(T)
e–qj(T)x(T–τj(T))
–x T–τj(T)
e–qj(T)x(T–τj(T))
eλT
=
λ–α(T)z(T)+β(T)z(T)+
m
j=
pj(T)
qj(T)
×qj(T)x T–τj(T)
e–qj(T)x(T–τj(T))
–qj(T)x T–τj(T)
e–qj(T)x(T–τj(T))
≤ λ–α(T)z(T)eλT+β(T)z(T)eλT
+
m
j=
pj(T)z T–τj(T)eλ(T–τj(T))eλτj(T)
≤
λ–α–+β++
m
j=
p+jeλτ+
M.
Thus,
λ–α–+β++
m
j=
p+jeλτ+≥,
which contradicts (.). Hence, (.) holds.
In case , in view of (.) and (.), (.) yields that
≤D+ V(T) –M
≤
λ–α(T)z(T)+β(T)z(T)
+
m
j=
pj(T)x T–τj(T)
e–qj(T)x(T–τj(T))
–x T–τj(T)
e–qj(T)x(T–τj(T))
eλT
=
λ–α(T)z(T)+β(T)z(T)+
m
j=
pj(T)
qj(T)
×qj(T)x T–τj(T)
e–qj(T)x(T–τj(T))
–qj(T)x T–τj(T)
e–qj(T)x(T–τj(T))
eλT
≤ λ–α(T)z(T)eλT+β(T)z(T)eλT
+
m
j=
pj(T)z T–τj(T)eλ(T–τj(T))eλτj(T)
≤
λ–α–+β++
m
j=
p+jeλτ+
M.
Thus,
λ–α–+β++
m
j=
p+jeλτ+≥,
which contradicts (.). Hence, (.) holds. It follows that
In view of (.) and the periodicity ofz(t), we have
zi(t) =xi(t) –xi(t) = for allt∈[t–τ,∞),i= , .
This completes the proof.
Remark . In Theorems . and ., the conditions that ensure the existence and
unique-ness of a positiveω-periodic solution for Nicholson-type delay systems with and without impulses are simple and easily to test, which is less conservative than the conditions re-quired in some previous works [, ]. Moreover, the main results in this paper are totally different from that of [].
4 An example
Example . Consider the following impulsive Nicholson-type system with delays
⎧
which implies thatf(t) is a periodic function with period .
Figure 1 The dynamic behavior of the system (4.1) with the initial condition (4.2). (a)Time-series of the
y1,y2of system (4.1) without impulsive effects fort∈[0, 20].(b)Phase portrait of solutions of system (4.1)
without impulsive effects fort∈[3, 20].(c)Times-series of they1,y2of impulsive system (4.1) fort∈[0, 20]. (d)Phase portrait of solutions of impulsive system (4.1) fort∈[5, 20].
p(t) =
t–e|cosπt|≤tk<t
sinπk
–
|cosπt|
,
q(t) =
<tk<t–e|cosπt|
sinπk
+|sinπt|
,
q(t) =
<tk<t–e|sinπt|
sinπk
+|cosπt|
,
q(t) =
<tk<t–e|sinπt|
sinπk
+|cosπt|
,
q(t) =
<tk<t–e|cosπt|
sinπk
+|sinπt|
.
Therefore,
α–i –βi+–
j=
p+ij=
> , i= , .
Remark . System (.) is a simple form of impulsive Nicholson-type system with delays. Sinceq–
=q–=> ,q–=q–=> , it is clear that the condition of Theorem . in [] and Theorem . in [] are not satisfied. Therefore, all the results obtained in [, ] and the references therein cannot be applicable to system (.). This implies that the results of this paper are essentially new.
5 Conclusion
In this paper, a class of Nicholson-type delay systems with impulsive effects are investi-gated. First, an equivalence relation between the solution (or positive periodic solution) of a type delay system with impulses and that of the corresponding Nicholson-type delay system without impulses is established. Then, by applying the cone fixed point theorem, some criteria are established for the existence and uniqueness of a positive pe-riodic solution of the given system. The fixed point theorem in cones is very popular in investigation of positive periodic solutions to impulsive functional differential equations [, ]. Our results imply that under the appropriate linear periodic impulsive pertur-bations, the Nicholson-type delay systems with impulses preserve the original periodic property of the Nicholson-type delay systems without impulses. Finally, an example and its simulation are provided to illustrate the main results. It is worth noting that there are only very few results [] on Nicholson-type delay systems with impulses, and our results extend and improve greatly some earlier works reported in the literature. Furthermore, our results are important in applications of periodic oscillatory Nicholson-type delay sys-tems with impulsive control.
Competing interests
The authors have declared that no competing interests exist.
Authors’ contributions
RZ came up with the main idea of the theorems and gave an example. Moreover, RZ and YH completed the proofs of the results, and TW designed a MATLAB program to simulate the results of example. RZ and YH wrote the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11171374) and the Scientific Research Fund of Shandong Provincial of P.R. China (Grant No. ZR2011AZ001).
Received: 4 June 2015 Accepted: 19 November 2015
References
1. Nicholson, AJ: An outline of the dynamics of animal populations. Aust. J. Zool.2(1), 9-65 (1954) 2. Gurney, WSC, Blythe, SP, Nisbet, RM: Nicholson’s blowflies revisited. Nature287(5777), 17-21 (1980)
3. Kulenovi´c, MRS, Ladas, G, Sficas, YG: Global attractivity in Nicholson’s blowflies. Appl. Anal.43(1-2), 109-124 (1992) 4. Saker, SH, Agarwal, S: Oscillation and global attractivity in a periodic Nicholson’s blowflies model. Math. Comput.
Model.35(7-8), 719-731 (2002)
5. Chen, Y: Periodic solutions of delayed periodic Nicholson’s blowflies models. Can. Appl. Math. Q.11(1), 23-28 (2003) 6. Gyori, I, Trofimchuk, SI: On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation. Nonlinear
Anal., Real World Appl.48(7), 1033-1042 (2002)
7. Li, J, Du, C: Existence of positive periodic solutions for a generalized Nicholson’s blowflies model. J. Comput. Appl. Math.221(1), 226-233 (2008)
8. Berezansky, L, Braverman, E, Idels, L: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model.34(6), 1405-1417 (2010)
9. Hou, X, Duan, L, Huang, Z: Permanence and periodic solutions for a class of delay Nicholson’s blowflies models. Appl. Math. Model.37(3), 1537-1544 (2013)
10. Berezansky, L, Idels, L, Troib, L: Global dynamics of Nicholson-type delay systems with applications. Nonlinear Anal., Real World Appl.12(1), 436-445 (2011)
11. Wang, W, Wang, L, Chen, W: Existence and exponential stability of positive almost periodic solution for Nicholson-type delay systems. Nonlinear Anal., Real World Appl.12(4), 1938-1949 (2011)
13. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
14. Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993)
15. Samoilenko, AM, Perestyuk, NA: Differential Equations with Impulsive Effect. World Scientific, Singapore (1995) 16. Benchohra, M, Henderson, J, Ntouyas, S: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing
Corporation, New York (2006)
17. Zhang, R, Lian, F: The existence and uniqueness of positive periodic solutions for a class of Nicholson-type systems with impulses and delays. Abstr. Appl. Anal. (2013). doi:10.1155/2013/980935
18. Yan, J, Zhao, A: Oscillation and stability of linear impulsive delay differential equations. J. Math. Anal. Appl.227(1), 187-194 (1998)
19. Guo, D: Nonlinear Functional Analysis. Shandong Science and Technology Press, Jinan (2001) (in Chinese) 20. Zhang, N, Dai, B, Qian, X: Periodic solutions for a class of higher-dimension functional differential equations with
impulses. Nonlinear Anal.68, 629-638 (2008)
![Figure 1 The dynamic behavior of the system (4.1) with the initial condition (4.2). (a) Time-series of they1, y2 of system (4.1) without impulsive effects for t ∈ [0,20]](https://thumb-us.123doks.com/thumbv2/123dok_us/981104.1120930/14.595.119.477.80.389/figure-dynamic-behavior-initial-condition-series-impulsive-eects.webp)