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Positive periodic solution for Nicholsontype delay systems with impulsive effects
Advances in Difference Equations volume 2015, Article number: 371 (2015)
Abstract
In this paper, a class of Nicholsontype delay systems with impulsive effects is considered. First, an equivalence relation between the solution (or positive periodic solution) of a Nicholsontype delay system with impulsive effects and that of the corresponding Nicholsontype 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.
Introduction
To describe the population of the Australian sheepblowfly and to agree with the experimental data obtained in [1], Gurney et al. [2] proposed the following Nicholson blowflies model:
where \(N(t)\) is the size of the population at time t, P is the maximum per capita daily egg production, \(\frac{1}{a}\) is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ 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 [3–9]. Recently, in order to describe the models of marine protected areas and Bcell chronic lymphocytic leukemia dynamics, which are examples of Nicholsontype delay differential systems, Berezansky et al. [10], Wang et al. [11], and Liu [12] studied the following Nicholsontype delay systems:
where \(\alpha_{i}(t), \beta_{i}(t), c_{ij}(t), \gamma_{ij}(t), \tau_{ij}(t)\in C(R, (0,\infty))\), \(i=1,2\), \(j=1,2,\ldots, m\). For constant coefficients and delays, Berezansky et al. [10] presented several results for the permanence and globally asymptotic stability of system (1.2). Supposing that \(\alpha_{i}(t)\), \(\beta_{i}(t)\), \(c_{ij}(t)\), \(\gamma_{ij}(t)\), and \(\tau_{ij}(t)\) are almost periodic functions, Wang et al. [11] obtained some criteria to ensure that the solutions of system (1.2) converge locally exponentially to a positive almost periodic solution. Furthermore, Liu [12] established some criteria for the existence and uniqueness of a positive periodic solution of system (1.2) by applying the method of the Lyapunov function.
However, species living in certain medium might undergo abrupt change of state at certain 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 differential equations has been extensively investigated due to its widespread applications [13–16].
Therefore, it is more realistic to investigate Nicholsontype delay systems with impulsive effects. However, to the best of our knowledge, few authors [17] have considered the conditions for existence and uniqueness of positive periodic solution for system (1.2) with impulsive effects. Thus, techniques and methods on the existence and uniqueness of a positive periodic solution for system (1.2) with impulsive effects should be developed and explored.
In this paper, we consider the following class of Nicholsontype delay systems with impulsive effects:
where \(\alpha_{i}(t),\beta_{i}(t),c_{ij}(t),\gamma_{ij}(t),\tau_{ij}(t)\in C([0,\infty),(0,\infty))\), \(i=1,2\), \(j=1,2,\ldots,m\). \(\triangle y_{i}(t_{k})=y_{i}(t^{+}_{k})y_{i}(t^{}_{k})\) are the impulses at moments \(t_{k}\).
In Equation (1.3), we shall use the following hypotheses:
 (H_{1}):

\(0< t_{0}< t_{1}< t_{2}< \cdots\), \(t_{i}\), \(i=1,2,\ldots\) are fixed impulsive points with \(\lim_{k\rightarrow\infty}t_{k}=\infty\);
 (H_{2}):

\(\{b_{k}\}\) is a real sequence, and \(b_{k}>1\), \(k=1,2,\ldots\) ;
 (H_{3}):

\(\alpha_{i}(t)\), \(\beta_{i}(t)\), \(c_{ij}(t)\), \(\gamma_{ij}(t)\), \(\tau_{ij}(t)\), and \(\prod_{0< t_{k}< t}(1+b_{k})\) are periodic functions with common period \(\omega>0\), \(i=1,2\), \(j=1,2,\ldots,m\), \(k=1,2,\ldots\) .
Here and in the sequel, we assume that a product equals unit if the number of factors is equal to zero.
Let \(\tau=\max\{\tau_{ij}^{+}\}\), \(\tau_{ij}^{+}=\max_{0\leq t\leq\omega}\tau_{ij}(t)\), \(i=1,2\), \(j=1,2,\ldots,m\). If \(y_{i}(t)\) is defined on \([t_{0}\tau, \sigma]\) with \(t_{0}, \sigma\in R\), then we define \(y_{t}\in C([\tau, 0], R)\) as \(y_{t}=(y_{t}^{1}, y_{t}^{2})\) where \(y_{t}^{i}(\theta)=y_{i}(t+\theta)\) for \(\theta\in[\tau,0]\) and \(i=1,2\).
Due to the biological interpretation of system (1.3), only positive solutions are meaningful and admissible. Thus, we shall only consider the admissible initial conditions:
where \(\varphi_{i}(s)\in C([\tau,0],(0,\infty))\). We write \(y(t)=y_{t}(t_{0},\varphi)\) for a solution of the initial value problems (1.3) and (1.4).
The remaining parts of this paper is organized as follows. In Section 2, we introduce some notation, definitions, and lemmas. In Section 3, we first establish the equivalence between the solution (or positive periodic solution) of a Nicholsontype delay system with impulses and that of the corresponding Nicholsontype delay system without impulses. Then, we give some criteria ensuring the existence and uniqueness of positive periodic solutions of Nicholsontype delay systems with and without impulses. In Section 4, an example and its simulation are provided to illustrate our results obtained in the previous sections. Finally, some conclusions are drawn in Section 5.
Preliminaries
For convenience, in the following discussion, we always use the notation
where g is a continuous ωperiodic function defined on R.
Definition 2.1
A function \(y(t)=(y_{1}(t),y_{2}(t))^{T}\) defined on \([t_{0}\tau,\infty)\) is said to be a solution of Equation (1.3) with initial condition (1.4) if

(i)
\(y(t)\) is absolutely continuous on the intervals \((t_{0},t_{1}]\) and \((t_{k},t_{k+1}]\), \(k=1,2,\ldots\) ;

(ii)
for all \(t_{k}\), \(k=1,2,\ldots\) , \(y(t_{k}^{+})\) and \(y(t_{k}^{})\) exist, and \(y(t_{k}^{})=y(t_{k})\);

(iii)
\(y(t)\) satisfies the differential equation of (1.3) in \([t_{0},\infty) \backslash\{t_{k}\}\) and the impulsive conditions for all \(t=t_{k}\), \(k=1,2,\ldots\) ;

(iv)
\(y_{i_{t_{0}}}(s)=\varphi_{i}(s)\), \(s\in[\tau,0]\).
Under hypotheses (H_{1})(H_{3}), we consider the following Nicholsontype delay systems without impulsive effects:
with initial conditions
where
\(i=1,2\), \(j=1,2,\ldots,m\).
By a solution \(x(t)\) of Equation (2.1) with initial condition (2.2) we mean an absolutely continuous function \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) defined on \([t_{0},\infty)\) satisfying Equation (2.1) for \(t\geq t_{0}\) and initial condition (2.2) on \([\tau,0]\).
Similarly to the method of [18], we have the following:
Lemma 2.1
Assume that (H_{1})(H_{3}) hold. Then
(i) if \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (2.1) with initial condition (2.2), then \(y(t)=(\prod_{0< t_{k}< t}(1+b_{k})x_{1}(t),\prod_{0< t_{k}< t}(1+b_{k})x_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (1.3) with initial condition (1.4) on \([\tau,\infty)\);
(ii) if \(y(t)=(y_{1}(t),y_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (1.3) with initial condition (1.4), then \(x(t)=(\prod_{0< t_{k}< t}(1+b_{k})^{1}y_{1}(t), \prod_{0< t_{k}< t}(1+b_{k})^{1}y_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (2.1) with initial condition (2.2) on \([\tau,\infty)\).
Proof
(i) If \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (2.1) on \([t_{0},\infty)\), then it is easy to see that \(y(t)\) is absolutely continuous on all intervals \((t_{0},t_{1}]\) and \((t_{k},t_{k+1}]\), \(k=1,2,\ldots\) , and for any \(t\neq t_{k}\),
Similarly, we have
On the other hand, for every \(t=t_{k}\), \(k=1,2,\ldots\) , and \(t_{k}\) situated in \([0,\infty)\),
and
Thus, for every \(t=t_{k}\), \(k=1,2,\ldots\) ,
Therefore, we arrive at the conclusion that \(y(t)\) is the solution (or positive ωperiodic solution) of Equation (1.3) with initial condition (1.4). In fact, if \(x(t)\) is the solution (or positive ωperiodic solution) of Equation (2.1) with initial condition (2.2), then \(y_{i}(t)=\prod_{0< t_{k}< t}(1+b_{k})x_{i}(t)=x_{i}(t)=\varphi_{i}(t)\) on \([\tau, 0]\), \(i=1,2\).
(ii) Since \(y(t)=(y_{1}(t),y_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (1.3) with initial condition (1.4), it follows that \(y(t)\) is absolutely continuous on all intervals \((t_{0},t_{1}]\) and \((t_{k},t_{k+1}]\), \(k=1,2,\ldots\) . Therefore, \(x_{i}(t)=\prod_{0< t_{k}< t}(1+b_{k})^{1}y_{i}(t)\) is absolutely continuous on all intervals \((t_{0},t_{1}]\) and \((t_{k},t_{k+1}]\), \(k=1,2,\ldots\) . Moreover, it follows that, for any \(t=t_{k}\), \(k=1,2,\ldots\) ,
and
which implies that \(x(t)\) is continuous and easy to prove absolutely continuous on \([0,\infty)\). Now, similarly to the proof in case (i), we can easily check that \(x(t)=\prod_{0< t_{k}< t}(1+b_{k})^{1}y(t)\) is a solution (or positive ωperiodic solution) of Equation (2.1) with initial condition (2.2) on \([\tau, \infty]\).
From the above analysis we know that the conclusion of Lemma 2.1 is true. This completes the proof. □
Lemma 2.2
Suppose that
 (H_{4}):

\(\frac{\beta_{1}^{+}\beta_{2}^{+}}{\alpha_{1}^{}\alpha _{2}^{}}<1\).
Proof
Clearly, by Lemma 2.1, we only need to prove that every solution \(x(t)\) of Equation (2.1) with (2.2) is positive and bounded on \([t_{0},\infty)\). In order to show that, we only need to see Lemma 2.3 in [11].
Furthermore, from the proof of Lemma 2.3 in [11] we also obtain the following conclusions: Under the condition (H_{4}), for every solution \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) of Equation (2.1) with (2.2), when \(t>t_{0}\),
and
□
Lemma 2.3
(Cone fixed point theorem [19])
Suppose that \(\Omega_{1}\), \(\Omega_{2}\) are open bounded subsets in Banach space X, and \(\theta\in\Omega_{1}\), \(\overline{\Omega_{1}}\subset\Omega_{2}\). Let P be a cone in X, and \(T:P\cap(\overline{\Omega_{2}}\setminus \Omega_{1})\rightarrow P\) be a completely continuous operator. If

(i)
\(\Tx\\leq\x\\) for \(x\in P\cap\partial\Omega_{1}\) and \(\Tx\\geq\x\\) for \(x\in P\cap\partial\Omega_{2}\), or

(ii)
\(\Tx\\leq\x\\) for \(x\in P\cap\partial\Omega_{2}\) and \(\Tx\\geq\x\\) for \(x\in P\cap\partial\Omega_{1}\),
then the operator T has at least one fixed point in \(P\cap (\overline{\Omega_{2}}\setminus\Omega_{1})\).
Existence and uniqueness of positive periodic solution
For ease of exposition, throughout this paper, we adopt the following notation:
We denote by X the set of all continuously ωperiodic functions \(x(t)\) defined on R, i.e., \(X=\{ x(t)x(t)=(x_{1}(t),x_{2}(t))^{T}\in C(R, R^{2}), x(t+\omega)=x(t)\} \), and denote
Then, X endowed with the norm \(\x\\) is a Banach space. Let P be the cone of X defined by \(P=\{ x(t)\in X x(t)\geq0, t\in[t_{0}, t_{0}+\omega]\}\).
Define the operator T by
where
It is easy to check that Equation (2.1) has positive ωperiodic solution if and only if the operator T has a fixed point in \(P^{0}=\{x(t)\in X x(t)>0, t\in[t_{0}, t_{0}+\omega]\}\). In addition, we have \(0< N_{i}\triangleq \frac{1}{e^{\int_{0}^{\omega}\alpha_{i}(u)\,du}1}=G_{i}(t,t)\leq G_{i}(t,s) \leq G_{i}(t,t+\omega)=\frac{e^{\int_{0}^{\omega}\alpha_{i}(u)\,du}}{e^{\int _{0}^{\omega}\alpha_{i}(u)\,du}1} \triangleq M_{i}\), \(i=1,2\).
Lemma 3.1
Assume that (H_{1})(H_{4}) hold. Then \(T:P\rightarrow P\) is completely continuous.
Proof
First, we prove \(T:P\rightarrow P\). From (H_{3}) we know that \(\alpha_{i}(t)\), \(i=1,2\), are continuous ωperiodic functions. Further, we find
In view of (H_{3}), (3.1), (3.2), and the definition of P, for any \(x\in P\) and \(t\in R\), we have
Similarly, we have
In addition, it is clear that \(Tx\in C(R,R^{2})\) and \((Tx)(t)\geq0\) for any \(x\in P\), \(t\in R\). Hence, \(Tx\in P\) for any \(x\in P\). Thus, \(T:P\rightarrow P\).
Second, we show that \(T:P\rightarrow P\) is completely continuous. Obviously, \(T:P\rightarrow P\) is continuous. Since \(\sup_{u\geq0}ue^{u}=\frac{1}{e}\), by (2.8) and (2.9), for any \(x\in P\) and \(t\in[t_{0},t_{0}+\omega]\), we have
and
Moreover,
Similarly, we have
In view of (3.3)(3.6), \(\{Tx:x\in P\}\) is a family of uniformly bounded and equicontinuous functions on \([t_{0},t_{0}+\omega]\). By the AscoliArzela theorem, \(T:P\rightarrow P\) is compact. Therefore, \(T:P\rightarrow P\) is completely continuous. The proof of Lemma 3.1 is complete. □
Theorem 3.1
Assume that (H_{1})(H_{4}) hold. Then Equation (1.3) with (1.4) has at least one positive ωperiodic solution.
Proof
By (3.3) and (3.4), for any \(x\in P \) and \(t>t_{0}\), we have
Therefore,
For any \(x\in P\) and \(t>t_{0}\), we have
Let \(\tau^{}=\min_{j=1,2,\ldots,m}\{\tau_{1j}^{},\tau_{2j}^{}\} \). There are two possible cases to consider.
Case 1. \(\tau^{}\geq\omega\). In view of (3.8), we have
where \(\varphi_{1}^{}=\min_{\tau\leq s\leq0}\varphi_{1}(t)\), \(\varphi_{1}^{+}=\max_{\tau\leq s\leq0}\varphi_{1}(t)\).
Case 2. \(\tau^{}< \omega\). In view of (3.8), we have
Therefore,
Similarly, we have
where \(A_{21}=N_{2} \omega\sum_{j=1}^{m} p_{2j}^{}\varphi_{2}^{} e^{q_{2j}^{+}\varphi_{2}^{+}}\), \(A_{22}=N_{2} \tau^{} \sum_{j=1}^{m} p_{2j}^{}\varphi_{2}^{} e^{q_{2j}^{+}\varphi_{2}^{+}}\), \(\varphi_{2}^{}=\min_{\tau\leq s\leq0}\varphi_{2}(t)\), \(\varphi_{2}^{+}=\max_{\tau\leq s\leq0}\varphi_{2}(t)\).
Then, for any \(x\in P\) and \(t>t_{0}\),
Let
and
Clearly, \(\Omega_{1}\) and \(\Omega_{2}\) are open bounded subsets in X, and \(\theta\in X\), \(\overline{\Omega_{1}}\subset\Omega_{2}\). By Lemma 3.1, \(T:P\cap (\overline{\Omega_{2}} \setminus\Omega_{1}) \rightarrow P\) is completely continuous.
If \(x\in P\cap\partial\Omega_{2}\), which implies that \(\x\=B\), then from (3.7) we have \(\Tx\\leq B\), and hence \(\Tx\\leq\x\\) for \(x\in P\cap\partial\Omega_{2}\).
If \(x\in P\cap\partial\Omega_{1}\), which implies that \(\x\=A\), then from (3.9) we have \(\Tx\\geq A\), and hence \(\Tx\\geq\x\\) for \(x\in P\cap\partial\Omega_{1}\).
By Lemma 2.3 the operator T has at least one fixed point in \(P\cap(\overline{\Omega_{2}} \setminus\Omega_{1})\), i.e., Equation (2.1) with (2.2) has at least one ωperiodic solution. Since \(\theta \overline{\in} P\cap(\overline{\Omega_{2}} \setminus\Omega_{1})\), Equation (2.1) with (2.2) has at least one positive ωperiodic solution. Therefore, Equation (1.3) with (1.4) has at least one positive ωperiodic solution by Lemma 2.1. This completes the proof of Theorem 3.1. □
Theorem 3.2
Let (H_{1})(H_{4}) hold. Suppose further that the following condition holds:
 (H_{5}):

\(\alpha_{i}^{}\beta_{i}^{+}\sum_{j=1}^{m} p_{ij}^{+}>0\), \(i=1,2\).
Proof
By Theorem 3.1 we know that Equation (2.1) with (2.2) has at least one positive ωperiodic solution. Thus, in order to prove Theorem 3.2, we only need to prove the uniqueness of a positive ωperiodic solution for Equation (2.1) with (2.2).
The following proof is similar to that of Theorem 3.2 in [11].
Assume that \(x(t)\) and \(\widetilde{x}(t)\) are two positive ωperiodic solutions of Equation (2.1). Set \(z_{i}(t)=x_{i}(t)\widetilde{x}_{i}(t)\), where \(t\in[t_{0}\tau, \infty)\), \(i=1,2\). Then
Set
Clearly, \(\Gamma_{i}(u)\), \(i=1,2\), are continuous functions on \([0,1]\). From (H_{5}) we have
Hence, we can choose two constants \(\eta>0\) and \(0< \lambda\leq1\) such that
Consider the Lyapunov functions
Calculating the upper right derivative of \(V_{i}(t) \) (\(i=1,2\)) along the solution \(z(t)\) of (3.10), we obtain
and
We claim that there is \(M>0\) such that
Otherwise, one of the following cases must occur.
Case 1. There exists \(T_{1}>t_{0}\) such that
Case 2. There exists \(T_{2}>t_{0}\) such that
We will need the inequality
Indeed, by the mean value theorem we have
For \(\xi>1\), we have \(\frac{1\xi}{e^{\xi}}=\frac{\xi1}{e^{\xi}}\leq \frac{1}{e^{2}}<1\), and for \(0\leq\xi\leq1\), we have \(\frac{1\xi }{e^{\xi}}=\frac{1\xi}{e^{\xi}}\leq1\). Therefore, inequality (3.17) holds.
In case 1, in view of (3.12) and inequality (3.17), (3.15) implies that
Thus,
which contradicts (3.11). Hence, (3.14) holds.
In case 2, in view of (3.13) and (3.17), (3.16) yields that
Thus,
which contradicts (3.11). Hence, (3.14) holds. It follows that
In view of (3.18) and the periodicity of \(z(t)\), we have
This completes the proof. □
Remark 3.1
In Theorems 3.1 and 3.2, the conditions that ensure the existence and uniqueness of a positive ωperiodic solution for Nicholsontype delay systems with and without impulses are simple and easily to test, which is less conservative than the conditions required in some previous works [11, 12]. Moreover, the main results in this paper are totally different from that of [17].
An example
Example 4.1
Consider the following impulsive Nicholsontype system with delays
with initial condition
where \(b_{k}=2^{\sin\frac{\pi}{2}k}1\), and \(t_{k}=k\), \(k=1,2,\ldots\) .
Let \(f(t)=\prod_{0< t_{k}< t}(1+b_{k})=\prod_{0< t_{k}< t}2^{\sin \frac{\pi}{2}k}\). Then
which implies that \(f(t)\) is a periodic function with period 4.
Since \(\alpha_{1}(t)=9+\sin^{2}\pi t\), \(\alpha_{2}(t)=9+\cos^{2}\pi t\), \(\beta _{1}(t)=5+\cos^{2}\pi t\), \(\beta_{2}(t)=5+\sin^{2}\pi t\), we have \(\alpha_{1}^{}=\alpha_{2}^{}=9\), \(\beta_{1}^{+}=\beta_{2}^{+}=6\), and thus \(\frac{ \beta_{1}^{+}\beta_{2}^{+}}{\alpha_{1}^{}\alpha_{2}^{}}=\frac {4}{9}<1\).
It is obvious that
Therefore,
It follows from Theorem 3.2 that Equation (4.1) with initial condition (4.2) has a unique 4periodic solution. This fact is verified by the numerical simulation in Figure 1.
Remark 4.1
System (4.1) is a simple form of impulsive Nicholsontype system with delays. Since \(q_{11}^{}=q_{21}^{}=\frac{7}{6}>1\), \(q_{12}^{}= q_{22}^{}=\frac{5}{4}>1\), it is clear that the condition of Theorem 3.1 in [11] and Theorem 2.1 in [12] are not satisfied. Therefore, all the results obtained in [11, 12] and the references therein cannot be applicable to system (4.1). This implies that the results of this paper are essentially new.
Conclusion
In this paper, a class of Nicholsontype delay systems with impulsive effects are investigated. First, an equivalence relation between the solution (or positive periodic solution) of a Nicholsontype delay system with impulses and that of the corresponding Nicholsontype 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 periodic 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 [20, 21]. Our results imply that under the appropriate linear periodic impulsive perturbations, the Nicholsontype delay systems with impulses preserve the original periodic property of the Nicholsontype 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 [17] on Nicholsontype 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 Nicholsontype delay systems with impulsive control.
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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).
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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.
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Zhang, R., Huang, Y. & Wei, T. Positive periodic solution for Nicholsontype delay systems with impulsive effects. Adv Differ Equ 2015, 371 (2015). https://doi.org/10.1186/s1366201507052
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Keywords
 Nicholsontype systems
 positive periodic solutions
 delay
 impulsive effect
 cone fixed point theorem