Permanence and Existence of Periodic Solutions for Nicholson’s Blowflies Model with Feedback Control and Delay on Time Scales

Abstract

Sufficient conditions are established for the permanence in a delayed Nicholson’s blowflies model with feedback control on time scales. Our investigation confirms that the bounded feedback terms do not have any influence on the permanence of this system.

1. Introduction

In 2008, we considered the following discrete Nicholson’s blowflies model with feedback control (see [1]):where , , , , and : (integer number set) (nonnegative real number set) are all bounded sequences, is also a bounded sequence and , is a nonnegative integer, and is the first-order former difference operator and obtained the following.

Theorem 1. Assume thathold true; then system (1) is permanent.

The continuous and discrete systems always appear separately, until in 1988, the theory of time scales, which has recently received much attention, was initiated by Hilger [2] in his Ph.D. thesis to unify both difference and differential calculus in a consistent way. Since then many authors have investigated the dynamic equations on time scales (see [3–6]). This theory provides a powerful tool for applications to economics, population models, quantum physics, among others. In fact, the progressive field of dynamic equations on time scales contains, links, and extends the classical theory of differential and difference equations.

For the origin of mathematical model for Nicholson’s blowflies, one can see [7]. For further study on equations with feedback control, we refer to [8–14] and references therein. And for investigation on delay differential equations, we refer to [15, 16] and so on.

In this paper, we will discuss the permanence of the following system:on time scales , where stands for the delta-derivative andin which and is the forward jump operator on . To make (3) meaningful, we suppose that, for any ,

We assume that , and, without loss of generality, suppose . In view of the biological significance, we also assume that the coefficient functions , , , , , , , and : are all bounded rd-continuous and (the definition of rd-continuous function will be given in Section 2).

When , let , , and then (3) can be rewritten as

When , if we let , then (3) can be rewritten asObviously, (7) includes (1).

In what follows we shall use the notations where is a bounded rd-continuous function in . Throughout this paper, we assume thatIn this case, for any ,

2. Preliminary

Before giving our main result, first we list some basic properties about time scales which could be found in ([2, 17, 18]).

Definition 2. A time scale is an arbitrary nonempty closed subset of the real number .

Definition 3. For we define the forward jump operator and the backward jump operator , by respectively.

Definition 4. Define the interval in by Other types of intervals are defined similarly.

Definition 5 (Definition 1.58 in [18], P22). A function is called rd-continuous provided that it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions is denoted by

Definition 6. Assume and let , where Then we define to be the number (provided it exists) with property that given any , there is a neighborhood of such that for all . We call the delta (or Hilger) derivative of and it turns out that is the usual derivative if and is the usual forward difference operator if .

Lemma 7 (Theorem 1.76 in [18], P28). If , then is increasing.

Definition 8 (Definition 2.25 in [18], P58). We say that a function is regressive provided thatholds. The set of all regressive and rd-continuous functions will be denoted by

Definition 9. If , we define the exponential function bywhere the cylinder transformation , for .
For , we define . In this case, , , and .
When , we know that , and then for ,

Definition 10. If , the function is defined by

Lemma 11. Suppose ; then(1);(2);(3);(4);(5).

Lemma 12 (Theorem 2.44 in [18], P66). Assume that and . If on , then for all .

Lemma 13 (Theorem 2.77 in [18], P77). Suppose and is rd-continuous. Let and . The unique solution of the initial value problem is given by

In order to give our main result, we also need to establish the following definitions and lemmas. The first definition is the generalized version of the semicycle in discrete situation [19].

Definition 14. Let be a constant and ; a positive semicycle relative to   of is defined as follows, as it consists of a “string” of terms:all greater than or equal to , and a negative semicycle relative to   of is defined as follows, as it is a “string” of terms:all less than or equal to .
The following two lemmas could be found in [20].

Lemma 15. Assume that and ; furthermore suppose that
(1)and then(2)and there exists a constant , such that ; then

Lemma 16. Assume that are bounded rd-continuous functions and ; furthermore suppose that the –differentiable function satisfies
(1)and is –differentiable; then there exists a constant , such that for , we haveEspecially, if is bounded above ultimately with respect to , then(2)and there exists a constant , such that, for , we haveEspecially, if is bounded below ultimately with respect to , thenBefore giving our main result, we list the definition of uniform ultimate boundedness.

Definition 17. Solutions for system (3) are said to be uniformly ultimate bounded if there exist two constants and such that, for any initial conditionwe havewhere and are independent of .

3. Main Results

First, we give a lemma which will be useful for our further discussion.

Lemma 18. Let be any solution of system (3) with initial condition (35); then

Proof. The exponential form ensures that for all Now we consider the second equation of system (3); by Lemma 13 and (4), we haveThe proof is complete.

In the sequel, we assume that

Theorem 19. Assume that (39) holds true; then system (3) is uniformly ultimate bounded.

We now prove the following result before proving Theorem 19. In fact, the two theorems are equivalent to each other.

Theorem 20. Assume that (39) holds true; then there exist two positive constants and such that andfor any solutions of (3) with initial condition (35).

Proof. We divided the proof into four claims.
Claim 1. There exists a positive constant such thatwhere can be chosen as Proof. From the first equation of system (3) and the positivity of , we haveFrom the first equation of (3), we havethuswhich implies thatand by (44), we haveand for simplicity, we setobviously, In the following, we divided the proof into three cases:
(1) There exists some such that for all , and then we have(2) There exists some such that for all , and then from (48), we have which implies that exists and if we denote , and thenand thus (48) shows thatthat is,while from (52) and (54), we get therefore(3) For simplicity, set Assume that oscillates about . By (48), we know that implies that Thus, by Lemma 7, if we let be the first element of a th positive semicycle relative to of , thenNow we divided the proof into two cases: is left-dense and is left-scattered. If the former holds, thenIf the latter holds, then , by (45), we haveand thereforewhile from (58) and (60), we haveBy (50), (56), and (61), we complete the proof of Claim 1.
Claim 2. There exists a positive constant such thatProof. From (42), we know that, for any arbitrary positive number , there exists an such that for all , and then from the second equation of system (3), we have by Lemma 16 (1),Let , and then we can obtainThe proof of Claim 2 is complete.
Claim 3. There exists a constant such thatBy Claims 1 and 2 and the first equation of system (3), we havefor sufficiently large, where is a constant. Then From the second equation of system (3), we have and then Lemma 16 (1) implies that, for any ,noting thatand hence there exists a positive integer such that, for any solution of system (3), , as , where . Fixing , we get Setand thenNotice that , for , and then from the first equation of system (3), we have for . According to the choosing of , we haveBy Lemma 15 (2), we havewhereand choosingthis shows that the conclusion holds true.
Claim 4. There exists a positive constant such thatNotice that (66) implies that there exists an such that for all , and then from the second equation of system (3), we have by Lemma 16 (2),The proof of Claim 4 is complete.
Chooseobviously, AndThus we complete the proof of Theorem 20.

Theorem 21. If the coefficient functions , , , , , , , and are all –periodic rd-continuous functions and also satisfy and (9), then system (3) has at least an –periodic solution with strictly positive for its second component.

Corollary 22. If all the conditions in Theorem 21 hold, then (6) or (7) has at least a positive –periodic solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by NSF of China (11201213, 11371183), NSF of Shandong Province (ZR2015AM026), and the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07).