Abstract
This paper deals with feedback control systems on time scales. Firstly, we generalize the semicycle concept to time scales and then establish some differential inequalities on time scales. Secondly, as applications of these inequalities, we study the uniform ultimate boundedness of solutions of these systems. We give a new method to investigate the permanence of ecosystem on time scales. And some known results have been generalized. Finally, an example is given to support the result.
1. Introduction
The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988. The study of dynamic equations on time scales has recently attracted a lot of attention, because it reveals many discrepancies and helps to avoid proving results twice, for differential equations and difference equations, respectively. The time scales calculus has wide applications in various fields like biology, engineering, economics, physics, neural networks, social sciences, and so on [2].
On the other hand, the permanence of a system is very important. As we know, permanence is affected by such factors as environment, food supplies, attack rates, and so on. As an example, Schreiber and Patel [3] showed that ecoevolutionary feedbacks can mediate permanence at intermediate trade-offs in the attack rates. So it is worthwhile considering the permanence of biological models. The permanence of biological systems has been discussed by many authors [4–17]. In the process of studying these problems, the comparison method plays a paramount role. Such as in [4, 6], some sufficient conditions of permanence for biological systems were established by the comparison method. Permanence is also applied in physics, chemistry, physiology, diseases, economics, and so on. Such as in [18], sufficient conditions of the permanence for a SEIRS system were obtained.
To the best of our knowledge, the permanence of ecosystem on time scales with feedback control was first investigated in [13]. The authors considered the -species cooperation system with time delays and feedback controlwhere , with the initial condition on time scales (which is -periodic; i.e., there exits such that if ; then ), where stands for the delta-derivative and , , as , is defined in the second section (Definition 8).
For system (1), they assumed the following: (H1), , , , , , , , and are all rd-continuous, real-valued functions which are bounded above and below by positive constants; are positive constants. , where can be found in [13]. (H2) are all positively regressive ( is defined as in the second section (Definition 12)), in which they used the notation where is a bounded function in .
They first give the definition of the permanence of ecosystem (1), and the definition of the permanence of other ecosystems mentioned in this paper is similar.
Definition 1 (see [13]). System (1) is said to be permanent if there exist -dimensional positive constant vectors , , , and , which are independent on the initial condition, such that, for any solution , ,
Then the main results in [13] can be rewritten as follows.
Theorem 2. Assume that (H1) and (H2) hold, and , where can be found in [13]. Then system (1) is permanent.
If , one refers the readers to [14, 17]. Chen and Xie [14] considered the following system:where , , and are all nonnegative continuous functions such that , , and , . It is easy to see that the integrations have no influence on the permanence of system (5). They cancelled the additional limitations of [17] and obtained the following.
Theorem 3. Assume that (H1) holds. Then system (5) is permanent.
If , one refers the readers to [15, 16]. Let ; they discussed the discrete form of cooperative system (1):In [16], Li and Zhang used the techniques in [11] instead of the comparison theorem in [15] and obtained the following.
Theorem 4. Assume that (H1) holds. Then system (6) is permanent.
Obviously, in Theorem 2, if (or , the conditions are stronger than that in Theorem 3 (or Theorem 4).
This paper deals with system (1) with the initial conditionand one assumes that , for , which implies that system (1) is valid; , which ensures the ultimate boundedness of the solutions; and there exists a positive constant such thatone can see that (8) includes the condition: is -periodic (in [13]). In fact, if is -periodic, then
Our aim is to unify Theorems 3 and 4 completely by cancelling the additional limitations of Theorem 2.
For simplicity, one first considers a special case (logistic system) of system (1)with the initial condition assume that and (H3) are all bounded, nonnegative, rd-continuous, and real-valued functions.
Let ; if , then system (9) reduces to the continuous system
In this continuous case, one refers the readers to [7, 19, 20]. They considered the system with feedback control in periodic case or boundedness case and obtained permanence, stability, and existence of periodic solutions for the system.
Let ; if , then system (9) is reformulated asthis system or its other forms attracted much interest; one refers the readers to [10–12, 21]. They also considered this system in periodic case or boundedness case and obtained permanence and existence of periodic solutions for this system. Chen [10] discussed the permanence of system (12). Using the comparison theorem, he obtained the following.
Theorem 5. Assume (H3) and (H4), where and hold. Then system (12) is permanent.
But for system (12) itself, (H4) may not be necessary. Fan and Wang [11] used a method instead of the comparison theorem and obtained the following.
Theorem 6. Assume (H3) holds. Then system (12) is permanent.
One generalizes the method of semicycle [11] to time scales instead of the comparison theorem. By establishing some differential inequalities on time scales, one proves that solutions of systems (1) and (9) are uniformly ultimate bounded. Our result is a unification and extension of Theorems 3 and 4.
The remainder of this paper is arranged as follows. In Section 2, we give some basic properties about time scales and generalize the semicycle concept to time scales. Two differential inequalities on time scales are proved in Section 3. Section 4 is devoted to uniform ultimate boundedness of solutions of systems (1) and (9). An example is presented in Section 5 to support the result. Conclusions are presented in Section 6.
2. Preliminaries
First we give some properties about time scales before our main result (see [1, 2]).
Definition 7 (see [1, 2]). A time scale is an arbitrary nonempty closed subset of the real number .
Definition 8 (see [1, 2]). For we define the forward jump operator and the backward jump operator , by respectively.
Throughout the paper we make an assumption that , are points in .
Definition 9 (see [2]). Define the interval in by and as usual, we assume , for all , is a constant.
We note that , for , where .
Lemma 10 (see [2]). If , where , then
Lemma 11 ((existence of antiderivatives) (see [2])). Every rd-continuous function has an antiderivative. In particular if , then is an antiderivative of .
Definition 12 (see [2]). We say that a function is regressive provided holds, where throughout the paper The set of all regressive and rd-continuous functions will be denoted in this paper by
Definition 13 (see [2]). If , we define the exponential function by where the cylinder transformation , for .
For , we define . In this case, , , and .
When , we know that ; then for ,
Definition 14 (see [2]). If , the function is defined by
Lemma 15 (see [2]). Suppose ; then(1);(2);(3);(4);(5).
Similar to the definition of semicycle in discrete case [22], one gives the following.
Definition 16. Let be a constant and ; a positive semicycle relative to of is defined as follows: it consists of a “string" of terms all greater than or equal to ; and a negative semicycle relative to of is defined as follows: it is a “string" of terms all less than or equal to .
3. Differential Inequalities
By using the semicycle concept and the comparison theorem, we give some differential inequalities as follows.
Lemma 17. Assume that and , and further suppose that satisfies the following:
(i)then(ii)and there exists a constant , such that ; then
Proof. (i) According to the oscillating property of , we divided the proof into two cases. Firstly we assume that does not oscillate ultimately about . If there exists a constant , such that , for , then thus (26) holds true. If there exists a constant , such that , for , then , which means that is eventually decreasing; therefore, exists. Then (25) yields , which leads to Finally, we assume that oscillates about , by (25), we know that implies . Thus, by semicycle concept, we assume that , for , , where is an index set, and the interval satisfies the following:(a)For any , if , .(b)If is left-scattered, then .(c)If is left-dense, then there exists a hollow left neighbourhood of such that , for .(d)If is right-scattered, then .(e)If is right-dense, then there exists a hollow right neighbourhood of such that , for .Notice that . If is left-scattered, by integrating (25) over the set , we have by (b), it follows thatand from (8), we know ; then .
If is left-dense, we choose , such that . By integrating (25) over the set , we have notice that , and it follows thatThen which completes the proof of (i).
(ii) The proof is similar to (i) of Lemma 17.
Lemma 18. Assume that are bounded and rd-continuous functions, , and ; further suppose that satisfies the following:
(iii) then there exists a constant , such that, for , one has Particularly, if is bounded above ultimately with respect to , then (iv) there exists a constant , such that for, , one has Particularly, if is bounded below ultimately with respect to , then
Proof. (iii) From , we have . Note that , when . Hence, there exists a , such that , as . We use the product rule to calculate noticing that is rd-continuous, from Lemma 11, and by integrating both sides over the set , we have then Thus (iv) From (iii) of Lemma 18, we know that there exists a constant , such that and ; then for , then Thus, Then the proof of Lemma 18 is completed.
4. Applications
Before giving our main result, we list the definition of uniform ultimate boundedness.
Definition 19. The solution of system (9) is said to be uniformly ultimate bounded if there exist two constants and such that, for any initial condition , where and are independent on .
The definition of the uniform ultimate boundedness of solutions of -species system is similar.
Then we give the applications of these differential inequalities in this section.
Theorem 20. Assume that (H3) holds, . Then the solutions for system (9) are uniformly ultimate bounded.
Proof. From , we have . From (9) and , we can see then from Lemma 17, we haveFrom (9) and (53), there exists a constant , such that, for , from Lemma 18, we haveFrom (9) and Lemma 18, then Note that , when . Hence, there exists , such that , for . Then we getfor .
Note that , from (9) and (58), when , noticing that , from Lemma 17, we haveFrom (9) and (60), there exists a constant , such that, for , from Lemma 18, we haveFrom (53), (55), (60), and (62), we obtain then the solutions for system (9) are uniformly ultimately bounded.
This completes the proof of Theorem 20.
Theorem 21. Assume the assumption (H5), , , , , , , , and are all bounded, nonnegative, rd-continuous, and real-valued functions; are positive constants. holds; . Then the solutions of system (1) are uniformly ultimate bounded.
Proof. By similar analysis as that in Theorem 20 and the initial condition (7), we have . From (1), we can see , by integrating both sides over the set ,so ; then the first equation of (1) implies from Lemma 17, we haveFrom (1) and (66), there exists a constant , such that, for , from Lemma 18, we haveFrom (1), (66), and (68), we have by integrating both sides over the set , we have ; thus from (1) and Lemma 18, then Note that , when . Hence, there exists such that for . Then we getfor .
Let and ; noticing that and , from (1) and (73), when ,noticing that , denote ; from Lemma 17, we haveFrom (1) and (75), there exists a constant , such that, for , from Lemma 18, we haveFrom (66), (68), (75), and (77), we obtain then the solutions for system (1) are uniformly ultimate bounded.
Therefore, the proof of Theorem 21 is completed.
Noticing the differences of Definitions 1 and 19, due to the exponential transformations in Section 1, when solutions of system (1) or (9) are uniformly ultimate bounded, systems (5) and (6) or (11) and (12) are permanent, and the contrary also holds true. Then we have the following remarks.
Remark 22. If , then Theorems 20 and 21 reduce to Theorems 6 and 4, respectively.
If , we could get a new method to solve the continuous result from Theorem 20. Thus, our result is a well unification of continuous and discrete cases.
Remark 23. If , then the result implies Theorem 3.
It is obvious that the conditions of Theorem 21 are weaker than that in Theorem 2, and it includes the known results of [14, 16] on or , respectively. Thus, Theorem 21 is a well unification of Theorems 3 and 4. And our result shows that the additional conditions in [15, 17] are not necessary.
5. An Example
In this section, we give a numerical example to support our main result. Let ; we assumeit is easy to see that the conditions in Theorem 20 are verified. In this case, we know that here . By applying numerical analysis from Matlab, we obtain Figure 1. It is obvious that the solutions for system (79) are uniformly ultimate bounded. Our numerical simulation supports our theoretical findings (see Figure 1).
6. Conclusion
In this paper, we consider two systems with feedback control on time scales. It is shown that these systems contain the differential and difference systems as special cases. By constructing some differential inequalities on time scales, the uniform ultimate boundedness of solutions for these systems is obtained. Compared with some previous relative results, our results could unify the continuous and discrete cases sufficiently (see Theorems 3 and 4). This provides a new method to get the permanence of ecosystems. Our results generalize and extend some known results. And a numerical example shows the verification of our main results.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
Research was supported by the National Natural Science Foundation of China (11201213 and 11371183), Natural Science Foundation of Shandong Province (ZR2015AM026 and ZR2013AM004), and the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07).