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ISRN Mathematical Analysis
Volume 2014 (2014), Article ID 347201, 10 pages
Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse
1Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China
2Department of Mathematics, National University of Defense Technology, Changsha 410073, China
Received 2 October 2013; Accepted 20 November 2013; Published 17 February 2014
Academic Editors: N. Shanmugalingam and T. Tran
Copyright © 2014 Zhenguo Luo 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.
We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria.
One of the powerful and effective methods on the existence of periodic solutions to periodic systems is the continuation method, which gives easily verifiable sufficient conditions. See Gaines and Mawhin  for detailed description of this method. In , Chen studied the following periodic predator-prey system with a Holling type IV functional response: where , , , , and , , are -periodic functions with and , and , are positive constants. The results on the existence of multiple periodic solutions have been obtained by employing the continuation method. There are some works following this direction. See, for example, [3–6].
To generalize Chen’s results, Ding and Jiang  considered the following periodic Gause-type predator-prey system with time delays: where , , , and are continuous -periodic functions with . They also afforded verifiable criteria for the existence of multiple positive periodic solutions for the system (2) when the numerical response function is nonmonotonic. Their results improve and supplement those in .
As we know, in population dynamics, many evolutionary processes experience short-time rapid change after undergoing relatively long smooth variation. For example, the harvesting and stocking occur at fixed time, and some species usually immigrate at the same time every year. Incorporating these phenomena gives us impulsive differential equations. For theory of impulsive differential equations, we refer to [7–16].
Based on the previous ideas, in , Wang, Dai, and Chen considered the following impulse predator-prey system with a Holling type IV functional response: where the assumptions on , , , , , , , , , are the same as (1), (, , ), is a strictly increasing sequence with , and . Further, there exist a such that (, ) and for . By employing the continuation theorem, they presented sufficient conditions on the existence of two positive periodic solutions to system (3).
In this paper, we will consider the following Gause-type predator-prey systems with impulse and time delays: where the assumptions on , , , , , and are the same as (3). and are the prey and the predator population size, respectively. The function is the growth rate of the prey in the absence of the predator and is the death rate of the predator. The function , called functional response of predator to prey, describes the change in the rate of exploitation of prey by a predator as a result of a change in the prey density. The function , called numerical response of predator to prey, describes the change in reproduction rate with change in the prey density.
In general, the response function is monotone (see [18–20]). However, there is nonmonotonic response occurrence; see Kuang and Beretta . The so-called Monod-Haldane function in (5) has been proposed and used to model; see . Sokol and Howell  proposed a simplified Monod-Haldane function of the form in (6)
Throughout this paper, we assume the following:, , and are continuous functions and -periodic with respect to ; , , and are also continuous functions;there exists a positive constant such that for ; there also exists a continuous -periodic function such that and for ;, for , ; there exists a positive constant such that for , ;, ; there exists a positive constant such that for ;; .
Remark 1. The assumption demonstrates that numerical response function is non-monotonic. The function in (5) and the function in (6) both satisfy the condition .
The main purpose of the present paper is, by using the coincidence theory developed by Gaines and Mawhin , to establish the sufficient conditions for the existence of multiple positive periodic solutions of system (4) when is a nonmonotonic numerical response function. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria.
The method to be used in this paper involves the applications of the continuation theorem of coincidence degree. For the readers’ convenience, we introduce a few concepts and results about the coincidence degree as follows.
Let , be two real Banach spaces, let be a linear mapping, and let be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , it follows that is invertible; we denote the inverse of that map by . If is an open bounded subset of , the mapping will be called - on if is bounded and is compact. Since is isomorphic to , there exist isomorphisms . Let denote the space of - periodic functions which are continuous for , are continuous from the left for , and have discontinuities of the first kind at point . We also denote .
Lemma 2 (see continuation theorem ). Let and be two Banach spaces and let be a Fredholm operator with index zero. is an open bounded set, and let be L-compact on . Suppose that(a) of for such , and ;(b) for each ;(c). Then, the equation has at least one solution lying in .
For convenience, we make change of variables , and the system (4) is reformulated as It is easy to see that if system (8) has one -periodic solution , then is a positive -periodic solution of system (4). Therefore, to complete the proof, it suffices to show that system (8) has multiple -periodic solutions.
We take and define where is the Euclidean norm of . Then and are Banach spaces.
Let and withIt is not difficult to show that and . So, is closed in , and is a Fredholm mapping of index zero. Take
It is trivial to show that , are continuous projectors such that and, hence, the generalized inverse exists. In the following part, we first devote ourselves to deriving the explicit expression of . Taking , then exists an such that
Then direct integration produces
Note that ; that is, , which, together with (17), implies then that is,
Thus, for ,
Clearly, and are continuous. By applying Ascoli-Arzela theorem, one can easily show that , are relatively compact for any open bounded set . Moreover, is obviously bounded. Thus, is -compact on for any open bounded set .
In what follows, we shall use the notations where is a continuous -periodic function, is a continuous function, and -periodic with respect to .
We also set
3. Existence of Multiple Positive Periodic Solutions
By and , we have Then is strictly increasing on and strictly decreasing on . By and , if ,
then, the equation has two distinct positive solutions, namely, . Without loss of generality, we suppose that , and then .
Now, we are ready to state and prove our main result.
Theorem 4. In addition to the conditions , suppose further that the following conditions hold:;.
Then, system (4) has at least two -periodic solutions with strictly positive components.
Proof. By Lemma 2, we need to search for two appropriate open bounded subsets . Considering the corresponding operator equation , , we have
Let be a -periodic solution of system (26) for a certain , integrating both sides of the first and second equations of (26) over the interval ; we obtain
From the first equation of (27), we have
It follows from (26)–(28) that
Note that ; then there exist such that
By (27), (30), and the monotonicity of and , we will show that and can not simultaneously lie in , , or . In fact, if , then
This is a contradiction. If , then
This is a contradiction. If , then
This is also a contradiction. Consequently, the distributions of and only have the following two cases.
Case 1 . From the first equation of (29) and Lemma 3, we have
Case 2 . From the first equation of (29) and Lemma 3, we also have By , we know Denote that By , and , one can easily see that , and are positive constants. Noticing that it follows from the first equation of (27), (30), and that which implies, by , Similarly, we also obtain which implies It follows from the second equation of (29), (40), (42), and Lemma 3 that In view of (43), we have Clearly, , , , , and are independent of .
It is easy to show that algebraic equations have two distinct solutions . Choose such that We now define Then both and are bounded open subsets of . It follows from (36) and (46) that , and . With the help of (34), (35), (44), and (46), it is easy to see that satisfies condition in Lemma 2. When , is a constant vector in . Thus, we have that is, the condition in Lemma 2 holds. In order to verify the condition in Lemma 2 and since the algebraic equations (45) have only one root in , let given by , where ; in view of the assumptions in Theorem 4, it is easy to see that and a direct computation gives By now we have proved that satisfies all the requirements of Lemma 2. Consequently, the system (8) has at least two -periodic solutions in , that is , and , respectively. Set , ; , ; then , are two positive -periodic solutions of the system (4). This completes the proof.
In a weaker condition, we have the following result for the existence of one positive periodic solution.
Theorem 5. In addition to , suppose further that the following holds:.Then, system (4) has at least one -periodic solution with strictly positive components.
Proof. The proof is similar to the proof of Theorem 4. Under condition , (36) is no longer true and . So, we have to make a corresponding change. By , we can know that
By , , and , one can easily see that , and are positive constants. By a similar analysis as that in Theorem 4, when , we have
Clearly, , , and are independent of .
It is easy to show that algebraic equations (45) have at least one solution . We now take . It follows from (46) and (51) that . By a similar analysis as that in Theorem 4, it is easy to see that satisfies condition and in Lemma 2. In order to verify the conditions in Lemma 2, a direct calculation shows that Hence, satisfies all the requirements in Lemma 2. Consequently, the system (8) has at least one -periodic solution in , say . Set ; then is one positive -periodic solution of the system (4). This completes the proof.
Remark 6. When there is no impulse, that is, , . The conditions and are automatically satisfied and , and reduce to;;.
Hence, we have the following corollaries.
Corollary 7. In addition to (H1)–(H5), suppose further that and hold. Then system (2) has at least two positive periodic solutions.
Corollary 8. In addition to , suppose further that holds. Then system (2) has at least one positive periodic solution.
Remark 9. In , Ding and Jiang got the following results.
Theorem A. In addition to the conditions , suppose further that the following conditions hold:;.Then, system (2) has at least two -periodic solutions with strictly positive components.
Theorem B. In addition to , suppose further that the following holds:. Then, system (2) has at least one -periodic solution with strictly positive components.
Obviously, the conditions , , and are weaker than the corresponding , , and , respectively. Hence, our results generalize and improve the corresponding results of .
In this section, we will list some applications of the previous results.
Theorem 10. Suppose that the following conditions hold:(1); (2), ;(3).Then, system (3) has at least two -periodic solutions with strictly positive components.
Remark 11. In Theorem 3.2 of , Wang et al. proved that system (3) has at least two -periodic solutions with strictly positive components under the conditions:;
+ .Obviously, implies (2). Notice that
Thus, also implies (3). Hence, Theorem 10 improves Theorem 3.2 in .
By Theorem 4, we have the following result.
Theorem 12. Suppose that the following conditions hold:(1); (2); (3).Then, system (59) has at least two -periodic solutions with strictly positive components.
Application 3. Consider the following system: which is a special case of (4) by letting where functions , , , , , , , and constant are defined as above, is a positive continuous -periodic function, and the prey population follows the Allee effect  model. By Theorem 4, we have the following result.
Theorem 13. Suppose that the following conditions hold:(1); (2); (3),Then, system (61) has at least two -periodic solutions with strictly positive components, where
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research is supported by NSF of China (nos. 11161015, 11371367, and 11361012), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, 12C0541, and 13C084), and the Construct Program of the Key Discipline in Hunan province.
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