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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 713503, 14 pages
Multiple Periodic Solutions of a Ratio-Dependent Predator-Prey Discrete Model
College of Science, Hunan Agricultural University, Changsha, Hunan 410128, China
Received 29 September 2012; Accepted 6 December 2012
Academic Editor: Xiang Ping Yan
Copyright © 2012 Tiejun Zhou 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.
A delayed ratio-dependent predator-prey discrete-time model with nonmonotone functional response is investigated in this paper. By using the continuation theorem of Mawhins coincidence degree theory, some new sufficient conditions are obtained for the existence of multiple positive periodic solutions of the discrete model. An example is given to illustrate the feasibility of the obtained result.
It is known that one of important factors impacted on a predator-prey system is the functional response. Holling proposed three types of functional response functions, namely, Holling I, Holling II, and Holling III, which are all monotonously nondescending . But for some predator-prey systems, when the prey density reaches a high level, the growth of predator may be inhibited; that is, to say, the predator’s functional response is not monotonously increasing. In order to describe such kind of biological phenomena, Andrews proposed the so-called Holling IV functional response function  which is humped and declines at high prey densities . Recently, many authors have explored the dynamics of predator-prey systems with Holling IV type functional responses [3–11]. For example, Ruan and Xiao considered the following predator-prey model : where and represent predator and prey densities, respectively. In (1.2), the functional response function is a special case of Holling IV functional response.
The functional response functions mentioned previously only depend on the prey . But some biologists have argued that the functional response should be ratio dependent or semi-ratio dependent in many situations. Based on biological and physiological evidences, Arditi and Ginzburg first proposed the ratio-dependent predator-prey model  where the functional response function is ratio dependent. Many researchers have putted up a great lot of works on the ratio-dependent or semi-ratio-dependent predator-prey system [13–19].
Recently, some researchers incorporated the ratio-dependent theory and the inhibitory effect on the specific growth rate into the predator-prey model [3, 7, 11, 15]. Ding et al. considered a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay ; they obtained some sufficient conditions for the existence and global stability of a positive periodic solution to this system. Hu and Xia considered a functional response function [7, 15]: With the functional response function, Xia and Han proposed the following periodic ratio-dependent model with nonmonotone functional response : where , , , , and are all positive periodic continuous functions with period , is a positive real constant, and is a delay kernel function. Based on Mawhins coincidence degree, they obtained some sufficient conditions for the existence of two positive periodic solutions of the ratio-dependent model (1.5).
It is well known that discrete population models are more appropriate than the continuous models when the populations do not overlap among generations. Therefore, many scholars have studied some discrete population models [3, 4, 14, 16–19]. For example, Lu and Wang considered the following discrete semi-ratio-dependent predator-prey system with Holling type IV functional response and time delay : They proved that the system (1.6) is permanent and globally attractive under some appropriate conditions. Furthermore, they also obtained some sufficient conditions which guarantee the existence and global attractivity of positive periodic solution.
Motivated by the mentioned previously, this paper is to investigate the existence of multiple periodic solutions of the following discrete ratio-dependent model with nonmonotone functional response: for , where , , and are all -periodic sequences, is a positive integer, is a positive real constant, and satisfies , where , , , , , and denote the sets of all integers, nonnegative integers, positive integers, real numbers, nonnegative real numbers, and positive real numbers, respectively. The model (1.7) is created from the continuous-time system (1.5) by employing the semidiscretization technique.
The initial conditions associated with (1.7) are of the form where for and .
For convenience, we will use the following notations in the discussion: where is a -periodic sequence of real numbers defined for .
In the system (1.7), the time delay kernel sequence satisfies . Therefore, if we define then is uniformly convergent with respect to , and it satisfies .
Proof. Let ; then the system (1.7) can be rewritten as
Therefore, is a positive -periodic solution of system (1.7) if and only if is a -periodic solution of the system (2.4).
Notice that If is -periodic, then we have Because is uniformly convergent with respect to , so we have Therefore, is a -periodic solution of the system (2.3) if and only if it is a -periodic solution of the system (2.4). This completes the proof.
Throughout this paper, we assume that(H1);(H2).
Under the assumption (H1), there exist the following six positive numbers: Obviously,
In this paper, we adopt coincidence degree theory to prove the existence of multiple positive periodic solutions of (1.7). We first summarize some concepts and results from the book by Gaines and Mawhin . Let and be normed vector spaces. Define an abstract equation in , where is a linear mapping, and is a continuous mapping. The mapping is called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, then there exist continuous projectors and such that and . It follows that is invertible, and its inverse is denoted by . If is a bounded open subset of , the mapping is called -compact on if is bounded and is compact. Because is isomorphic to , there exists an isomorphism .
In our proof of the existence, we also need the following two lemmas.
Lemma 2.2 (continuation theorem ). Let be a Fredholm mapping of index zero and let be -compact on . Suppose that(a)for each , , ;(b)for each ;(c). Then the operator equation has at least one solution in .
Lemma 2.3 (see ). If is a -periodic sequence, then for any fixed ,, one has
3. Existence of Two Positive Periodic Solutions
We are ready to state and prove our main theorem.
Theorem 3.1. Suppose that (H1) and (H2) hold. Then model (1.7) has at least two positive -periodic solutions.
Proof. It is easy to see that if the system (2.3) has a -periodic solution , then is a positive -periodic solution to the system (1.7). Therefore, to complete the proof, it suffices to show that the system (2.3) has at least two -periodic solutions.
We take and define the norm of and for or . Then and are Banach spaces when they are endowed with the previous norm .
For any , because of its periodicity, it is easy to verify that are -periodic with respect to .
Set Obviously, , is closed in , and . Therefore, is a Fredholm mapping of index zero.
Define two mappings and as It is easy to prove that and are two projectors such that and . Furthermore, by a simple computation, we find that the inverse of has the form Evidently, and are continuous by the Lebesgues convergence theorem. Moreover, by Arzela Ascolis theorem, and are relatively compact for the open bounded set . Therefore, is -compact on for the open bounded set .
Corresponding to the operator equation (2.11), we get the following system: where . Suppose that is an arbitrary solution of system (3.8) for a constant . Summing (3.8) over , we obtain From system (3.8), we have By using (3.9) and (3.10), we obtain
Obviously, there exist ,, such that From (3.10), it follows that therefore By using Lemma 2.3 and (3.12), we obtain In particular, we have or The assumption (H1) implies that . So we have From (3.10), we also have it follows that By using Lemma 2.3 and (3.12) again, we have In particular, we have or Therefore,
From (3.12) and (3.20), we have Similarly, from (3.12) and (3.26), we have
By using (3.14), (3.27), and (3.28), it follows from (3.9) that From (3.29), we have In view of (3.12), we obtain Under the assumption (H2), it follows from (3.30) that By using (3.12), we obtain again It follows from (3.32) and (3.34) that
Notice that for . Under the conditions (H1) and (H2), we can obtain two distinct solutions of After choosing a constant such that we can define two bounded open subsets of as follows: It follows from (2.10) and (3.38) that and . Because of , it is easy to see that is empty, and satisfies the condition (a) in Lemma 2.2 for . Moreover, for . This shows that the condition (b) in Lemma 2.2 is satisfied.
Because , we can take the isomorphic as the identity mapping, then we have From (3.37), has two solutions and . Therefore we have Similarly, we can obtain that So the condition (c) in Lemma 2.2 is also satisfied.
By now we know that satisfies all the requirements of Lemma 2.2. Hence the system (2.3) has at least two -periodic solutions. This completes the proof.
4. An Example
In the system (1.7), let , let , let , let , let , and let . Obviously, they are positive periodic sequences with period . The time delay kernel sequence , which satisfies . It is easy to obtain that , , . Therefore, the conditions (H1) and (H2) are satisfied. From Theorem 3.1, the system (1.7) has at least two 3-periodic solutions.
In , Lu and Wang investigated a discrete time semi-ratio-dependent predator-prey system (1.6) with Holling type IV functional response and time delay. They established sufficient conditions which guarantee the existence and global attractivity of a positive periodic solution of the system. In this paper, a ratio-dependent predator-prey discrete-time model with discrete distributed delays and nonmonotone functional response is investigated. By using the continuation theorem of Mawhins coincidence degree theory, we prove that the system (1.7) has at least two positive periodic solutions under conditions (H1) and (H2). As , we would like to know the local stability of the two positive periodic solutions of system (1.7), which is our future work.
This work was supported by the Project of Hunan Provincial Department of Finance ((2010)51).
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