Research Article | Open Access
Double Periodic Solutions for a Ratio-Dependent Predator-Prey System with Harvesting Terms on Time Scales
We investigate a nonautonomous ratio-dependent predator-prey model with Beddington-DeAngelis functional response and multiple harvesting (or exploited) terms on time scales. By means of using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of at least two periodic solutions for the system. Moreover, when the time scale is chosen as or , the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.
In recent years, the existence of periodic solutions for the predator-prey model has been widely studied. Models with harvesting (or exploited) terms are often considered (see, e.g., [1–4]). Generally, the model with harvesting (or exploited) terms is described as follows: where and are functions of time representing densities of prey and predator, respectively; and are harvesting (or exploited) terms standing for the harvests (see ). Considering the inclusion of the effect of changing environment, Zhang and Hou  considered the following model of ordinary differential equations with Holling-type II functional response and harvesting (or exploited) terms: where the parameters in system (1.2) are continuous positive -periodic functions. Authors discussed the existence of positive periodic solutions of system (1.2) in the region .
On the other hand, the theory of calculus on time scales unifies continuous and discrete analysis, many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on general time scales can reveal such discrepancies and help avoid proving results twice-once for differential equations and once again for difference equations. The two main features of the calculus on time scales are unification and extension. To prove a result for a dynamic equation on a time scale is not only related to the set of real numbers or set of integers but those pertaining to more general time scales.
The principle aim of this paper is to systematically unify the existence of multiple periodic solutions of population models modelled by ordinary differential equations and their discrete analogues in form of difference equations and to extend these results to more general time scales. The approach is based on a continuation theorem in coincidence degree, which has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations. Therefore, we consider the following ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms on time scales : where and represent the prey and the predator population, respectively; and are all rd-continuous positive -periodic functions denoting the prey intrinsic growth rate, death rate, capture rate, conversion rate of predator, death rate of predator, harvesting rate, and Beddington-DeAngelis functional response parameters, respectively.
Remark 1.1. In (1.3), set . If , then (1.3) reduces to the ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms governed by the ordinary differential equations If , then (1.3) is reformulated as which is the discrete time ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms and is also a discrete analogue of (1.3).
A time scale is an arbitrary nonempty closed subset of the real numbers . Throughout this paper, we assume that the time scale is unbounded above and below, such as , and . The following definitions and lemmas can be found in .
Definition 2.1 (see ). One says that a time scale is periodic if there exists such that if , then . For , the smallest positive is called the period of the time scale.
Definition 2.2 (see ). Let be a periodic time scale with period . One says that the function is periodic with period if there exists a natural number such that , for all and is the smallest number such that . If , one says that is periodic with period if is the smallest positive number such that for all .
Notation 2.3. To facilitate the discussion below, we now introduce some notation to be used throughout this paper. Let be -periodic; that is, implies , where is an -periodic function.
Notation 2.4. Let , be two Banach spaces, let be a linear mapping, and let be a continuous mapping. If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , then the restriction is invertible. Denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .
Lemma 2.5 (Continuation theorem ). Let , be two Banach spaces, and let be a Fredholm mapping of index zero. Assume that is -compact on with being open bounded in . Furthermore assume the following:(a)for each , , ;(b)for each , ;(c).Then the operator equation has at least one solution in .
3. Existence of Periodic Solutions
Our main result on the existence of two positive periodic solutions for system (1.3) is stated in the following theorem.
Theorem 3.1. Assume that the following holds:(i),(ii),(iii), then system (1.3) has at least two positive -periodic solutions.
where is the Euclidean norm. Then and are both Banach spaces with the above norm . Let
and . Since is closed in , then is a Fredholm mapping of index zero. It is easy to show that , are continuous projectors such that and . Furthermore, the generalized inverse (to ) exists and is given by
Obviously, , are continuous. Since is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that is compact for any open bounded set . Moreover, is bounded, thus, is -compact on for any open bounded set . Corresponding to the operator equation , , we have
Suppose that is a solution of (3.7) for certain . Then there exist , such that It is clear that . From this and system (3.7), we have It follows from (3.9a) that From (3.9b), we obtain which leads to Equation (3.10a) yields Equation (3.10b) deduces which implies From (3.9a), we also have which implies that Set then From (3.10a), a parallel argument to (3.20) gives From (3.9b), we also obtain which implies Set then From (3.11), (3.14), (3.20), and (3.21), we obtain for all , or From (3.13), (3.16), and (3.25), we obtain that for all , Obviously, , , , , are independent of . Now take then and are bounded open subsets of , and . Thus and satisfy the requirement (a) in Lemma 2.5.
Now we show that (b) of Lemma 2.5 holds; that is, we need to prove when , , . If it is not true, then when , , constant vector with , , satisfies . Thus there exist two points such that From this and following the arguments of (3.26)–(3.28), we have Thus , or . This contradicts the fact that , . Hence condition (b) in Lemma 2.5 holds.
Finally, we show that (c) in Lemma 2.5 is valid. Noticing that the system of algebraic equations has two distinct solutions and since the conditions hold in Theorem 3.1, and we have where It is easy to verify that Therefore, , .
Define the homotopy by where and is a parameter, , , and is a constant vector in . It is easy to show that . In fact, if there are a certain and a certain such that , where , then there exist such that By carrying out similar arguments as above, we also obtain conclusions as same as (3.26)–(3.28), that is, This contradicts the fact that .
Note that (identical mapping), since , according to the invariance property of homotopy, direct calculation produces where is the Brouwer degree, and , are positive solution for (3.32) in , respectively. And otherwise, we have After test and verification, it is not possible. Thus
By now we have proved that verifies all requirements of Lemma 2.5. Therefore, system (1.3) has at least two -periodic solutions in , respectively. The proof is complete.
Remark 3.3. If in system (1.3), then the system is a ratio-dependent predator-prey model with Holling-type functional response and harvesting terms on time scales: Therefore, we have the following results.
Corollary 3.4. If the conditions in Theorem 3.1 hold, then system (3.43) has at least two -periodic solutions. Specifically, both the corresponding continuous model and the discrete model of the system (3.43) have at least two -periodic solutions.
Remark 3.5. If , in system (1.3), then the system is a general ratio-dependent predator-prey model with Beddington-DeAngelis functional response on time scales: Using Lemma 2.5, we can obtain another important conclusion as follows.
Theorem 3.6. Assume that the following holds:(i),(ii),(iii), then system (3.44) has at least a positive -periodic solutions.
The author is very grateful to his supervisor Professor M. Fan for his many valuable comments and suggestions which greatly improved the presentation of this paper. In addition, he would like to express his gratitude to the reviewers and editor for their valuable comments and careful reading and the time and efforts he/she has spent in the review. This work is supported by the Foundation for Subjects Development of Harbin University (no. HXK200716).
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Copyright © 2009 Zhenjie Liu. 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.