Journal of Applied Mathematics

Volume 2014 (2014), Article ID 592543, 21 pages

http://dx.doi.org/10.1155/2014/592543

Research Article

Existence and Uniqueness of Positive Periodic Solutions for a Delayed Predator-Prey Model with Dispersion and Impulses

Zhenguo Luo,1,2 Liping Luo,¹ Liu Yang,¹ Zhenghui Gao,¹ and Yunhui Zeng¹

¹Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China

²China Department of Mathematics, National University of Defense Technology, Changsha, Hunan 410073, China

Received 28 September 2013; Revised 6 January 2014; Accepted 9 January 2014; Published 27 March 2014

Academic Editor: Zhijun Liu

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.

Abstract.

An impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays is investigated, where we assume the model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches. By applying the continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system. Some known results subject to the underlying systems without impulses are improved and generalized. As an application, we also give two examples to illustrate the feasibility of our main results.

1. Introduction

The aim of this paper is to investigate the existence and uniqueness of the positive periodic solution of the following impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays:

with the following initial conditions:

where represents the prey population in the ith patch and represents the predator population for both patches. is the intrinsic growth rate of the prey in the ith patch and are the density-dependent coefficients of the prey at the ith patch. and are the capturing rates of the predator in patches 1 and 2, respectively, and and are the conversion rates of nutrients into the reproduction of the predator. is the death rate of the predator and denotes the dispersal rate of the prey in the ith patch . is the delay due to gestation; that is, mature adult predators can only contribute to the production of predator biomass. In addition, we have included the term in the dynamics of the predator to incorporate the negative feedback of predator crowding, where represent the population at regular harvest pulse.

As was pointed out by Xu and Chen [1], dispersal between patches often occurs in ecological environments, and more realistic models should include the dispersal process. During the last decade, many scholars had done excellent works on the predator-prey system with dispersal; see [2–16] and the references cited therein. In [5], Cui proposed the following two species predator-prey system with prey dispersal:

where and represent the population density of prey species and predator species in patch 1 and is the density of prey species in patch 2. Predator species is confined to patch 1, while the prey species can diffuse between two patches. is strictly positive functions that can be viewed as the dispersal rate or inverse barrier strength. By giving a thoroughly analysis on the right hand side of the system (3), Cui obtained a sufficient and necessary condition to guarantee the predator and prey species to be permanent.

It is unlike system (3), where the predator species is confined on patch 1. In [10], the authors proposed a model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches; that is, they considered the following predator-prey system in two-patch environment:

where represents the prey population in the ith patch, , at time . stands for the total predator population for both patches. The predator population is assumed to have no barriers between patches. is the specific growth rate for the prey population in the absence of predation when it is restricted to the ith patch. is the predator functional response of the predator population on the prey in the ith patch. is a positive constant that can be viewed as the dispersal rate or inverse barrier strength. is the density-dependent death rate of the predator in the absence of prey. is the conversion ratio of prey into predator. Conditions have been established in [10] for the existence, uniform persistence, and local and global stability of positive steady states of system (4).

The model (4), however, as was pointed out by Yang [11], is not perfect. Therefore, Xu et al. [12] had considered the following delayed periodic Lotka-Volterra type predator-prey system with prey dispersal in two-patch environments:

with initial conditions:

by using Gaines and Mawhins continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, they obtained a set of easily verifiable sufficient conditions to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system (5).

On the other hand, impulsive differential equations [17–19] arise frequently in the modeling of many physical systems whose states are subjects to sudden change at certain moments, for example, in population biology, the diffusion of chemicals, the spread of heat, the radiation of electromagnetic waves, the maintenance of a species through instantaneous stocking, and harvesting. There has been an increasing interest in the investigation for such equations during the past few years. There are many researchers who introduced impulsive differential equations in population dynamics [20–28]. However, to the best of the authors’ knowledge, to this day, no scholars had done works on the existence, uniqueness, and global stability of positive periodic solution of (1). Based on the idea of [10–15], we propose and study the system (1) in this paper.

For the sake of generality and convenience, we always make the following fundamental assumptions:

, and are all positive periodic continuous functions with period , and ; satisfies and , are constants with and there exists a positive integer such that , . Without loss of generality, we can assume that and , then .

In what follows, we will use the notation.

Throughout this paper, we make the following notation and assumptions.

Let be a constant and

  , , with the norm defined by ;  , , with the norm defined by ;  , , if ,  , exists, ;  ,  ;  ,  , with the norm defined by ;  ,  , with the norm defined by .

Then those spaces are all Banach spaces. We also denote

The aim of this paper is to obtain a set of easily verifiable sufficient conditions to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system (1) by further developing the analysis technique of [10–15]. The organization of this paper is as follows. In the next section, first, the necessary knowledge and lemmas are provided. Second, by using continuation theorem developed by Gaines and Mawhin [29], we establish the existence of at least one periodic solution of system (1). In Section 3, the uniqueness and global attractivity of periodic solution of system (1) are presented. Finally, we give two examples to show our results.

2. Existence of Positive Periodic Solutions

In this section, by using the continuation theorem which was proposed in [29] by Gaines and Mawhin, we will establish the existence conditions of at least one positive periodic solution to system (1). In doing so, we will introduce the following definitions and lemmas.

Let , be a real Banach space, 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 -compact 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 .

Definition 1 (see [18]). The set is said to be quasiequicontinuous in if for any there exists such that if , , , and , then .

Lemma 2 (Gaines and Mawhin [29]). Let and be two Banach spaces and let be a Fredholm operator with index zero. is an open bounded set, and is L-compact on . Suppose

(a)for each , every solution of is such that ;(b) for each ;(c).

Then, the equation has at least one solution lying in .

Lemma 3 (see [30]). Assume that , are continuous nonnegative functions defined on the interval . Then there exists such that .

Lemma 4 (see [20, 27, 28]). Assume that , . Then the following inequality holds:

Lemma 5. The region ,  ,   is the positive invariable region of the system (1).

Proof. In view of biological population, we obtain , . By the system (1), we have

Therefore, the conclusion is true.

Lemma 6 (see [27, 28, 31]). Suppose and , . Then the function has a unique inverse satisfying with , . If , ; then .

Proof. Since , and is continuous on , it follows that has a unique inverse function on . Hence, it suffices to show that ,   . For any , by the condition , one can find that, for the equation , exists a unique solution and, for the equation , exists a unique solution ; that is and , that is, and . As

it follows that . Since , thus, we have and . We can easily obtain that if , , then , , where is the unique inverse function of , which together with implies that . The proof of Lemma 6 is completed.

We denote by the inverse of , .

Theorem 7. In addition to ()(), assume the following conditions hold:

, .

Then, system (1) has at least one positive -periodic solution, where

Proof. We carry out the change of variable , ; then (1) can be transformed to

It is easy to see that if system (12) has one -periodic solution , then is a positive -periodic solution of system (1). Therefore, it suffices to prove system (12) has a -periodic solution. Let

and define

where is the Euclidean norm of . Then and are Banach spaces.

Let

and with

It 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, we first devote ourselves to deriving the explicit expression of . Taking , then exists an such that

Then direct integration produces

that ; that is , which, together with (20), 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 . Now, we reach the position to search for an appropriate open bounded set for the application of Lemma 2. Considering the operate equation , , we have

Since are -periodic functions, we need only to prove the result in the interval . Integrating (26) over the interval leads to

Hence, we have

It follows from (26)–(28) that

Multiplying the first equation of (26) by and integrating over we have

Since , we obtain

which yields

Similarly, multiplying the second equation of (26) by and integrating over gives

By using the inequalities

it follows from (32)–(34) that

If , then it follows from the second equation of (35) that

which implies

If , similarly, we obtain

Set

Then it follows from (37)–(39) that

Note that ; then there exists such that

Then it follows from (40) and (41), that

Since , we can let , that is, ; then

According to Lemma 6, we know . Thus,

Similarly, we have

On the other hand, by Lemma 6, we can see that , so we can derive

therefore, we can derive from (27) and (46) that

which implies

that is

It follows from (29), (42), and (49) that

From (42), (50), and Lemma 4, it follows that, for ,

It follows from (28) and (48) that

which deduces

This, together with (53) and Lemma 4, leads to

Let It follows from (51) and (54) that

From (28), (41) and (48) we have

which deduces

where , implies

This, together with (41) and Lemma 4, leads to

Set

It follows from (51) and (59) that

Noting that

it follows from (28) and (46) that

which yields

This, together with (41) and Lemma 4, leads to

Set

It follows from (51) and (65) that

Thus, we obtain

Clearly, are independent of .

In order to use the invariance property of homotopy, we need to consider the following algebraic equations:

for , where . Carrying out similar arguments as above, one can easily show that any solution of (69) with also satisfies

Choose and define , , ; it is clear that satisfies the condition (a) of Lemma 2. Let ; then is a constant vector in with . Then

That is, the condition (b) of Lemma 2 holds. Finally, for the convenience of computing the Brouwer degree, we consider a homotopy

where

By (69) and (70), it follows that for , . In addition, it is clear that the algebraic equation has a unique solution in . Choose the isomorphism to be the identity mapping; by a direct computation and the invariance property of homotopy, one has

By now we have proved that all the requirements in Lemma 2 are satisfied. Hence system (12) has at least one -periodic solution, say . Set , , ; then has at least one positive -periodic solution of system (1). The proof of Theorem 7 is complete.

Remark 8. If , , then (1) is translated to (5). In this case, the conditions , are the same as , of Theorem 2.1 in [12], but we see that of Theorem 2.1 in [12] is not needed here. Hence our result improves and generalizes the corresponding result of [12].

Remark 9. If , then (1) is translated to in [14]. In this case, the conditions (₁)–(₄) are the same as (₁)–(₄) in [14]. Hence our result generalizes the corresponding result of [14].

3. Uniqueness and Global Stability

We now proceed to the discussion on the uniqueness and global stability of the -periodic solution in Theorem 14. It is immediate that if is globally asymptotically stable then is unique in fact. Under the hypotheses , , we consider the nonimpulsive delay differential equation

with the initial conditions

where

The following lemmas will be used in the proofs of our results. The proof of the first lemma is similar to that of Theorem 1 in [23].

Lemma 10. Suppose that (), () hold; then

(i)if is a solution of (75) on , then is a solution of (1) on ;(ii)if is a solution of (1) on , then is a solution of (75) on .

Proof. (i) It is easy to see that is absolutely continuous on every interval ; , ,

On the other hand, for any , ,

Thus

which implies that is a solution of (1); similarly, we can prove that , are also solutions of (i). Therefore, are solutions of (1) on .

(ii) Since is absolutely continuous on every interval ;  , , and in view of (79), it follows that, for any ,

which implies that is continuous on . It is easy to prove that is absolutely continuous on . Similarly, we can prove that are absolutely continuous on . Similar to the proof of , we can check that are solutions of (75) on . The proof of Lemma 10 is completed.

Lemma 11. Let denote any positive solution of system (75) with initial conditions (76). Then there exists a such that , for , where

Proof. Let . Calculating the upper-right derivative of along the positive solution of system (75), we have the following:

if   in some intervals, then

if in other intervals, similarly, we have

It follows from and that

By (85) we can derive the following.

(A)If , then , .(B)If , let ( by the condition (82)). We consider the following two cases:

(a);(b).

If (a) holds, then there exists such that if , then , and we have

If (b) holds, then there exists such that if , then , and we also have

From what has been discussed above, we can conclude that if , then is strictly monotone decreasing with speed at least . Therefore there exists a such that if , then

From the third equation of system (75) and (88) we can deduce that, for ,

A standard comparison argument shows that

Thus, there exists a such that

The proof of Lemma 11 is completed.

Lemma 12. Let denote any positive solution of system (75) with initial conditions (76). Then there exists a such that , for , where

and are defined in Lemma 11.

Proof. Let . Calculating the lower-right derivative of along the positive solution of system (75), similar to the discussion for inequality (85), for any , where is defined in Lemma 11, we easily obtain:

if , in some intervals, then

if , in other intervals, similarly, we have

From and , we can reduce the following.

(C)If , then , .(D)If , and let . There are three cases:

(c);(d);(e).

If (c) holds, then there exists such that if , we have and .

If (d) holds, similar to (c), there exists such that if , we have and .

If (e) holds, in the same way also there exists such that if , we have and , .

From (c)–(e), we know that if , will strictly monotonically increase with speed . So there exists such that if , we have .

From the third equation of system (75), for any , we know that

and using the fact that

therefore, for , we get

A standard comparison argument shows that

Thus, there exists a such that

The proof of Lemma 12 is completed.

Lemma 13 (see Barbălat’s Lemma [32]). Let be a nonnegative function defined on such that is integrable and uniformly continuous on ; then .

We now formulate the uniqueness and global stability of the positive -periodic solution of system (1). It is immediate that if is globally asymptotically stable then is in fact unique.

Theorem 14. In addition to () and (), assume further that

, then, system (1) has a unique positive -periodic solution which is globally asymptotically stable, where

Proof. Let be a positive -periodic solution of (1); then , is the positive -periodic solution of system (75), and let be any positive solution of system (75) with the initial conditions (76). It follows from Lemmas 11 and 12 that there exist positive constants , , and , such that, for all ,

We define

Calculating the upper right derivative of along solutions of (75), it follows that

where

We estimate under the following two cases:

(i)if , then

(ii)if , then

Combining the conclusions in (i)-(ii), we obtain

A similar argument as in the discussion above shows that

It follows from (103), (107), and (108) that

We define . Calculating the upper right derivative of along solutions of (75), it follows that

By substituting (75) into (110), we obtain

Define

It follows from (111) and (112) that, for any ,

We also define

It follows from (112)–(114) that, for any ,

We now define a Lyapunov functional as

Then it follows from (109), (115), and (116) that, for any ,

where are defined in (100). By hypothesis, there exist positive constants , and such that if

Integrating both sides of (117) on interval , we have

It follows from (118) and (119) that

Therefore, is bounded on and also . Since and are bounded for , therefore, are uniformly continuous on . By Lemma 13, we have

Therefore

By Theorems 7.4 and 8.2 in [33], we know that the periodic positive solution is uniformly asymptotically stable. The proof of Theorem 14 is completed.

4. Some Examples

The following illustrative examples will demonstrate the effectiveness of our results.

Example 1. We consider the following delayed periodic Lotka-Volterra predator-prey system with prey dispersal and impulse:

We fix the parameters , , . It is easy to see that , , , , , , , , , , , , , . Thus we have

According to Theorem 7, we see that system (123) has at least one positive -periodic solution.

Example 2. We consider another delayed periodic Lotka-Volterra diffusive predator-prey model with impulse:

We fix the parameters , , ,. It is easy to see that , , , , , , , , , , , , , , , , , , , , , . Thus we have

According to Theorem 7, we see that model (125) has at least one positive -periodic solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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 Science Foundation of Hengyang Normal University (no. 11B36), and the construct program of the key discipline in Hunan Province.

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