Journal of Applied Mathematics

Volume 2013 (2013), Article ID 101238, 9 pages

http://dx.doi.org/10.1155/2013/101238

Research Article

An Impulsive Periodic Single-Species Logistic System with Diffusion

Chenxue Yang,1,2 Mao Ye,1,2 and Zijian Liu3,4

¹School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

²State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, China

³School of Science, Chongqing Jiaotong University, Chongqing 400074, China

⁴Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 11 January 2013; Accepted 17 April 2013

Academic Editor: Michael Meylan

Copyright © 2013 Chenxue Yang 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.

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.

1. Introduction

In the practical world, on the one hand, owing to natural enemy, severe competition, or deterioration of the patch environment, species dispersal in two or more patches becomes one of the most prevalent phenomena of nature. Many empirical works and monographies on population dynamics in a spatial heterogeneous environment have been done (see [1–9] and the references cited therein). On the other hand, many natural and man-made factors (e.g., fire, drought, flooding deforestation, hunting, harvesting, breeding, etc.) always lead to rapid decrease or increase of population number at fixed moment. Such sudden changes can often be characterized mathematically in the form of impulses. With the development of the theory of impulsive differential equations [10], various population dynamical models of impulsive differential equations have been proposed and studied extensively. For example, many important and interesting results on the permanence, persistence, extinction, global stability, the existence of positive periodic solutions, bifurcation and dynamical complexity, and so forth can be found in [11–17] and the references cited therein.

Although considerable researches on the dispersal and impulses of species have been reported in the literature, there are few papers that investigate the dynamical behavior of population systems under the circumstances in which both dispersal and impulse exist. However, dispersal species which undergoes impulses is also one of the most prevalent phenomena of nature. In our previous paper [18], an impulsive periodic predator-prey system with diffusion is studied, and some conditions for the permanence, extinction, and existence of a unique globally stable periodic solution are established. In this paper, we will present and study a single-species periodic logistic system with impulses and dispersal in different patches. Our model takes the form

where and represent the intrinsic growth rates and the density-dependent coefficients in patch , respectively. denotes the dispersal rate of the species from patch to patch . is the regular pulse at time of species in patch . Throughout this paper, we always assume the following.

, and are continuously periodic functions with common period , defined on and for all and . for all and there exists a positive integer such that and for any .

The organization of this paper is as follows. In the next section, some sufficient conditions for the permanence and extinction of system (1) are obtained. In Section 3, conditions for the existence of a unique globally attractively positive periodic solution are also established. Finally, some numerical simulations are proposed to illustrate the feasibility of our results and discuss the model further.

2. Permanence and Extinction

In this section, applying inequality estimation technique, we get some sufficient conditions on the permanence and extinction of system (1).

Theorem 1. There exists a positive constant such that for all if

where .

Proof. Let for any , then we have and there exists a positive constant such that function

for all and . Choose , and is bounded for all . Then from conditions (2) and (3), we have two positive constants and such that

Define the function . For any , there is an such that . Calculating the upper-right derivative of , we obtain

When , we have .

Consider the following auxiliary system:

with the initial condition . If there is a constant such that

for any positive solution of system (6), then, according to the comparison theorem of impulsive differential equations [10], we have for all . Therefore, choose , and we will finally have for all .

Next, we will prove that (7) holds. In fact, for any positive solution of system (6), we only need to consider the following three cases.

Case 1. There is a such that for all .

Case 2. There is a such that for all .

Case 3. is oscillatory about for all .

We first consider Case 1. Since for all , then for , where is any positive integer, integrating system (6) from to , from (4) we have

Hence, as , which leads to a contradiction.

Then, we consider Case 3. From the oscillation of about , we can choose two sequences and satisfying and such that

For any , if for some integer , then we can choose integer and constant such that . Since for all and , integrating this inequality from to , by (3) and (4) we obtain

where . If there is an integer such that , then we have . Therefore, for Case 3 we always have for all .

Lastly, if Case 2 holds, then we directly have for all .

Choose constant , then we see that (7) holds. This completes the proof.

Remark 2. It can be seen from Theorem 1 that, in one time period , if the density-dependent coefficient in patch () is strictly greater than zero and the impulsive coefficient is bounded in the same time period, the dispersal species is always ultimately bounded.

Theorem 3. Assume that all conditions of Theorem 1 hold. In addition, there is an such that

Then system (1) is permanent.

Proof. The ultimate boundedness of system (1) has been proved in Theorem 1. In the following, we mainly prove the permanence of the system; that is, there is a constant such that

for each and any positive solution of system (1).

From assumption , we obtain that there exists a constant such that function

for any and .

For , by condition (11) and the boundedness of , there are two positive constants and such that

Let be any positive solution of system (1). Since

by the comparison theorem of impulsive differential equations, we obtain for all , where is the positive solution of system

with initial condition .

In the following, we first prove that there is a constant such that

for any positive solution of system (16). We only need to consider the following three cases.

Case 1. There is a such that for all .

Case 2. There is a such that for all .

Case 3. is oscillatory about for all .

For Case 1, let , where is any integer. From (14), we obtain

Hence, as , which leads to a contradiction.

For Case 3, we choose two sequences and satisfying and such that

For any , if for some integer , then we can choose integer and constant such that . Since, for all and , we have , integrating this inequality from to , then from (13) and (14) we obtain

where . If there is an integer such that , obviously we have . Therefore, for Case 3 we always have for all . Let constant . Then is independent of any positive solution of system (16) and we finally have that (17) holds.

Lastly, if Case 2 holds, then from for all , we directly have that (17) holds.

From the fact that for all , then we have

It follows immediately from (21) that there is a such that for all . Then for any , when and we have

Obviously, there is a constant and is independent of any positive solution of system (1), such that for any , , and all the following inequality holds

Hence, for any , by (22) we have

for all and , where .

In order to prove that (12) holds, we only need to consider the following three cases.

Case 1. There is a such that for all .

Case 2. There is a such that for all .

Case 3. is oscillatory about for all .

Equation (12) is obviously true if Case 1 holds.

For Case 2, there exists an impulsive time for some integer . For any , there is an integer such that , and from system (24) we have

where . Therefore, we obtain

for any .

Then, we consider Case 3. We choose two sequences and satisfying and such that

For any , if for some integer , we first of all show that must be an impulsive time. Otherwise, there exists a positive constant such that there is no pulse in the interval . Then for any , from system (24) we have , which leads to a contradiction.

If there is only one impulsive time in the interval which must be ), then from system (24) we get

If there are at least twice pulses in , then we can denote , where , , and are some positive integers. Hence, for any , there is from system (24). Moreover, if , we have

If , then

However, when , then

It follows from (28)–(31) that

for all and any .

If there is an integer such that , obviously we have . Therefore, for Case 3 we always have

for all and any .

From (26) and (33), choose . Then we finally have that (12) holds. System (1) is permanent. The proof of Theorem 3 is completed.

Remark 4. It follows from Theorem 3 that system (1) is permanent if there is a positive average growth rate (which does not include the dispersal entrance) in one time period in any patch ). In paper [4], the authors showed an interesting result that the dispersal species without impulses is permanent in all other patches if it is permanent in a patch. However, we extend this result to a periodic case with impulses.

Remark 5. In paper [18], we studied an impulsive periodic predator-prey system with Holling type III functional response and diffusion, and the paper mainly considers the influences of Holling type functional response and impulses. The conditions of the main result Theorem 3 require the minimum of the coefficients. However, in this paper, we consider a single-species logistic system. Although the predator is not involved in the model, which is simple than the model in [18], a more accurate and reasonable condition is established in the present paper; that is, in dispersal system, species with impulses is permanent in all patches if it is permanent in a patch, which improves the minimum conditions in the previous paper.

Theorem 6. System (1) is extinct if conditions

hold, where for all , and and are defined in Theorem 1.

Proof. In fact, from (34), for any constant , there is a positive constant such that

Define . When , calculating the right-upper derivative of , we have

When , we obtain . From this and (35), a similar argument as in the proof of (7), we can obtain for . Then from the arbitrariness of , we obtain as . Finally, we have as for all . This completes the proof of Theorem 6.

Remark 7. It can be seen from Theorem 6 that system (1) is always extinct if, in one time period , there are a positive density-dependent coefficient and a nonpositive average growth rate (which includes the dispersal entrance) in the time period in each patch .

3. Periodic Solutions

In this section, by constructing an appropriate Lyapunov function, sufficient conditions for the existence of the unique globally attractively positive -periodic solution of system (1) are established.

Let and be any two positive solutions of system (1). From Theorem 3, we can obtain that there are constants and such that

Theorem 8. Suppose all the conditions of Theorem 3 hold. Moreover, if

where , , then system (1) has a unique globally attractively positive -periodic solution ; that is, any positive solution of system (1) satisfies

Proof. Choose Lyapunov function . Since for any impulsive time we have

then is continuous for all . On the other hand, from (37) we can obtain that for any and

For any and , calculating the derivative of , we obtain

where

For all , we estimate under the following two cases.

(i)If , then .(ii)If , then .

It follows from the estimation of and (41) that

From this and condition (38), we have as . Further more, from (41) we have that (39) holds.

Now let us consider the sequence , where and . It is compact in the domain since for all and . Let be a limit point of this sequence, with . Then . Indeed, since and as , we get

The sequence has a unique limit point. On the contrary, let the sequence have two limit points and . Then, taking into account (39) and , we have

and hence . The solution is the unique periodic solution of system (1). By (39), it is globally attractive. This completes the proof of Theorem 8.

4. Numerical Simulation and Discussion

In this paper, we have investigated a class of single-species periodic logistic system with impulses and dispersal in different patches. By means of inequality estimation technique and Lyapunov function, we gave the criteria for the permanence, extinction, and existence of a unique globally stable positive periodic solution of system (1).

In order to testify the validity of our results and present a more in-depth problem for further discussion, we discuss the following two patches -periodic dispersal system:

We take , , , , , , , , and , . Obviously, , , , and system (47) is periodic with period . For , we have , , and for all . It is easy to verify that and are bounded for all and . Further more, since

all the conditions of Theorem 3 are satisfied. Hence, system (47) is permanent. See Figures 1 and 2.

Figure 1: The time series of the permanence of species .

Figure 2: The phase of the permanence of species .

However, if the survival environment of the two patches is austere, the intrinsic growth rates will be negative. Hence, if we take , and all other parameters are retained, then we obtain and

hence conditions of Theorem 6 are satisfied. From Theorem 6 we find that any positive solution of system (47) will be extinct. See Figure 3.

Figure 3: The time series and phase of the extinction of species .

From the illustrations of the theorems, we note that there is a great difference on the choice of the intrinsic growth rates , which guarantee that the system is permanent or extinct. These differences make us want to know what results will be if all the parameters satisfy

For this aim, we choose , and all other parameters are retained; then

and all the conditions of Theorems 3 and 6 are not satisfied. But from Figures 4 and 5 we find that the system is permanent.

Figure 4: The time series of the permanence of species .

Figure 5: The phase of the permanence of species .

Furthermore, if we choose , and keep all other parameters, then we have

which do not satisfy conditions of any theorem. But, from Figure 6 we see that any positive solution of system (1) is extinct.

Figure 6: The time series and phase of the extinction of species .

Remark 9. Through the above analysis, we realize that there is a little flaw of the finding conditions of the theorems. A challenging problem is to find some sufficient and necessary conditions (if the conditions hold, then the system will be permanent, otherwise, it will be extinct) to guarantee the permanence and extinction of the system.

Throughout Figures 1–6, we always take the initial condition .

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (60702071), Program for New Century Excellent Talents in University (NCET-06-0811), 973 National Basic Research Program of China (2010CB732501), Foundation of Sichuan Excellent Young Talents (09ZQ026-035), and Open Project of State Key Laboratory for Novel Software Technology of Nanjing University and Zhejiang Provincial Natural Science Foundation of China (Q13A010080).

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