Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 393729, 9 pages

http://dx.doi.org/10.1155/2013/393729

Research Article

Permanence and Global Attractivity of the Discrete Predator-Prey System with Hassell-Varley-Holling III Type Functional Response

Runxin Wu and Lin Li

Department of Mathematics and Physics, Fujian University of Technology, Fuzhou, Fujian 350108, China

Received 6 November 2012; Revised 18 February 2013; Accepted 19 February 2013

Academic Editor: M. De la Sen

Copyright © 2013 Runxin Wu and Lin Li. 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.

By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling III type functional response are obtained. An example together with its numerical simulation shows that the main results are verifiable.

1. Introduction

Recently, there were many works on predator-prey system done by scholars [1–6]. In particular, since Hassell-Varley [7] proposed a general predator-prey model with Hassell-Varley type functional response in 1969, many excellent works have been conducted for the Hassell-Varley type system [1, 7–13].

Liu and Huang [8] studied the following discrete predator-prey system with Hassell-Varley-Holling III type functional response:

where , denote the density of prey and predator species at the th generation, respectively. , , , , , are all periodic positive sequences with common period . Here represents the intrinsic growth rate of prey species at the th generation, and measures the intraspecific effects of the th generation of prey species on their own population; is the death rate of the predator; is the capturing rate; is the maximal growth rate of the predator. Liu and Huang obtained the necessary and sufficient conditions for the existences of positive periodic solutions by applying a new estimation technique of solutions and the invariance property of homotopy. As we know, the persistent property is one of the most important topics in the study of population dynamics. For more papers on permanence and extinction of population dynamics, one could refer to [2–5, 14–17] and the references cited therein. The purpose of this paper is to investigate permanence and global attractivity of this system.

We argue that a general nonautonomous nonperiodic system is more appropriate, and thus, we assume that the coefficients of system (1) satisfy the following:

(A) , , , , , are nonnegative sequences bounded above and below by positive constants.

By the biological meaning, we consider (1) together with the following initial conditions as

For the rest of the paper, we use the following notations: for any bounded sequence , set and .

2. Permanence

Now, let us state several lemmas which will be useful to prove our main conclusion.

Definition 1 (see [5]). System (1) said to be permanent if there exist positive constants and , which are independent of the solution of system (1), such that for any positive solution of system (1) satisfies

Lemma 2 (see [14]). Assume that satisfies and

for , where and are all nonnegative sequences bounded above and below by positive constants. Then,

Lemma 3 (see [14]). Assume that satisfies

, and , where and are all nonnegative sequences bounded above and below by positive constants and . Then,

Theorem 4. Assume that

hold, then system (1) is permanent, that is, for any positive solution of system (1), one has

where

Proof. We divided the proof into four steps.

Step 1. We show

From the first equation of (1), we have

By Lemma 2, we have

Previous inequality shows that for any , there exists a , such that

Step 2. We prove by distinguishing two cases.

Case 1. There exists a , such that .

By the second equation of system (1), we have

which implies

The previous inequality combined with (13) leads to . Thus, from the second equation of system (1), again we have

We claim that

By a way of contradiction, assume that there exists a such that . Then . Let be the smallest integer such that . Then . The previous argument produces that , a contradiction. This proves the claim. Therefore, . Setting in it leads to .

Case 2. Suppose for all . Since is nonincreasing and has a lower bound , we know that exists, denoted by , we claim that

By a way of contradiction, assume that .

Taking limit in the second equation in system (1) gives

however,

which is a contradiction. It implies that . By the fact , we obtain that

Therefore, we have

Then,

Step 3. We verify

Conditions imply that for enough small positive constant , we have

For the previous , it follows from Steps 1 and 2 that there exists a such that for all

Then, for , it follows from (26) and the first equation of system (1) that

According to Lemma 3, one has

where

Setting in (28) leads to

By the fact that , we see that .

This ends the proof of Step 3.

Step 4. We present two cases to prove that

For any small positive constant , from Step 1 to Step 3, it follows that there exists a such that for all

Case 1. There exists a such that , then

Hence,

and so,

Set

We claim that

By a way of contradiction, assume that there exists a , such that . Then . Let be the smallest integer such that . Then , which implies that , a contradiction, this proves the claim. Therefore, , setting in it leads to .

Case 2. Assume that for all , then, exists, denoted by , then . We claim that

By a way of contradiction, assume that . Taking limit in the second equation in system (1) gives

which is a contradiction since

This proves the claim, then we have

So,

Obviously, . This completes the proof of the theorem.

3. Global Attractivity

Definition 5 (see [18]). System (1) is said to be globally attractive if any two positive solutions and of system (1) satisfy

Theorem 6. Assume that and hold. Assume further that there exist positive constants , , and such that

where

Then, system (1), with initial condition (2), is globally attractive, that is, for any two positive solutions and of system (1), we have

Proof. From conditions and , there exists an enough small positive constant such that

where

Since and hold, for any positive solutions and of system (1), it follows from Theorem 4 that

For the previous and (48), there exists a such that for all ,

Let

Then from the first equation of system (1), we have

Using the mean value theorem, we get

where lies between and , lies between and .

It follows from (51) and (52) that

And so, for ,

Let

Then, from the second equation of system (1), we have

Using the mean value theorem, we get

where lies between and , respectively. Then, it follows from (56) and (57) that for ,

Now, we define a Lyapunov function as follows:

Calculating the difference of along the solution of system (1), for , it follows from (54) and (58) that

Summating both sides of the previous inequalities from to , we have

which implies

It follows that

Using the fundamental theorem of positive series, there exists small enough positive constant such that

which implies that

that is,

This completes the proof of Theorem 6.

4. Extinction of the Predator Species

This section is devoted to study the extinction of the predator species .

Theorem 7. Assume that

Then, the species will be driven to extinction, and the species is permanent, that is, for any positive solution of system (1),

where

Proof. For condition , there exists small enough positive , such that

for all , from (69) and the second equation of the system (1), one can easily obtain that

Therefore,

which yields

From the proof of Theorem 4, we have

For enough small positive constant ,

For the previous , from (72) and (73) there exists a such that for all ,

From the first equation of (1), we have

By Lemma 3, we have

Setting in (72) leads to

The proof of Theorem 7 is completed.

5. Example

The following example shows the feasibility of the main results.

Example 8. Consider the following system:

One could easily see that

Clearly, conditions and are satisfied. It follows from Theorem 4 that the system is permanent. Numerical simulation from Figure 1 shows that solutions do converge and system is permanent and globally attractive.

Figure 1: Dynamics behaviors of system (1) with initial conditions , respectively.

6. Conclusion

In this paper, a discrete predator-prey model with Hassell-Varley-Holling III type functional response is discussed. The main topics are focused on permanence, global attractivity, and extinction of predator species. The numerical simulation shows that the main results are verifiable.

The investigation in this paper suggests the following biological implications. Theorem 4 shows that the coefficients, such as the death rate of the predator, the capturing rate, and the intraspecific effects of prey species, influence permanence. Conditions and imply that the higher the intraspecific effects of prey species are, the more favourable permanence is. Those results have further application on predator-prey population dynamics. However, the conditions for global attractivity in Theorem 4 is so complicated that its application is very difficult. A further study is required to simplify the application.

Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (JA11193).

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