Research Article | Open Access

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

**Academic Editor:**M. De la Sen

#### 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.

#### 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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#### Copyright

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.