#### 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 type functional response are obtained. Example together with its numerical simulation shows that the main results are verifiable.

#### 1. Introduction

The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. The most popular predator-prey model is the one with Holling type II functional response [2]: where , denote the density of prey and predator species at time , respectively. The constants , , , , , are all positive constants that stand for prey intrinsic growth rate, carrying capacity of prey species, capturing rate, half saturation constant, maximal predator growth rate, predator death rate, respectively.

Standard Lotka-Volterra type models, on which a large body of existing predator-prey theory is built, assume that the per capita rate of predation depends on the prey numbers only. There is growing explicit biological and physiological evidence [3β8] that in many situations, especially when predators have to search and share or compete for food, a more suitable general predator-prey model should be based on the βratio-dependentββ theory.

Arditi and Ginzburg [9] proposed the following predator-prey model with ratio-dependent type functional response:

It was known that the functional response can depend on predator density in other ways. One of the more widely known ones is due to Hassell and Varley [10]. A general predator-prey model with Hassell-Varley tape functional response may take the following form: This model is appropriate for interactions, where predators form groups and have applications in biological control. System (1.3) can display richer and more plausible dynamics. In a typical predator-prey interaction where predators do not form groups, one can assume that , producing the so-called ratio-dependent predator-prey dynamics [11]. For terrestrial predators that form a fixed number of tight groups, it is often reasonable to assume that For aquatic predators that form a fixed number of tight groups, may be more appropriate. Recently, Hsu [11] presents a systematic analysis on the above system.

On the other hand, when the size of the population is rarely small or the population has nonoverlapping generation, the discrete time models are more appropriate than the continuous ones [12β24]. This motivated us to propose and study the discrete analogous of predator-prey system (1.3): where ; , , , , , are all bounded nonnegative sequences. For the rest of the paper, we use the following notations: for any bounded sequence , set

By the biological meaning, we will focus our discussion on the positive solution of system of (1.3). Thus, we require that

#### 2. Permanence

In order to establish the persistent result for system (1.4), we make some preparations.

*Definition 2.1. *System (1.4) said to be permanent if there exist positive constants and , which are independent of the solution of system (1.4), such that for any positive solution of system (1.4) satisfies

Lemma 2.2 (see [23]). *Assume that satisfies and
**
for , where and are all nonnegative sequences bounded above and below by positive constants. Then
*

Lemma 2.3 (see [23]). *Assume that satisfies
** and , where and are all nonnegative sequences bounded above and below by positive constants and . Then
*

Theorem 2.4. *Assume that
**
hold, then system (1.4) is permanent, that is, for any positive solution of system (1.4), one has
**
where
*

*Proof. *We divided the proof into four claims.*Claim 1. *From the first equation of (1.4), we have
By Lemma 2.2, we have
Above inequality shows that for any , there exists a , such that
*Claim 2. *We divide it into two cases to prove that
*Case (i)*

There exists an , such that . Then by the second equation of system (1.4), we have
Hence,
therefore,
and so,
It follows that
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 above argument produces that , a contradiction. This prove the claim.*Case (ii)*

We assume that for all . Since is nonincreasing and has a lower bound , we know that exists, denoted by , then
We claim that
By a way of contradiction, assume that . Taking limit in the second equation in system (1.4) gives
which is a contradiction since for
This prove the claim, then we have
Combining Cases (i) and (ii), we see that
Let , we have
*Claim 3 (). *Conditions () imply that for enough small positive constant , we have
For above , it follows form Claims 1 and 2 that there exists a such that for all
From the first equation of (1.4), we have
By applying Lemma 2.3 to above inequality, we have
Setting in (2.28) leads to
This ends the proof of Claim 3.*Claim 4. *For any small positive constant , from Claims 1β3, it follows that there exists a such that for all
We present two cases to prove that
*Case (i)*

There exists an such that , then
Hence
and so,
Set
We claim that for 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.*Case (ii)*

We assume that for all . According to (2.30), exists, denoted by , then
We claim that
By the way of contradiction, assume that . Taking limit in the second equation in system (1.4) gives
which is a contradiction since for
The above analysis show that
Letting , we have
where
According to Claims 1β4, we can easily find that the result of Theorem 2.4 holds.

#### 3. Global Attractivity

Theorem 3.1. *Assume that () and () hold. Assume further that there exist positive constants , and such that
**
Then system (1.4) with initial condition (1.6) is globally attractive, that is, for any two positive solutions and of system (1.4), one has
*

*Proof. *From conditions () and (), there exists an enough small positive constant such that
Since () and () hold, for any positive solutions and of system (1.4), it follows from Theorem 2.4 that
For above and (3.3), there exists a such that for all ,
Let
Then from the first equation of system (1.3), we have
Using the Mean Value Theorem, we get
where lies between and , lies between and .

It follows from (3.6), (3.7) that
And so, for
Let
Then from the second equation of system (1.4), we have
Using the Mean Value Theorem, we get
where lie between and , respectively. Then, it follows from (3.11), (3.12) that for
Now we define a Lyapunov function as follows:
Calculating the difference of along the solution of system (1.4), for , it follows from (3.9) and (3.13) that
Summating both sides of the above 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 3.1.

#### 4. Extinction of the Predator Species

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

Theorem 4.1. *Assume that
**
Then, the species will be driven to extinction, and the species is permanent, that is, for any positive solution of system (1.4),
**
where
*

*Proof. * For condition (), there exists small enough positive , such that
for all from (4.3) and the second equation of the system (1.4), one can easily obtain that
Therefore,
which yields
From the proof of Theorem 3.1, we have
For enough small positive constant ,
For above , from (2.9) and (4.6), there exists a such that for all ,
From the first equation of (1.4), we have
By Lemma 2.3, we have
Setting in (4.11) leads to
The proof of Theorem 4.1 is completed.

#### 5. Example

The following example shows the feasibility of the main results.

*Example 5.1. *Consider the following system:

One could easily see that there exist positive constants such that
Clearly, conditions ()β() are satisfied. It follows from Theorems 2.4 and 3.1, that the system is permanent and globally attractive. Numerical simulation from Figure 1 shows that solutions do converge and system is permanent.

#### 6. Conclusion

In this paper, we have obtained sufficient conditions for the permanence and global attractivity of the system (1.4), where . If in the system (1.4), the system (1.4) is a discrete ratio-dependent predator-prey model with Holling-II functional response, in this case, HUO and LI gave sufficient conditions for the permanence of the system in [24], however, they did not provide the condition for the extinction of the predator species . In this paper, Theorem 2.4 gives the same conditions as that of Huo and Li's condition for the permanence of the system. Furthermore, Theorem 4.1 gives sufficient conditions which ensure the extinction the predator of the system (1.4) when . If holds, then the prey species is permanence. If in the system of (1.4), the system is a discrete predator-prey model with Holling-II function response, Theorem 4.1 also holds for the case .