Discrete Dynamics in Nature and Society

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.