Discrete Dynamics in Nature and Society

VolumeΒ 2008, Article IDΒ 186539, 19 pages

http://dx.doi.org/10.1155/2008/186539

## Dynamics Behaviors of a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China

Received 24 April 2008; Revised 16 August 2008; Accepted 2 October 2008

Academic Editor: Juan JoseΒ Nieto

Copyright Β© 2008 Jinghui Yang. 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

A ratio-dependent predator-prey system with Holling type III functional response and feedback controls is proposed. 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 system are obtained. After that, under some suitable conditions, we show that the predator species will be driven to extinction. Examples together with their numerical simulations show that the main results are verifiable.

#### 1. Introduction

Wang and Li [1, 2] established verifiable criteria for the existence of globally attractive positive periodic solutions of the following delayed predator-prey model with Holling type III functional response: where are the densities of the prey population and predator population at time , respectively; are continuous functions of period and ; is a nonnegative constant. For more works on the predator-prey system with Holling type functional response, one could refer to [3β11] and the references cited therein. But, recently, lots of scholars found that when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be the so-called ratio-dependent functional response. This is strongly supported by numerous fields and laboratory experiments and observations [12, 13]. In [14], Wang and Li proposed the following ratio-dependent predator-prey system with Holling type III functional response:where are the densities of the prey population and predator population at time , respectively, , and are all positive periodic continuous functions, and are real constants. They found that the criteria for the permanence are exactly the same as those for the existence of positive periodic solution of (1.2). For more works on the ratio-dependent predator-prey system, one could refer to [15, 16] and the references cited therein.

On the other hand, when the size of the population is rarely small or the population has nonoverlapping generations, the discrete time models are more appropriate than the continuous ones [17, 18]. For the discrete ratio-dependent predator-prey model with Holling type III functional response, Fan and Li [19] considered the following system: sufficient conditions which ensure the permanence of system (1.3) are obtained.

However, as pointed out by Huo and Li [20], βecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. The question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time is of practical interest in ecology . In the language of control variables, we call the disturbance functions as control variables.β For this direction, in [15], Chen and Ji proposed the following system:where stand for the densities of the prey and predator, respectively, is the control variable, , and are continuous and strictly positive functions. In this paper, they considered the almost periodic solution of system (1.4). For more work on this direction, one could refer to [15, 21β28].

To the best of the author's knowledge, so far no scholar has considered system (1.3) with feedback controls. This motivates us to propose and study the following discrete ratio-dependent predator-prey system with Holling type III and feedback controls: where are the densities of the prey population and predator population at time , respectively, for , and are all bounded nonnegative sequences such thatHere, for any bounded sequence .

By the biological meaning, we will focus our discussion on the positive solutions of system (1.5). So, it is assumed that the initial conditions of system (1.5) are of the formIt is not difficult to see that solutions of (1.5) and (1.6) are well defined and satisfy

The main purpose of this paper is to derive sufficient conditions for the permanence, global attractivity, and extinction of system (1.5).

The organization of this paper is as follows. In Section 2, we introduce some useful lemmas. In Section 3, we will study the permanence and global attractivity of system (1.5). In Section 4, we will study the extinction of the predator species . In the last section, numerical simulation is presented to illustrate the feasibility of our main results.

#### 2. Preliminaries

Now, let us state several lemmas which will be useful in proving the main results.

First, let us consider the first-order difference equationwhere are positive constants. The following Lemma 2.1 is a direct corollary of Theorem 6.2 of L. Wang and M. Q. Wang [29, page 125].

Lemma 2.1. *
Assume that .
For any initial ,
there exists a unique solution of (2.1), which can be expressed as follows:
**where .
Thus, for any solutions of
system
(2.1),
*

The following comparison theorem for difference equation is Theorem 2.1 of [29, page 241].

Lemma 2.2. * Let and .
For any fixed is a nondecreasing function with respect to ,
and for ,
the following inequalities hold:
**If ,
then for all .*

The following Lemmas 2.3 and 2.4 can be found in [21].

Lemma 2.3 (see [21]). * Assume that satisfies and
**for ,
where and are nonnegative sequences bounded above and
below by positive constants. Then *

Lemma 2.4 (see [21]). * Assume that satisfies
** and ,
where and are nonnegative sequences bounded above and
below by positive constants and .
Then *

#### 3. Permanence and Global Attractivity

Now, we investigate the permanence and global attractivity of system (1.5).

Theorem 3.1. *
Assume that
**hold, then the species and are permanent, that is, for any positive
solution
of system (1.5) with the initial conditions
(1.6), **where *

*Proof. * We divided the proof into five steps.*Step 1. *
We show
From the first
equation of system (1.5),Then (3.3) follows immediately
from Lemma 2.3. Thus, for any positive ,
there exists an such that*Step 2. *We prove by distinguishing two cases.*Case 1. *
There exists an such that .
Then, by the second equation of system (1.5), we havewhich impliesThe above inequality combined
with (3.5) leads to .
Thus from the second equation of system (1.5) again we haveWe claim that for all .
In fact, suppose there exists such that .
Let be the smallest integer between and such that and .
Then implies ,
a contradiction. This proves the claim. Setting in (3.8) leads to .*Case 2. *
Suppose for all .
In this case, exists, denoted by .
We claim that .
If not, suppose .
Choose such that .
Taking limit in the second equation of system (1.5) produceswhich is impossible asfor sufficiently large .
This proves the claim. Since
(*H _{2}*) implies ,
we have proved .

*Step 3.*We verifyFor any positive , there exists such thatFor , (3.12) combined with the third equation of (1.5) givesThusWith the help of Lemmas 2.1 and 2.2, we obtainLetting , we immediately getSimilarly, one can show .

*Step 4.*We checkConditions (

*H*) and (

_{1}*H*) imply thathold for all small enough positive constant . For any such , there exists such thatThen, for , it follows from (3.19) and the first equation of system (1.5) thatAccording to Lemma 2.4, one hasLetting leads to

_{2}Now, for any small positive , it follows from (3.19) and (3.22) that there exists such thatfor . This, combined with the second equation of system (1.5), givesApplying Lemma 2.4, one easily obtainsBecause of the arbitrariness of , it is not difficult to see that .

*Step 5.*Finally, we only show as the proof of is similar. For any such that , there exists such thatThis and the third equation of system (1.5) imply thatfor . Then, for ,It follows from Lemmas 2.1 and 2.2 immediately thatLetting gives . This completes the proof of the theorem.

Theorem 3.2. * Assume that ( H_{1}) and (H_{2}) hold. Assume further that there exist positive
constants ,
and such that
*

*Then the species and are globally attractive, that is, for any positive solutions and of system (1.5) with the initial conditions (1.6),*

*Proof. *
From conditions
(*H _{3}*)β(

*H*), there exists small enough positive constant such thatSince (

_{6}*H*) and (

_{1}*H*) hold, for any positive solutions and of system (1.5) with the initial conditions (1.6), it follows from Theorem 3.1 thatThen, there exists an such that, for all ,LetThen, from the first equation of system (1.5), we haveUsing the mean value theorem, we getwhere lies between and .

_{2}It follows from (3.35) and (3.36) thatSo, for ,LetThen, from the second equation of system (1.5), we haveUsing the mean value theorem, we getwhere lies between and .

Now, it follows from (3.40) and (3.41) that, for ,Let Then, from the third equation of system (1.5), one can easily obtain thatIt follows from (3.44) that, for ,Let Similar to the analysis of (3.44) and (3.45), we can getNow, we define a Lyapunov function as follows:For , it follows from (3.38), (3.42), (3.45), and (3.47) thatSummating both sides of the above inequalities from to , we havewhich impliesIt follows thatLetting giveswhich implies thatthat is,This completes the proof of Theorem 3.2.

#### 4. Extinction of the Predator Species

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

Theorem 4.1. *
Assume that
**Then, for any solution of system (1.5), *

*Proof. *
From condition (*H _{7}*),
there exists small enough positive constant such that
For all ,
from (4.2) and the second equation of system (1.5), one can easily obtain
thatTherefore,which yieldsThe proof of Theorem 4.1 is complete.

#### 5. Examples

The following two examples show the feasibility of the main results.

*Example 5.1. *Consider the following
system:One could easily see that there
exist ,
and such thatClearly, conditions (*H _{1}*)β(

*H*) are satisfied. From Theorems 3.1 and 3.2, the system is permanent and globally attractive. Numeric simulation (Figure 1) strongly supports our results.

_{6}*Example 5.2. *
Consider the following system:By simple computation, we can
easily haveThus, condition (*H _{7}*) is satisfied; from Theorem 4.1, it follows
that Numeric simulation (Figure 2) strongly
supports our result.

#### Acknowledgment

This work was supported by the Program for New Century Excellent Talents in Fujian Province University (0330-003383).

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