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Discrete Dynamics in Nature and Society
VolumeΒ 2008Β (2008), Article IDΒ 186539, 19 pages
http://dx.doi.org/10.1155/2008/186539
Research Article

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 𝑦 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: π‘ξ…ž1(𝑑)=𝑁1𝑏(𝑑)1(𝑑)βˆ’π‘Ž1(𝑑)𝑁1(π‘‘βˆ’πœ1𝛼(𝑑))βˆ’1(𝑑)𝑁1(𝑑)1+π‘šπ‘21𝑁(𝑑)2,𝑁(π‘‘βˆ’πœŽ(𝑑))ξ…ž2(𝑑)=𝑁2ξ‚€(𝑑)βˆ’π‘2(𝑑)βˆ’π‘Ž2(𝑑)𝑁2𝛼(𝑑)+2(𝑑)𝑁21(π‘‘βˆ’πœ2(𝑑))1+π‘šπ‘21(π‘‘βˆ’πœ2,(𝑑))(1.1) where 𝑁1(𝑑),𝑁2(𝑑) are the densities of the prey population and predator population at time 𝑑, respectively; π‘π‘–βˆΆβ„β†’β„,π‘Žπ‘–,πœπ‘–,𝜎,π›Όπ‘–βˆΆβ„β†’[0,+∞)(𝑖=1,2) are continuous functions of period 𝑇 and βˆ«π‘‡0𝑏𝑖(𝑑)𝑑𝑑>0,𝛼𝑖(𝑑)β‰ 0; π‘š is a nonnegative constant. For more works on the predator-prey system with Holling type functional response, one could refer to [3–11] 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:π‘₯ξ…žξ‚€βˆ«(𝑑)=π‘₯(𝑑)π‘Ž(𝑑)βˆ’π‘(𝑑)π‘‘βˆ’βˆžξ‚βˆ’π‘˜(π‘‘βˆ’π‘ )π‘₯(𝑠)𝑑𝑠𝑐(𝑑)π‘₯2(𝑑)𝑦(𝑑)π‘š2𝑦2(𝑑)+π‘₯2,𝑦(𝑑)ξ…žξ‚€(𝑑)=𝑦(𝑑)𝑒(𝑑)π‘₯2(π‘‘βˆ’πœ)π‘š2𝑦2(π‘‘βˆ’πœ)+π‘₯2,(π‘‘βˆ’πœ)βˆ’π‘‘(𝑑)(1.2)where π‘₯(𝑑),𝑦(𝑑) are the densities of the prey population and predator population at time 𝑑, respectively, π‘Ž(𝑑),𝑏(𝑑),𝑐(𝑑),𝑑(𝑑), and 𝑒(𝑑) are all positive periodic continuous functions, and π‘š>0,𝜏β‰₯0 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: 𝑁1(π‘˜+1)=𝑁1𝑏(π‘˜)exp1(π‘˜)βˆ’π‘Ž1(π‘˜)𝑁1(π‘˜βˆ’[𝜏1π‘Ž])βˆ’1(π‘˜)𝑁1(π‘˜)𝑁2(π‘˜)𝑁21(π‘˜)+π‘š2(π‘˜)𝑁22,𝑁(𝐾)2(π‘˜+1)=𝑁2(π‘˜)expβˆ’π‘2π‘Ž(π‘˜)+2(π‘˜)𝑁21(π‘˜βˆ’[𝜏2])𝑁21(π‘˜βˆ’[𝜏1])+π‘š2(π‘˜)𝑁22(π‘˜βˆ’[𝜏2,])(1.3) sufficient conditions which ensure the permanence of system (1.3) are obtained.

However, as pointed out by Huo and Li [20], β€œecosystem 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.” For this direction, in [15], Chen and Ji proposed the following system:Μ‡π‘₯1=π‘₯1ξ‚ƒπ‘Ž1𝑐(𝑑)βˆ’π‘(𝑑)π‘₯βˆ’1(𝑑)π‘₯1+𝛼(𝑑)π‘₯2βˆ’π‘’1(𝑑)𝑒1ξ‚„,Μ‡π‘₯2=π‘₯2ξ‚ƒβˆ’π‘Ž2𝑐(𝑑)+2(𝑑)π‘₯1+𝛼(𝑑)π‘₯2+𝑒2(𝑑)𝑒2ξ‚„,̇𝑒1=β„Ž1(𝑑)βˆ’π‘‘1(𝑑)𝑒1+𝑓1(𝑑)π‘₯1,̇𝑒2=β„Ž2(𝑑)βˆ’π‘‘2(𝑑)𝑒2+𝑓2(𝑑)π‘₯2,(1.4)where π‘₯𝑖(𝑑) stand for the densities of the prey and predator, respectively, 𝑒𝑖(𝑑)(𝑖=1,2) is the control variable, 𝛼(𝑑),𝑏(𝑑),π‘Žπ‘–(𝑑),𝑐𝑖(𝑑),β„Žπ‘–(𝑑),𝑑𝑖(𝑑),𝑓𝑖(𝑑), and 𝑒𝑖(𝑑)(𝑖=1,2) 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, 21–28].

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: π‘₯(𝑛+1)=π‘₯(𝑛)expπ‘Ž(𝑛)βˆ’π‘(𝑛)π‘₯(𝑛)βˆ’π‘(𝑛)π‘₯(𝑛)𝑦(𝑛)π‘₯2(𝑛)+π‘š2(𝑛)𝑦2(𝑛)βˆ’π‘’1(𝑛)𝑒1,(𝑛)𝑦(𝑛+1)=𝑦(𝑛)expβˆ’π‘‘(𝑛)+𝑓(𝑛)π‘₯2(𝑛)π‘₯2(𝑛)+π‘š2(𝑛)𝑦2(𝑛)βˆ’π‘’2(𝑛)𝑒2,(𝑛)Δ𝑒1(𝑛)=βˆ’πœ‚1(𝑛)𝑒1(𝑛)+π‘ž1(𝑛)π‘₯(𝑛),Δ𝑒2(𝑛)=βˆ’πœ‚2(𝑛)𝑒2(𝑛)+π‘ž2(𝑛)𝑦(𝑛),(1.5) where π‘₯(𝑑),𝑦(𝑑) are the densities of the prey population and predator population at time 𝑑, respectively, for 𝑖=1,2,{π‘Ž(𝑛)},{𝑏(𝑛)},{𝑐(𝑛)},{𝑑(𝑛)},{𝑓(𝑛)},{π‘š(𝑛)},{𝑒𝑖(𝑛)},{πœ‚π‘–(𝑛)}, and {π‘žπ‘–(𝑛)} are all bounded nonnegative sequences such that0<π‘ŽπΏβ‰€π‘Ž(𝑛)β‰€π‘Žπ‘ˆ,0<𝑏𝐿≀𝑏(𝑛)β‰€π‘π‘ˆ,0<𝑐𝐿≀𝑐(𝑛)β‰€π‘π‘ˆ,0<𝑑𝐿≀𝑑(𝑛)β‰€π‘‘π‘ˆ,0<𝑓𝐿≀𝑓(𝑛)β‰€π‘“π‘ˆ,0<π‘šπΏβ‰€π‘š(𝑛)β‰€π‘šπ‘ˆ,0<𝑒𝐿𝑖≀𝑒𝑖(𝑛)β‰€π‘’π‘ˆπ‘–,0<πœ‚πΏπ‘–β‰€πœ‚π‘–(𝑛)β‰€πœ‚π‘ˆπ‘–<1,0<π‘žπΏπ‘–β‰€π‘žπ‘–(𝑛)β‰€π‘žπ‘ˆπ‘–(<1.H0)Here, for any bounded sequence {π‘Ž(𝑛)},π‘ŽπΏ=infπ‘›βˆˆβ„•{π‘Ž(𝑛)},π‘Žπ‘ˆ=supπ‘›βˆˆβ„•{π‘Ž(𝑛)}.

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 formπ‘₯(0)>0,𝑦(0)>0,𝑒𝑖(0)>0,𝑖=1,2.(1.6)It is not difficult to see that solutions of (1.5) and (1.6) are well defined and satisfy π‘₯(𝑛)>0,𝑦(𝑛)>0,𝑒𝑖(𝑛)>0,forπ‘˜βˆˆβ„•+.(1.7)

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 𝑦. 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 equation𝑦(π‘˜+1)=𝐴𝑦(π‘˜)+𝐡,π‘˜=1,2…,(2.1)where 𝐴,𝐡 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 |𝐴|<1. For any initial 𝑦(0), there exists a unique solution 𝑦(π‘˜) of (2.1), which can be expressed as follows: 𝑦(π‘˜)=π΄π‘˜ξ‚€π‘¦(0)βˆ’π‘¦βˆ—ξ‚+π‘¦βˆ—,(2.2)where π‘¦βˆ—=𝐡/(1βˆ’π΄). Thus, for any solutions 𝑦(π‘˜) of system (2.1), limπ‘˜β†’+βˆžπ‘¦(π‘˜)=π‘¦βˆ—.(2.3)

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

Lemma 2.2. Let π‘˜βˆˆβ„•+π‘˜0={π‘˜0,π‘˜0+1,…,π‘˜0+𝑙,…} and π‘Ÿβ‰₯0. For any fixed π‘˜,𝑔(π‘˜,π‘Ÿ) is a nondecreasing function with respect to π‘Ÿ, and for π‘˜β‰₯π‘˜0, the following inequalities hold: 𝑦(π‘˜+1)≀𝑔(π‘˜,𝑦(π‘˜)),𝑒(π‘˜+1)β‰₯𝑔(π‘˜,𝑒(π‘˜)).(2.4)If 𝑦(π‘˜0)≀𝑒(π‘˜0), then 𝑦(π‘˜)≀𝑒(π‘˜) for all π‘˜β‰₯π‘˜0.

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

Lemma 2.3 (see [21]). Assume that π‘₯(𝑛) satisfies π‘₯(𝑛)>0 and π‘₯(𝑛+1)≀π‘₯(𝑛)exp(π‘Ÿ(𝑛)βˆ’π‘Ž(𝑛)π‘₯(𝑛))(2.5)for π‘›βˆˆβ„•, where π‘Ÿ(𝑛) and π‘Ž(𝑛) are nonnegative sequences bounded above and below by positive constants. Then limsup𝑛→+∞1π‘₯(𝑛)β‰€π‘ŽπΏexp(π‘Ÿπ‘ˆβˆ’1).(2.6)

Lemma 2.4 (see [21]). Assume that {π‘₯(𝑛)} satisfies π‘₯(𝑛+1)β‰₯π‘₯(𝑛)exp(π‘Ÿ(𝑛)βˆ’π‘Ž(𝑛)π‘₯(𝑛)),π‘˜β‰₯𝑁0,(2.7)limsup𝑛→+∞π‘₯(𝑛)≀π‘₯βˆ— and π‘₯(𝑁0)>0, where π‘Ÿ(𝑛) and π‘Ž(𝑛) are nonnegative sequences bounded above and below by positive constants and 𝑁0βˆˆβ„•. Then liminf𝑛→+βˆžπ‘Ÿπ‘₯(𝑛)β‰₯πΏξ‚†π‘ŸexpπΏβˆ’π‘Žπ‘ˆπ‘₯βˆ—ξ‚‡π‘Žπ‘ˆ.(2.8)

3. Permanence and Global Attractivity

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

Theorem 3.1. Assume that π‘ŽπΏβˆ’π‘π‘ˆ2π‘šπΏβˆ’π‘’π‘ˆ1π‘Š1>0,(H1)π‘“πΏβˆ’π‘‘π‘ˆβˆ’π‘’π‘ˆ2π‘Š2>0(H2)π‘₯hold, then the species 𝑦 and (π‘₯(𝑛),𝑦(𝑛),𝑒1(𝑛),𝑒2(𝑛)) are permanent, that is, for any positive solution π‘š1≀liminf𝑛→+∞π‘₯(𝑛)≀limsup𝑛→+∞π‘₯(𝑛)≀𝑀1,π‘š2≀liminf𝑛→+βˆžπ‘¦(𝑛)≀limsup𝑛→+βˆžπ‘¦(𝑛)≀𝑀2,𝑀1≀liminf𝑛→+βˆžπ‘’1(𝑛)≀limsup𝑛→+βˆžπ‘’1(𝑛)β‰€π‘Š1,𝑀2≀liminf𝑛→+βˆžπ‘’2(𝑛)≀limsup𝑛→+βˆžπ‘’2(𝑛)β‰€π‘Š2,(3.1) of system (1.5) with the initial conditions (1.6), π‘š1=π‘ŽπΏβˆ’π‘π‘ˆ/2π‘šπΏβˆ’π‘’π‘ˆ1π‘Š1π‘π‘ˆξ‚†π‘ŽexpπΏβˆ’π‘π‘ˆπ‘€1βˆ’π‘π‘ˆ2π‘šπΏβˆ’π‘’π‘ˆ1π‘Š1,π‘š2=2π‘š1(π‘“πΏβˆ’π‘‘π‘ˆβˆ’π‘’π‘ˆ2π‘Š2)π‘“πΏπ‘šπ‘ˆξ‚†expβˆ’π‘‘π‘ˆβˆ’π‘’2π‘Š2,𝑀1=π‘žπΏ1π‘š1πœ‚π‘ˆ1,𝑀2=π‘žπΏ2π‘š2πœ‚π‘ˆ2,𝑀1=1π‘πΏξ€½π‘Žexpπ‘ˆξ€Ύβˆ’1,𝑀2=π‘“π‘ˆπ‘€12π‘‘πΏπ‘šπΏξ€½π‘“expπ‘ˆβˆ’π‘‘πΏξ€Ύ,π‘Š1=π‘žπ‘ˆ1𝑀1πœ‚πΏ1,π‘Š2=π‘žπ‘ˆ2𝑀2πœ‚πΏ2.(3.2)where limsup𝑛→+∞π‘₯(𝑛)≀𝑀1.(3.3)

Proof. We divided the proof into five steps.
Step 1. We showξ€½ξ€Ύπ‘₯(𝑛+1)≀π‘₯(𝑛)expπ‘Ž(𝑛)βˆ’π‘(𝑛)π‘₯(𝑛).(3.4) From the first equation of system (1.5),πœ€Then (3.3) follows immediately from Lemma 2.3. Thus, for any positive 𝑛0>0, there exists an π‘₯(𝑛)≀𝑀1+πœ€βˆ€π‘›>𝑛0.(3.5) such thatlimsup𝑛→+βˆžπ‘¦(𝑛)≀𝑀2Step 2. We prove 𝑛1β‰₯𝑛0 by distinguishing two cases.
Case 1. There exists an 𝑦(𝑛1+1)β‰₯𝑦(𝑛1) such that βˆ’π‘‘(𝑛1)+𝑓(𝑛1)π‘₯2(𝑛1)π‘₯2(𝑛1)+π‘š2(𝑛1)𝑦2(𝑛1)βˆ’π‘’2(𝑛1)𝑒2(𝑛1)β‰₯0,(3.6). Then, by the second equation of system (1.5), we haveβˆ’π‘‘πΏ+π‘“π‘ˆπ‘₯(𝑛1)2π‘šπΏπ‘¦(𝑛1)β‰₯0.(3.7)which implies𝑦(𝑛1)β‰€π‘“π‘ˆ(𝑀1+πœ€)/2π‘šπΏπ‘‘πΏThe above inequality combined with (3.5) leads to 𝑦(𝑛1+1)≀𝑦(𝑛1)expβˆ’π‘‘(𝑛1)+𝑓(𝑛1)π‘₯2(𝑛1)π‘₯2(𝑛1)+𝑦2(𝑛1)≀𝑦(𝑛1)expβˆ’π‘‘πΏ+π‘“π‘ˆξ‚‡β‰€π‘“π‘ˆ(𝑀1+πœ€)2π‘šπΏπ‘‘πΏξ‚†expβˆ’π‘‘πΏ+π‘“π‘ˆξ‚‡def=π‘€πœ€2.(3.8). Thus from the second equation of system (1.5) again we have𝑦(𝑛)β‰€π‘€πœ€2We claim that 𝑛β‰₯𝑛1 for all π‘žβ‰₯𝑛1+2. In fact, suppose there exists 𝑦(π‘ž)>π‘€πœ€2 such that π‘ž0. Let 𝑛1 be the smallest integer between π‘ž and 𝑦(π‘ž0)>π‘€πœ€2 such that 𝑦(π‘ž0βˆ’1)β‰€π‘€πœ€2 and 𝑦(π‘ž0)>𝑦(π‘ž0βˆ’1). Then 𝑦(π‘ž0)β‰€π‘€πœ€2 implies πœ€β†’0, a contradiction. This proves the claim. Setting limsup𝑛→+βˆžπ‘¦(𝑛)≀𝑀2 in (3.8) leads to 𝑦(𝑛+1)≀𝑦(𝑛).Case 2. Suppose 𝑛β‰₯𝑛0 for all lim𝑛→+βˆžπ‘¦(𝑛). In this case, 𝑦 exists, denoted by π‘¦β‰€π‘“π‘ˆπ‘€1/2π‘šπΏπ‘‘πΏ. We claim that 𝑦>π‘“π‘ˆπ‘€1/2π‘šπΏπ‘‘πΏ. If not, suppose 𝜎>0. Choose 𝜎<π‘¦βˆ’π‘“π‘ˆπ‘€1/2π‘šπΏπ‘‘πΏ such that lim𝑛→+βˆžξ‚€βˆ’π‘‘(𝑛)+𝑓(𝑛)π‘₯2(𝑛)π‘₯2(𝑛)+π‘š2(𝑛)𝑦2(𝑛)βˆ’π‘’2(𝑛)𝑒2(𝑛)=0,(3.9). Taking limit in the second equation of system (1.5) producesβˆ’π‘‘(𝑛)+𝑓(𝑛)π‘₯2(𝑛)π‘₯2(𝑛)+π‘š2(𝑛)𝑦2(𝑛)βˆ’π‘’2(𝑛)𝑒2(𝑛)β‰€βˆ’π‘‘πΏ+π‘“π‘ˆπ‘₯(𝑛)2π‘šπΏπ‘¦(𝑛)<βˆ’π‘‘πΏ+π‘“π‘ˆπ‘€12π‘šπΏ(π‘¦βˆ’πœŽ)<0(3.10)which is impossible as𝑛for sufficiently large π‘“π‘ˆπ‘€1/2π‘šπΏπ‘‘πΏβ‰€π‘€2. This proves the claim. Since (H2) implies limsup𝑛→+βˆžπ‘¦(𝑛)≀𝑀2, we have proved limsup𝑛→+βˆžπ‘’1(𝑛)β‰€π‘Š1,limsup𝑛→+βˆžπ‘’2(𝑛)β‰€π‘Š2.(3.11).
Step 3. We verifyπœ€For any positive 𝑛2, there exists π‘₯(𝑛)≀𝑀1+πœ€,𝑦(𝑛)≀𝑀2+πœ€for𝑛β‰₯𝑛2.(3.12) such that𝑛β‰₯𝑛2For Δ𝑒1(𝑛)β‰€βˆ’πœ‚1(𝑛)𝑒1(𝑛)+π‘ž1(𝑛)(𝑀1+πœ€).(3.13), (3.12) combined with the third equation of (1.5) gives𝑒1ξ‚€(𝑛+1)≀1βˆ’πœ‚πΏ1𝑒1(𝑛)+π‘žπ‘ˆ1(𝑀1+πœ€)for𝑛β‰₯𝑛2.(3.14)Thuslimsup𝑛→+βˆžπ‘’1π‘ž(𝑛)β‰€π‘ˆ1(𝑀1+πœ€)πœ‚πΏ1.(3.15)With the help of Lemmas 2.1 and 2.2, we obtainπœ€β†’0Letting limsup𝑛→+βˆžπ‘’1(𝑛)β‰€π‘Š1.(3.16), we immediately getlimsup𝑛→+βˆžπ‘’2(𝑛)β‰€π‘Š2Similarly, one can show liminf𝑛→+∞π‘₯(𝑛)β‰₯π‘š1,limsup𝑛→+βˆžπ‘¦(𝑛)β‰₯π‘š2.(3.17).Step 4. We checkπ‘ŽπΏβˆ’π‘π‘ˆ2π‘šπΏβˆ’π‘’π‘ˆ1(π‘Š1+πœ€)>0,π‘“πΏβˆ’π‘‘π‘ˆβˆ’π‘’π‘ˆ2(π‘Š1+πœ€)>0(3.18)Conditions (H1) and (H2) imply thatπœ€hold for all small enough positive constant πœ€. For any such 𝑛3, there exists π‘₯(𝑛)≀𝑀1+πœ€,𝑒1(𝑛)β‰€π‘Š1+πœ€for𝑛β‰₯𝑛3.(3.19) such that𝑛β‰₯𝑛3Then, for ξ‚†π‘Žπ‘₯(𝑛+1)β‰₯π‘₯(𝑛)expπΏβˆ’π‘π‘ˆ2π‘šπΏβˆ’π‘’π‘ˆ1(π‘Š1+πœ€)βˆ’π‘π‘ˆξ‚‡π‘₯(𝑛).(3.20), it follows from (3.19) and the first equation of system (1.5) thatliminf𝑛→+βˆžπ‘Žπ‘₯(𝑛)β‰₯πΏβˆ’π‘π‘ˆ/2π‘šπΏβˆ’π‘’π‘ˆ1(π‘Š1+πœ€)π‘π‘ˆξ‚†π‘ŽexpπΏβˆ’π‘π‘ˆ2π‘šπΏβˆ’π‘’π‘ˆ1(π‘Š1+πœ€)βˆ’π‘π‘ˆ(𝑀1+πœ€).(3.21)According to Lemma 2.4, one hasπœ€β†’0Letting liminf𝑛→+∞π‘₯(𝑛)β‰₯π‘š1.(3.22) leads toπœ€1<π‘š1/2
Now, for any small positive 𝑛4>0, it follows from (3.19) and (3.22) that there exists π‘₯(𝑛)β‰₯π‘š1βˆ’πœ€1,𝑦(𝑛)≀𝑀2+πœ€1,𝑒2(𝑛)β‰€π‘Š2+πœ€1(3.23) such that𝑛β‰₯𝑛4for 𝑦(𝑛+1)β‰₯𝑦(𝑛)expβˆ’π‘‘π‘ˆ+π‘“πΏβˆ’π‘’π‘ˆ2(π‘Š2+πœ€1𝑓)βˆ’πΏπ‘šπ‘ˆπ‘¦(𝑛)2(π‘š1βˆ’πœ€1).(3.24). This, combined with the second equation of system (1.5), givesliminf𝑛→+∞β‰₯𝑦(𝑛)2(π‘š1βˆ’πœ€1)(π‘“πΏβˆ’π‘‘π‘ˆβˆ’π‘’π‘ˆ2(π‘Š2+πœ€1))π‘“πΏπ‘šπ‘ˆξ‚†π‘“expπΏβˆ’π‘‘π‘ˆβˆ’π‘’π‘ˆ2(π‘Š2+πœ€1𝑓)βˆ’πΏπ‘šπ‘ˆ(𝑀2+πœ€1)2(π‘š1βˆ’πœ€1).(3.25)Applying Lemma 2.4, one easily obtainsπœ€Because of the arbitrariness of liminf𝑛→+βˆžπ‘¦(𝑛)β‰₯π‘š2, it is not difficult to see that liminf𝑛→+βˆžπ‘’1(𝑛)β‰₯𝑀1.
Step 5. Finally, we only show liminf𝑛→+βˆžπ‘’2(𝑛)β‰₯𝑀2 as the proof of πœ€2>0 is similar. For any πœ€2<π‘š1 such that 𝑛5, there exists π‘₯(𝑛)β‰₯π‘š1βˆ’πœ€2for𝑛β‰₯𝑛5.(3.26) such thatΔ𝑒1(𝑛)β‰₯βˆ’πœ‚1(𝑛)𝑒1(𝑛)+π‘žπΏ1(π‘š1βˆ’πœ€2)(3.27)This and the third equation of system (1.5) imply that𝑛β‰₯𝑛5for 𝑛β‰₯𝑛5. Then, for 𝑒1(𝑛+1)β‰₯(1βˆ’πœ‚π‘ˆ1)𝑒1(𝑛)+π‘žπΏ1(π‘š1βˆ’πœ€2).(3.28),liminf𝑛→+βˆžπ‘’1π‘ž(𝑛)β‰₯𝐿1(π‘š1βˆ’πœ€2)πœ‚π‘ˆ1.(3.29)It follows from Lemmas 2.1 and 2.2 immediately thatπœ€2β†’0Letting liminf𝑛→+βˆžπ‘’1(𝑛)β‰₯𝑀1 gives 𝛼,𝛽,𝛾,𝜁. This completes the proof of the theorem.

Theorem 3.2. Assume that (H1) and (H2) hold. Assume further that there exist positive constants 𝛿, and 𝑏𝛼min𝐿,2𝑀1βˆ’π‘π‘ˆξ‚‡π‘βˆ’π›Όπ‘ˆ4(π‘šπΏ)2π‘š2π‘βˆ’π›Όπ‘ˆπ‘€14π‘š12π‘“βˆ’π›½π‘ˆπ‘€22π‘š2π‘š1βˆ’π›Ύπ‘žπ‘ˆ1>𝛿,(H3)𝛽min2𝑓𝐿(π‘šπΏ)2π‘š21π‘š2𝑀21+(π‘šπ‘ˆ)2𝑀222,2𝑀2βˆ’π‘“π‘ˆπ‘€12π‘š1π‘š2ξ‚‡π‘βˆ’π›Όπ‘ˆπ‘€14(π‘šπΏ)2π‘š22π‘βˆ’π›Όπ‘ˆ4π‘š1βˆ’πœπ‘žπ‘ˆ2(>𝛿,H4)π›Ύπœ‚πΏ1βˆ’π›Όπ‘’π‘ˆ1(>𝛿,H5)πœπœ‚πΏ2βˆ’π›½π‘’π‘ˆ2(>𝛿.H6) such that π‘₯𝑦(π‘₯1(𝑛),𝑦1(𝑛),𝑒1(𝑛),𝑒2(𝑛))(π‘₯2(𝑛),𝑦2(𝑛),𝑒1(𝑛),𝑒2(𝑛))Then the species lim𝑛→+∞|||π‘₯1(𝑛)βˆ’π‘₯2|||(𝑛)=0,lim𝑛→+∞|||𝑦1(𝑛)βˆ’π‘¦2|||(𝑛)=0,lim𝑛→+∞|||𝑒1(𝑛)βˆ’ξ‚π‘’1|||(𝑛)=0,lim𝑛→+∞|||𝑒2(𝑛)βˆ’ξ‚π‘’2|||(𝑛)=0.(3.30) and πœ€<min{π‘š1/2,π‘š2/2} are globally attractive, that is, for any positive solutions 𝑏𝛼min𝐿,2(𝑀1+πœ€)βˆ’π‘π‘ˆξ‚‡π‘βˆ’π›Όπ‘ˆ4(π‘šπΏ)2(π‘š2π‘βˆ’πœ€)βˆ’π›Όπ‘ˆ(𝑀1+πœ€)4(π‘š1βˆ’πœ€)2π‘“βˆ’π›½π‘ˆ(𝑀2+πœ€)2(π‘š1βˆ’πœ€)(π‘š2βˆ’πœ€)βˆ’π›Ύπ‘žπ‘ˆ1>𝛿,𝛽min2𝑓𝐿(π‘šπΏ)2(π‘š1βˆ’πœ€)2(π‘š2βˆ’πœ€)((𝑀1+πœ€)2+(π‘šπ‘ˆ)2(𝑀2+πœ€)2)2,2(𝑀2βˆ’π‘“+πœ€)π‘ˆ(𝑀1+πœ€)2(π‘š1βˆ’πœ€)(π‘š2ξ‚‡π‘βˆ’πœ€)βˆ’π›Όπ‘ˆ(𝑀1+πœ€)4(π‘šπΏ)2(π‘š2βˆ’πœ€)2π‘βˆ’π›Όπ‘ˆ4(π‘š1βˆ’πœ€)βˆ’πœπ‘žπ‘ˆ2>𝛿,π›Ύπœ‚πΏ1βˆ’π›Όπ‘’π‘ˆ1>𝛿,πœπœ‚πΏ2βˆ’π›½π‘’π‘ˆ2>𝛿.(3.31) and (π‘₯1(𝑛),𝑦1(𝑛),𝑒1(𝑛),𝑒2(𝑛)) of system (1.5) with the initial conditions (1.6), (π‘₯2(𝑛),𝑦2(𝑛),𝑒1(𝑛),𝑒2(𝑛))

Proof. From conditions (H3)–(H6), there exists small enough positive constant π‘š1≀liminf𝑛→+∞π‘₯𝑖(𝑛)≀limsup𝑛→+∞π‘₯𝑖(𝑛)≀𝑀1,π‘š2≀liminf𝑛→+βˆžπ‘¦π‘–(𝑛)≀limsup𝑛→+βˆžπ‘¦π‘–(𝑛)≀𝑀2,𝑀𝑖≀liminf𝑛→+βˆžπ‘’π‘–(𝑛)≀limsup𝑛→+βˆžπ‘’π‘–(𝑛)β‰€π‘Šπ‘–,𝑀𝑖≀liminf𝑛→+βˆžξ‚π‘’π‘–(𝑛)≀limsup𝑛→+βˆžξ‚π‘’π‘–(𝑛)β‰€π‘Šπ‘–,𝑖=1,2.(3.32) such that𝑛0>0Since (H1) and (H2) hold, for any positive solutions 𝑛>𝑛0 and π‘š1βˆ’πœ€β‰€π‘₯𝑖(𝑛)≀𝑀1+πœ€,π‘š2βˆ’πœ€β‰€π‘¦π‘–(𝑛)≀𝑀2𝑀+πœ€,π‘–βˆ’πœ€β‰€π‘’π‘–(𝑛)β‰€π‘Šπ‘–+πœ€,π‘€π‘–βˆ’πœ€β‰€ξ‚π‘’π‘–(𝑛)β‰€π‘Šπ‘–+πœ€,𝑖=1,2.(3.33) of system (1.5) with the initial conditions (1.6), it follows from Theorem 3.1 that𝑉1||(𝑛)=lnπ‘₯1(𝑛)βˆ’lnπ‘₯2||(𝑛).(3.34)Then, there exists an 𝑉1||(𝑛+1)=lnπ‘₯1(𝑛+1)βˆ’lnπ‘₯2||≀||(𝑛+1)lnπ‘₯1(𝑛)βˆ’lnπ‘₯2(𝑛)βˆ’π‘(𝑛)(π‘₯1(𝑛)βˆ’π‘₯2|||||π‘₯(𝑛))+𝑐(𝑛)1(𝑛)𝑦1(𝑛)π‘₯21(𝑛)+π‘š2(𝑛)𝑦21βˆ’π‘₯(𝑛)2(𝑛)𝑦2(𝑛)π‘₯22(𝑛)+π‘š2(𝑛)𝑦22|||(𝑛)+𝑒1||𝑒(𝑛)1(𝑛)βˆ’ξ‚π‘’1||.(𝑛)(3.35) such that, for all π‘₯1(𝑛)βˆ’π‘₯2(𝑛)=exp(lnπ‘₯1(𝑛))βˆ’exp(lnπ‘₯2(𝑛))=πœ‰1(𝑛)(lnπ‘₯1(𝑛)βˆ’lnπ‘₯2(𝑛)),(3.36),πœ‰1(𝑛)Letπ‘₯1(𝑛)Then, from the first equation of system (1.5), we haveπ‘₯2(𝑛)Using the mean value theorem, we get𝑉1||(𝑛+1)≀lnπ‘₯1(𝑛)βˆ’lnπ‘₯2||βˆ’ξ‚€1(𝑛)πœ‰1βˆ’|||1(𝑛)πœ‰1|||(𝑛)βˆ’π‘(𝑛)|π‘₯1(𝑛)βˆ’π‘₯2||+|||(𝑛)𝑐(𝑛)π‘₯1(𝑛)𝑦1(𝑛)π‘₯2(𝑛)(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22|||||π‘₯(𝑛))1(𝑛)βˆ’π‘₯2||+|||(𝑛))𝑐(𝑛)π‘š2(𝑛)𝑦21(𝑛)𝑦2(𝑛)(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22|||||π‘₯(𝑛))1(𝑛)βˆ’π‘₯2||+|||(𝑛))𝑐(𝑛)π‘₯21(𝑛)π‘₯2(𝑛)(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22|||||𝑦(𝑛))1(𝑛)βˆ’π‘¦2||+|||(𝑛))𝑐(𝑛)π‘š2(𝑛)𝑦1(𝑛)𝑦2(𝑛)π‘₯1(𝑛)(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22|||||𝑦(𝑛))1(𝑛)βˆ’π‘¦2||(𝑛))+𝑒1||𝑒(𝑛)1(𝑛)βˆ’ξ‚π‘’1||.(𝑛)(3.37)where 𝑛β‰₯𝑛0 lies between Δ𝑉1𝑏(𝑛)β‰€βˆ’min𝐿,2𝑀1+πœ€βˆ’π‘π‘ˆξ‚‡||π‘₯1(𝑛)βˆ’π‘₯2||+𝑐(𝑛)π‘ˆ4(π‘šπΏ)2(π‘š2||π‘₯βˆ’πœ€)1(𝑛)βˆ’π‘₯2||+𝑐(𝑛)π‘ˆ(𝑀1+πœ€)4(π‘š1βˆ’πœ€)2||π‘₯1(𝑛)βˆ’π‘₯2||+𝑐(𝑛)π‘ˆ(𝑀1+πœ€)4(π‘šπΏ)2(π‘š2βˆ’πœ€)2||𝑦1(𝑛)βˆ’π‘¦2||+𝑐(𝑛)π‘ˆ4(π‘š1||π‘¦βˆ’πœ€)1(𝑛)βˆ’π‘¦2||(𝑛)+π‘’π‘ˆ1||𝑒1(𝑛)βˆ’ξ‚π‘’1||.(𝑛)(3.38) and 𝑉2||(𝑛)=ln𝑦1(𝑛)βˆ’ln𝑦2||(𝑛).(3.39).
It follows from (3.35) and (3.36) that𝑉2=||(𝑛+1)ln𝑦1(𝑛+1)βˆ’ln𝑦2||=|||(𝑛+1)ln𝑦1(𝑛)βˆ’ln𝑦2ξ‚€π‘₯(𝑛)+𝑓(𝑛)21(𝑛)π‘₯21+π‘š2(𝑛)𝑦21βˆ’π‘₯(𝑛)22(𝑛)π‘₯22(𝑛)+π‘š2(𝑛)𝑦22(𝑛)βˆ’π‘’2(𝑛)(𝑒2(𝑛)βˆ’ξ‚π‘’2|||≀|||(𝑛))ln𝑦1(𝑛)βˆ’ln𝑦2(𝑛)βˆ’π‘“(𝑛)π‘š2(𝑛)π‘₯1(𝑛)(π‘₯1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)π‘₯2(𝑛))(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22(𝑛))Γ—(𝑦1(𝑛)βˆ’π‘¦2|||+(𝑛))𝑓(𝑛)π‘š2(𝑛)𝑦1(𝑛)(π‘₯1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)π‘₯2(𝑛))(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22Γ—||π‘₯(𝑛))1(𝑛)βˆ’π‘₯2||(𝑛)+𝑒2||𝑒(𝑛)1(𝑛)βˆ’ξ‚π‘’1||.(𝑛)(3.40)So, for 𝑦1(𝑛)βˆ’π‘¦2(𝑛)=exp(ln𝑦1(𝑛))βˆ’exp(ln𝑦2(𝑛))=πœ‰2(𝑛)(ln𝑦1(𝑛)βˆ’ln𝑦2(𝑛)),(3.41),πœ‰2(𝑛)Let𝑦1(𝑛)Then, from the second equation of system (1.5), we have𝑦2(𝑛)Using the mean value theorem, we get𝑛>𝑛0where Δ𝑉2ξ‚€1(𝑛)β‰€βˆ’πœ‰2βˆ’|||1(𝑛)πœ‰2βˆ’(𝑛)𝑓(𝑛)π‘š2(𝑛)π‘₯1(𝑛)(π‘₯1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)π‘₯2(𝑛))(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22|||||𝑦(𝑛))1(𝑛)βˆ’π‘¦2||+(𝑛)𝑓(𝑛)π‘š2(𝑛)𝑦1(𝑛)(π‘₯1(𝑛)𝑦2(𝑛)+𝑦1(𝑛)π‘₯2(𝑛))(π‘₯21(𝑛)+π‘š2(𝑛)𝑦21(𝑛))(π‘₯22(𝑛)+π‘š2(𝑛)𝑦22||π‘₯(𝑛))1(𝑛)βˆ’π‘₯2||(𝑛)+𝑒2||𝑒(𝑛)2(𝑛)βˆ’ξ‚π‘’2||(𝑛)β‰€βˆ’min2𝑓𝐿(π‘šπΏ)2(π‘š1βˆ’πœ€)2(π‘š2βˆ’πœ€)((𝑀1+πœ€)2+(π‘šπ‘ˆ)2(𝑀2+πœ€))2,2(𝑀2βˆ’π‘“+πœ€)π‘ˆ(𝑀1+πœ€)2(π‘š1βˆ’πœ€)(π‘š2×||π‘¦βˆ’πœ€)1(𝑛)βˆ’π‘¦2||+𝑓(𝑛)π‘ˆ(𝑀2+πœ€)2(π‘š1βˆ’πœ€)(π‘š2||π‘₯βˆ’πœ€)1(𝑛)βˆ’π‘₯2||(𝑛)+π‘’π‘ˆ2||𝑒2(𝑛)βˆ’ξ‚π‘’2||.(𝑛)(3.42) lies between 𝑉3|||𝑒(𝑛)=1(𝑛)βˆ’ξ‚π‘’1|||(𝑛).(3.43) and 𝑉3|||𝑒(𝑛+1)=1(𝑛+1)βˆ’ξ‚π‘’1|||(𝑛+1)≀(1βˆ’πœ‚1|||𝑒(𝑛))1(𝑛)βˆ’ξ‚π‘’1|||(𝑛)+π‘ž1||π‘₯(𝑛)1(𝑛)βˆ’π‘₯2||.(𝑛)(3.44).
Now, it follows from (3.40) and (3.41) that, for 𝑛>𝑛0,Δ𝑉3(𝑛)β‰€βˆ’πœ‚πΏ1|||𝑒1(𝑛)βˆ’ξ‚π‘’1|||(𝑛)+π‘žπ‘ˆ1|||π‘₯1(𝑛)βˆ’π‘₯2|||(𝑛).(3.45)Let𝑉4|||𝑒(𝑛)=2(𝑛)βˆ’ξ‚π‘’2|||(𝑛).(3.46) Then, from the third equation of system (1.5), one can easily obtain thatΔ𝑉4(𝑛)β‰€βˆ’πœ‚πΏ2|||𝑒2(𝑛)βˆ’ξ‚π‘’2|||(𝑛)+π‘žπ‘ˆ2|||𝑦1(𝑛)βˆ’π‘¦2|||(𝑛).(3.47)It follows from (3.44) that, for 𝑉(𝑛)=𝛼𝑉1(𝑛)+𝛽𝑉2(𝑛)+𝛾𝑉3(𝑛)+πœπ‘‰4(𝑛).(3.48),𝑛>𝑛0Let𝑏Δ𝑉(𝑛)β‰€βˆ’π›Όmin𝐿,2(𝑀1+πœ€)βˆ’π‘π‘ˆξ‚‡π‘βˆ’π›Όπ‘ˆ4(π‘šπΏ)2(π‘š2π‘βˆ’πœ€)βˆ’π›Όπ‘ˆ(𝑀1+πœ€)4(π‘š1βˆ’πœ€)2π‘“βˆ’π›½π‘ˆ(𝑀2+πœ€)2(π‘š1βˆ’πœ€)(π‘š2βˆ’πœ€)βˆ’π›Ύπ‘žπ‘ˆ1ξ‚„Γ—||π‘₯1(𝑛)βˆ’π‘₯2||βˆ’ξ‚ƒξ‚†(𝑛)𝛽min2𝑓𝐿(π‘šπΏ)2(π‘š1βˆ’πœ€)2(π‘š2βˆ’πœ€)((𝑀1+πœ€)2+(π‘šπ‘ˆ)2(𝑀2+πœ€)2)2,2(𝑀2βˆ’+πœ€)2π‘“π‘ˆ(𝑀1+πœ€)2(π‘š1βˆ’πœ€)(π‘š2ξ‚‡π‘βˆ’πœ€)βˆ’π›Όπ‘ˆ(𝑀1+πœ€)4(π‘šπΏ)2(π‘š2βˆ’πœ€)2π‘βˆ’π›Όπ‘ˆ4(π‘š1βˆ’πœ€)βˆ’πœπ‘žπ‘ˆ2ξ‚„Γ—||𝑦1(𝑛)βˆ’π‘¦2||(𝑛)βˆ’(π›Ύπœ‚πΏ1βˆ’π›Όπ‘’π‘’1|||𝑒)Γ—1(𝑛)βˆ’ξ‚π‘’1|||(𝑛)βˆ’(πœπœ‚πΏ2βˆ’π›½π‘’π‘ˆ2|||𝑒)Γ—2(𝑛)βˆ’ξ‚π‘’2|||||π‘₯(𝑛)β‰€βˆ’π›Ώ(1(𝑛)βˆ’π‘₯2||+||𝑦(𝑛)1(𝑛)βˆ’π‘¦2||+|||𝑒(𝑛)1(𝑛)βˆ’ξ‚π‘’1|||+||𝑒(𝑛)2(𝑛)βˆ’ξ‚π‘’2||(𝑛)).(3.49) Similar to the analysis of (3.44) and (3.45), we can get𝑛0Now, we define a Lyapunov function as follows:𝑛For 𝑛𝑝=𝑛0(𝑉(𝑝+1)βˆ’π‘‰(𝑝))β‰€βˆ’π›Ώπ‘›ξ“π‘=𝑛0ξ‚€||π‘₯1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||+|||𝑒(𝑝)1(𝑝)βˆ’ξ‚π‘’1|||+|||𝑒(𝑝)2(𝑝)βˆ’ξ‚π‘’2|||,(𝑝)(3.50), it follows from (3.38), (3.42), (3.45), and (3.47) that𝑉(𝑛0)β‰₯𝛿𝑛𝑝=𝑛0ξ‚€||π‘₯1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||+|||𝑒(𝑝)1(𝑝)βˆ’ξ‚π‘’1|||+|||𝑒(𝑝)2(𝑝)βˆ’ξ‚π‘’2|||(𝑝)+𝑉(𝑛+1).(3.51)Summating both sides of the above inequalities from 𝑉(𝑛0)𝛿β‰₯𝑛𝑝=𝑛0ξ‚€||π‘₯1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||+|||𝑒(𝑝)1(𝑝)βˆ’ξ‚π‘’1|||+|||𝑒(𝑝)2(𝑝)βˆ’ξ‚π‘’2|||.(𝑝)(3.52) to π‘›β†’βˆž, we have𝑉(𝑛0)𝛿β‰₯+βˆžξ“π‘=𝑛0ξ‚€|||π‘₯1(𝑝)βˆ’π‘₯2|||+|||𝑦(𝑝)1(𝑝)βˆ’π‘¦2|||+|||𝑒(𝑝)1(𝑝)βˆ’ξ‚π‘’1|||+|||𝑒(𝑝)2(𝑝)βˆ’ξ‚π‘’2|||,(𝑝)(3.53)which implieslimπ‘›β†’βˆžξ‚€|||π‘₯1(𝑛)βˆ’π‘₯2|||+|||𝑦(𝑛)1(𝑛)βˆ’π‘¦2|||+|||𝑒(𝑛)1(𝑛)βˆ’ξ‚π‘’1|||+|||𝑒(𝑛)2(𝑛)βˆ’ξ‚π‘’2|||(𝑛)=0,(3.54)It follows thatlimπ‘›β†’βˆž|||π‘₯1(𝑛)βˆ’π‘₯2|||(𝑛)=0,limπ‘›β†’βˆž|||𝑦1(𝑛)βˆ’π‘¦2|||(𝑛)=0,limπ‘›β†’βˆž|||𝑒1(𝑛)βˆ’ξ‚π‘’1|||(𝑛)=0,limπ‘›β†’βˆž|||𝑒1(𝑛)βˆ’ξ‚π‘’1|||(𝑛)=0.(3.55)Letting 𝑦 givesβˆ’π‘‘πΏ+π‘“π‘ˆ(<0.H7)which implies that(π‘₯(𝑛),𝑦(𝑛),𝑒1(𝑛),𝑒2(𝑛))that is,lim𝑛→+βˆžπ‘¦(𝑛)=0.(4.1)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 𝛾>0.

Theorem 4.1. Assume that βˆ’π‘‘πΏ+π‘“π‘ˆ<βˆ’π›Ύ<0.(4.2)Then, for any solution π‘›βˆˆβ„• of system (1.5), 𝑦(𝑛+1)=𝑦(𝑛)expβˆ’π‘‘(𝑛)+𝑓(𝑛)π‘₯2(𝑛)π‘₯2(𝑛)+π‘š2(𝑛)𝑦2(𝑛)βˆ’π‘’2(𝑛)𝑒2(𝑛)<𝑦(𝑛)expβˆ’π‘‘πΏ+π‘“π‘ˆξ€Ύξ€½<𝑦(𝑛)expβˆ’π›Ύ}.(4.3)

Proof. From condition (H7), there exists small enough positive constant 𝑦(𝑛+1)<𝑦(0)expβˆ’π‘›π›Ύ,(4.4) such thatlim𝑛→+βˆžπ‘¦(𝑛)=0.(4.5) For all ξ‚†ξ‚€βˆšπ‘₯(𝑛+1)=π‘₯(𝑛)exp1.225+0.025sinξ‚βˆ’2π‘›βˆ’(5.25+0.25cos(𝑛))π‘₯(𝑛)((0.0075+0.0025sin(𝑛))𝑦(𝑛)π‘₯(𝑛)π‘₯2ξ‚€ξ‚€βˆš(𝑛)+0.885+0.005cos3𝑛2𝑦2βˆ’ξ‚€ξ‚€βˆš(𝑛)0.002+0.001cos𝑒3𝑛1,ξ‚†βˆ’ξ‚€ξ‚€βˆš(𝑛)𝑦(𝑛+1)=𝑦(𝑛)exp0.1125+0.0025cos+ξ‚€ξ‚€βˆš2𝑛0.195+0.005sinπ‘₯3𝑛2(𝑛)π‘₯2ξ‚€ξ‚€βˆš(𝑛)+0.885+0.005cos3𝑛2𝑦2(𝑛)βˆ’(0.0015+0.0005sin(𝑛))𝑒2,(𝑛)Δ𝑒1(𝑛)=βˆ’(0.925+0.025sin(𝑛))𝑒1(𝑛)+(0.0375+0.0275cos(𝑛))π‘₯(𝑛),Δ𝑒2(𝑛)=βˆ’(0.925+0.025cos(𝑛))𝑒2(𝑛)+(0.025+0.005sin(𝑛))𝑦(𝑛).(5.1), from (4.2) and the second equation of system (1.5), one can easily obtain that𝛼=0.06,𝛽=0.05,𝛾=0.01,𝜁=0.001Therefore,𝛿=0.00001which yieldsπ‘ŽπΏβˆ’π‘π‘ˆ2π‘šπΏβˆ’π‘’π‘ˆ1π‘Š1π‘“β‰ˆ1.2>0,πΏβˆ’π‘‘π‘ˆβˆ’π‘’π‘ˆ2π‘Š2ξ‚†π‘β‰ˆ0.04>0,𝛼min𝐿,2𝑀1βˆ’π‘π‘ˆξ‚‡π‘βˆ’π›Όπ‘ˆ4(π‘šπΏ)2π‘š2π‘βˆ’π›Όπ‘ˆπ‘€14π‘š21π‘“βˆ’π›½π‘ˆπ‘€22π‘š1π‘š2βˆ’π›Ύπ‘žπ‘ˆ1ξ‚†β‰ˆ0.04512>𝛿,𝛽min2𝑓𝐿(π‘šπΏ)2π‘š21π‘š2𝑀21+(π‘šπ‘ˆ)2𝑀22,2𝑀2βˆ’π‘“π‘ˆπ‘€12π‘š1π‘š2ξ‚‡π‘βˆ’π›Όπ‘ˆπ‘€14(π‘šπΏ)2π‘š2π‘βˆ’π›Όπ‘ˆ4π‘š1βˆ’πœπ‘žπ‘ˆ2β‰ˆ0.000013>𝛿,π›Ύπœ‚πΏ1βˆ’π›Όπ‘’π‘ˆ1β‰ˆ0.0072>𝛿,πœπœ‚πΏ2βˆ’π›½π‘’π‘ˆ2β‰ˆ0.00087>𝛿.(5.2)The 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:(π‘₯(0),𝑦(0),𝑒1(0),𝑒2(0))𝑇=(0.31,0.26,0.02,0.08)𝑇One could easily see that there exist (0.18,0.18,0.01,0.1)𝑇, and π‘₯,𝑦 such that𝑒1,𝑒2Clearly, conditions (H1)–(H6) are satisfied. From Theorems 3.1 and 3.2, the system is permanent and globally attractive. Numeric simulation (Figure 1) strongly supports our results.

fig1
Figure 1: Dynamics behaviors of system (5.1) with initial conditions ξ‚†βˆ’π‘₯(𝑛+1)=π‘₯(𝑛)exp0.8+0.1sin(𝑛)βˆ’(11.5+3.5cos(𝑛))π‘₯(𝑛)(0.035+0.025sin(𝑛))𝑦(𝑛)π‘₯(𝑛)π‘₯2ξ‚€ξ‚€βˆš(𝑛)+5.75+0.35cos2𝑛2𝑦2(𝑛)βˆ’(0.025+0.005cos(𝑛))𝑒1,ξ‚†βˆ’ξ‚€ξ‚€βˆš(𝑛)𝑦(𝑛+1)=𝑦(𝑛)exp0.975+0.025cos+√2𝑛(0.195+0.005sin(3𝑛))π‘₯2(𝑛)π‘₯2√(𝑛)+(5.75+0.35cos(2𝑛))2𝑦2βˆ’ξ‚€ξ‚€βˆš(𝑛)0.0015+0.0005cos𝑒3𝑛2,(𝑛)Δ𝑒1(𝑛)=βˆ’(0.925+0.025sin(𝑛))𝑒1(𝑛)+(0.0375+0.0275cos(𝑛))π‘₯(𝑛),Δ𝑒2(𝑛)=βˆ’(0.925βˆ’0.025cos(𝑛))𝑒2(𝑛)+(0.025+0.005sin(𝑛))𝑦(𝑛).(5.3) and βˆ’π‘‘πΏ+π‘“π‘ˆ=βˆ’0.8<0.(5.4), respectively.

Example 5.2. Consider the following system:π‘₯By simple computation, we can easily have𝑦Thus, condition (H7) is satisfied; from Theorem 4.1, it follows that 𝑛 Numeric simulation (Figure 2) strongly supports our result.

fig2
Figure 2: Dynamics behaviors of system (5.2) with initial conditions .

Acknowledgment

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

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