Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 323065 | 17 pages | https://doi.org/10.1155/2009/323065

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

Academic Editor: Leonid Berezansky
Received25 Feb 2009
Accepted04 Apr 2009
Published14 Jun 2009

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]: 𝑑π‘₯ξ‚€π‘₯𝑑𝑑=π‘Žπ‘₯1βˆ’π‘˜ξ‚βˆ’π‘π‘₯𝑦,π‘š+π‘₯𝑑𝑦𝑑𝑑=π‘¦βˆ’π‘‘+𝑓π‘₯ξ‚Ά,π‘š+π‘₯π‘₯(0)>0,𝑦(0)>0,(1.1) where π‘₯, 𝑦 denote the density of prey and predator species at time 𝑑, respectively. The constants π‘Ž, π‘˜, 𝑐, π‘š, 𝑓, 𝑑 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 [3–8] 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 β€œratio-dependent’’ theory.

Arditi and Ginzburg [9] proposed the following predator-prey model with ratio-dependent type functional response: 𝑑π‘₯ξ‚€π‘₯𝑑𝑑=π‘Žπ‘₯1βˆ’π‘˜ξ‚βˆ’π‘π‘₯𝑦,π‘šπ‘¦+π‘₯𝑑𝑦𝑑𝑑=π‘¦βˆ’π‘‘+𝑓π‘₯ξ‚Ά,π‘šπ‘¦+π‘₯π‘₯(0)>0,𝑦(0)>0.(1.2)

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: 𝑑π‘₯ξ‚€π‘₯𝑑𝑑=π‘₯π‘Žβˆ’π‘˜ξ‚βˆ’π‘π‘₯π‘¦π‘šπ‘¦π‘Ÿ,+π‘₯𝑑𝑦𝑑𝑑=π‘¦βˆ’π‘‘+𝑓π‘₯π‘šπ‘¦π‘Ÿξ‚Ά+π‘₯,π‘Ÿβˆˆ(0,1),π‘₯(0)>0,𝑦(0)>0.(1.3) 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 𝛾=1, 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 𝛾=1/2. For aquatic predators that form a fixed number of tight groups, 𝛾=1/3 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 [12–24]. This motivated us to propose and study the discrete analogous of predator-prey system (1.3): ξ‚»π‘₯(π‘˜+1)=π‘₯(π‘˜)expπ‘Ž(π‘˜)βˆ’π‘(π‘˜)π‘₯(π‘˜)βˆ’π‘(π‘˜)𝑦(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿξ‚Ό,ξ‚»(π‘˜)+π‘₯(π‘˜)𝑦(π‘˜+1)=𝑦(π‘˜)expβˆ’π‘‘(π‘˜)+𝑓(π‘˜)π‘₯(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ(ξ‚Ό,π‘˜)+π‘₯(π‘˜)(1.4) where π‘Ÿβˆˆ(0,1); {π‘Ž(π‘˜)}, {𝑏(π‘˜)}, {𝑐(π‘˜)}, {𝑑(π‘˜)}, {π‘š(π‘˜)}, {𝑓(π‘˜)} are all bounded nonnegative sequences. For the rest of the paper, we use the following notations: for any bounded sequence {𝑔(π‘˜)}, set 𝑔𝑒=supπ‘˜βˆˆπ‘π‘”(π‘˜),𝑔𝑙=infπ‘˜βˆˆπ‘π‘”(π‘˜).(1.5)

By the biological meaning, we will focus our discussion on the positive solution of system of (1.3). Thus, we require that π‘₯(0)>0,𝑦(0)>0.(1.6)

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 π‘š and 𝑀, which are independent of the solution of system (1.4), such that for any positive solution {π‘₯(π‘˜),𝑦(π‘˜)} of system (1.4) satisfies π‘šβ‰€liminfπ‘˜β†’+∞{π‘₯(π‘˜),𝑦(π‘˜)}≀limsupπ‘˜β†’+∞{π‘₯(π‘˜),𝑦(π‘˜)}≀𝑀.(2.1)

Lemma 2.2 (see [23]). Assume that {π‘₯(π‘˜)} satisfies π‘₯(π‘˜)>0 and π‘₯(π‘˜+1)≀π‘₯(π‘˜)exp{π‘Ž(π‘˜)βˆ’π‘(π‘˜)π‘₯(π‘˜)}(2.2) for π‘˜βˆˆπ‘, where π‘Ž(π‘˜) and 𝑏(π‘˜) are all nonnegative sequences bounded above and below by positive constants. Then limsupπ‘˜β†’+∞1π‘₯(π‘˜)≀𝑏𝑙exp(π‘Žπ‘’βˆ’1).(2.3)

Lemma 2.3 (see [23]). Assume that {π‘₯(π‘˜)} satisfies π‘₯(π‘˜+1)β‰₯π‘₯(π‘˜)exp{π‘Ž(π‘˜)βˆ’π‘(π‘˜)π‘₯(π‘˜)},π‘˜β‰₯𝑁0,(2.4)limsupπ‘˜β†’+∞π‘₯(π‘˜)≀π‘₯βˆ— and π‘₯(𝑁0)>0, where π‘Ž(π‘˜) and 𝑏(π‘˜) are all nonnegative sequences bounded above and below by positive constants and 𝑁0βˆˆπ‘. Then liminfπ‘˜β†’+∞π‘₯π‘Ž(π‘˜)β‰₯π‘™ξ€½π‘Žexpπ‘™βˆ’π‘π‘’π‘₯βˆ—ξ€Ύπ‘π‘’.(2.5)

Theorem 2.4. Assume that π‘Žπ‘™βˆ’π‘π‘’π‘€21βˆ’π‘Ÿπ‘šπ‘™(𝐻>0,1ξ€Έ)𝑓𝑙>𝑑𝑒(𝐻2ξ€Έ) hold, then system (1.4) is permanent, that is, for any positive solution {π‘₯(π‘˜),𝑦(π‘˜)} of system (1.4), one has π‘š1≀liminfπ‘˜β†’+∞π‘₯(π‘˜)≀limsupπ‘˜β†’+∞π‘₯(π‘˜)≀𝑀1,π‘š2≀liminfπ‘˜β†’+∞π‘₯(π‘˜)≀limsupπ‘˜β†’+βˆžπ‘¦(π‘˜)≀𝑀2,(2.6) where π‘š1=π‘Žπ‘™βˆ’ξ€·π‘π‘’π‘€21βˆ’π‘Ÿ/π‘šπ‘™ξ€Έπ‘π‘’ξƒ―π‘Žexpπ‘™βˆ’π‘π‘’π‘€21βˆ’π‘Ÿπ‘šπ‘™βˆ’π‘π‘’π‘€1ξƒ°,π‘š2⎧βŽͺ⎨βŽͺβŽ©ξƒ―ξ€·π‘“=minπ‘™βˆ’π‘‘π‘’ξ€Έπ‘š1π‘šπ‘’π‘‘π‘’ξƒ°1/π‘Ÿ,ξƒ―ξ€·π‘“π‘™βˆ’π‘‘π‘’ξ€Έπ‘š1π‘šπ‘’π‘‘π‘’ξƒ°1/π‘Ÿξ‚»expβˆ’π‘‘π‘’+π‘“π‘™π‘š1π‘šπ‘’π‘€π‘Ÿ2+π‘š1ξ‚ΌβŽ«βŽͺ⎬βŽͺ⎭,𝑀1=1𝑏𝑙exp(π‘Žπ‘’π‘€βˆ’1),2=𝑓𝑒𝑀1π‘šπ‘™π‘‘π‘™ξ‚Ό1/π‘Ÿξ€½expβˆ’π‘‘π‘™+𝑓𝑒.(2.7)

Proof. We divided the proof into four claims.Claim 1. From the first equation of (1.4), we have π‘₯(π‘˜+1)≀π‘₯(π‘˜)exp{π‘Ž(π‘˜)βˆ’π‘(π‘˜)π‘₯(π‘˜)}.(2.8) By Lemma 2.2, we have limsupπ‘˜β†’+∞1π‘₯(π‘˜)≀𝑏𝑙exp(π‘Žπ‘’βˆ’1)def=𝑀1.(2.9) Above inequality shows that for any πœ€>0, there exists a π‘˜1>0, such that π‘₯(π‘˜+1)≀𝑀1+πœ€,βˆ€π‘˜β‰₯π‘˜1.(2.10)Claim 2. We divide it into two cases to prove that limsupπ‘˜β†’+βˆžπ‘¦(π‘˜)≀𝑀2.(2.11)Case (i)
There exists an 𝑙0β‰₯π‘˜1, such that 𝑦(𝑙0+1)β‰₯𝑦(𝑙0). Then by the second equation of system (1.4), we have ξ€·π‘™βˆ’π‘‘0ξ€Έ+𝑓𝑙0ξ€Έπ‘₯𝑙0ξ€Έπ‘šξ€·π‘™0ξ€Έπ‘¦π‘Ÿξ€·π‘™0𝑙+π‘₯0ξ€Έβ‰₯0.(2.12) Hence, ξ€·π‘™βˆ’π‘‘0ξ€Έ+𝑓𝑙0ξ€Έπ‘₯𝑙0ξ€Έπ‘šξ€·π‘™0ξ€Έπ‘¦π‘Ÿξ€·π‘™0ξ€Έβ‰₯0,(2.13) therefore, π‘¦π‘Ÿξ€·π‘™0≀𝑙0ξ€Έπ‘₯𝑙0ξ€Έπ‘šξ€·π‘™0𝑑𝑙0≀𝑓𝑒𝑀1ξ€Έ+πœ€π‘šπ‘™π‘‘π‘™,(2.14) and so, 𝑦𝑙0≀𝑓𝑒(𝑀1+πœ€)π‘šπ‘™π‘‘π‘™ξ‚Ό1/π‘Ÿ.(2.15) It follows that 𝑦𝑙0𝑙+1=𝑦0𝑙expβˆ’π‘‘0ξ€Έ+𝑓𝑙0ξ€Έπ‘₯𝑙0ξ€Έπ‘šξ€·π‘™0ξ€Έπ‘¦π‘Ÿξ€·π‘™0𝑙+π‘₯0≀𝑓𝑒(𝑀1+πœ€)π‘šπ‘™π‘‘π‘™ξ‚Ό1/π‘Ÿξ€½expβˆ’π‘‘π‘™+𝑓𝑒def=𝑀2πœ€.(2.16) We claim that 𝑦(π‘˜)≀𝑀2πœ€βˆ€π‘˜β‰₯𝑙0.(2.17) By a way of contradiction, assume that there exists a 𝑝0>𝑙0 such that 𝑦(𝑝0)>𝑀2πœ€. Then 𝑝0β‰₯𝑙0+2. Let 𝑦(̃𝑝0)β‰₯𝑙0+2 be the smallest integer such that 𝑦(̃𝑝0)β‰₯𝑀2πœ€. Then 𝑦(̃𝑝0)>𝑦(̃𝑝0βˆ’1). The above argument produces that 𝑦(̃𝑝0)≀𝑀2πœ€, a contradiction. This prove the claim.
Case (ii)
We assume that 𝑦(π‘˜+1)<𝑦(π‘˜) for all 𝐾β‰₯𝐾1. Since 𝑦(π‘˜) is nonincreasing and has a lower bound 0, we know that limπ‘˜β†’+βˆžπ‘¦(π‘˜) exists, denoted by 𝑦, then limπ‘˜β†’+βˆžπ‘¦(π‘˜)=𝑦.(2.18) We claim that 𝑓𝑦≀𝑒(𝑀1+πœ€)π‘šπ‘™π‘‘π‘™ξ‚Ό1/π‘Ÿ.(2.19) By a way of contradiction, assume that 𝑦>{𝑓𝑒(𝑀1+πœ€)/π‘šπ‘™π‘‘π‘™}1/π‘Ÿ. Taking limit in the second equation in system (1.4) gives limπ‘˜β†’+βˆžξ‚»βˆ’π‘‘(π‘˜)+𝑓(π‘˜)π‘₯(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿξ‚Ό(π‘˜)+π‘₯(π‘˜)=0,(2.20) which is a contradiction since for 𝐾>𝐾1βˆ’π‘‘(π‘˜)+𝑓(π‘˜)π‘₯(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ(π‘˜)+π‘₯(π‘˜)β‰€βˆ’π‘‘π‘™+𝑓𝑒𝑀1ξ€Έ+πœ€π‘šπ‘™π‘¦π‘Ÿ<0.(2.21) This prove the claim, then we have limsupπ‘˜β†’+βˆžπ‘¦(π‘˜)=limπ‘˜β†’+βˆžπ‘¦(π‘˜)=𝑓𝑦≀𝑒(𝑀1+πœ€)π‘šπ‘™π‘‘π‘™ξ‚Ό1/π‘Ÿ.(2.22) Combining Cases (i) and (ii), we see that limsupπ‘˜β†’+βˆžπ‘¦(π‘˜)≀𝑀2πœ€.(2.23) Let πœ€β†’0, we have limsupπ‘˜β†’+βˆžξ‚»π‘“π‘¦(π‘˜)≀𝑒𝑀1π‘šπ‘™π‘‘π‘™ξ‚Ό1/π‘Ÿξ€½expβˆ’π‘‘π‘™+𝑓𝑒=𝑀2.(2.24)
Claim 3 (liminfπ‘˜β†’βˆžπ‘₯(π‘˜)β‰₯π‘š1). Conditions (𝐻1) imply that for enough small positive constant πœ€, we have π‘Žπ‘™βˆ’π‘π‘’(𝑀2+πœ€)1βˆ’π‘Ÿπ‘šπ‘™>0.(2.25) For above πœ€, it follows form Claims 1 and 2 that there exists a π‘˜2 such that for all π‘˜>π‘˜2π‘₯(π‘˜)≀𝑀1+πœ€,𝑦(π‘˜)≀𝑀2+πœ€.(2.26) From the first equation of (1.4), we have ξƒ―π‘Žπ‘₯(π‘˜+1)β‰₯π‘₯(π‘˜)expπ‘™βˆ’π‘π‘’(𝑀2+πœ€)1βˆ’π‘Ÿπ‘šπ‘™βˆ’π‘π‘’ξƒ°π‘₯(π‘˜).(2.27) By applying Lemma 2.3 to above inequality, we have liminfπ‘˜β†’+βˆžπ‘Žπ‘₯(π‘˜)β‰₯π‘™βˆ’ξ€·π‘π‘’(𝑀2+πœ€)1βˆ’π‘Ÿ/π‘šπ‘™ξ€Έπ‘π‘’ξƒ―π‘Žexpπ‘™βˆ’π‘π‘’(𝑀2+πœ€)1βˆ’π‘Ÿπ‘šπ‘™βˆ’π‘π‘’ξ€·π‘€1ξ€Έξƒ°+πœ€.(2.28) Setting πœ€β†’0 in (2.28) leads to liminfπ‘˜β†’+βˆžπ‘Žπ‘₯(π‘˜)β‰₯π‘™βˆ’ξ€·π‘π‘’π‘€21βˆ’π‘Ÿ/π‘šπ‘™ξ€Έπ‘π‘’ξƒ―π‘Žexpπ‘™βˆ’π‘π‘’π‘€21βˆ’π‘Ÿπ‘šπ‘™βˆ’π‘π‘’π‘€1ξƒ°def=π‘š1.(2.29) This ends the proof of Claim 3.Claim 4. For any small positive constant πœ€<π‘š1/2, from Claims 1–3, it follows that there exists a π‘˜3>π‘˜2 such that for all π‘˜>π‘˜3π‘₯(π‘˜)β‰₯π‘š1βˆ’πœ€,π‘₯(π‘˜)≀𝑀1+πœ€,𝑦(π‘˜)≀𝑀2+πœ€.(2.30) We present two cases to prove that liminfπ‘˜β†’+βˆžπ‘¦(π‘˜)β‰₯π‘š2(2.31)Case (i)
There exists an 𝑛0β‰₯π‘˜3 such that 𝑦(𝑛0+1)≀𝑦(𝑛0), then ξ€·π‘›βˆ’π‘‘0ξ€Έ+𝑓𝑛0ξ€Έπ‘₯𝑛0ξ€Έπ‘šξ€·π‘›0ξ€Έπ‘¦π‘Ÿξ€·π‘›0𝑛+π‘₯0≀0.(2.32) Hence 𝑦𝑛0ξ€Έβ‰₯ξ‚»(π‘“π‘™βˆ’π‘‘π‘’)(π‘š1βˆ’πœ€)π‘šπ‘’π‘‘π‘’ξ‚Ό1/π‘Ÿdef=𝑐1πœ€,(2.33) and so, 𝑦𝑛0ξ€Έβ‰₯ξ‚»+1(π‘“π‘™βˆ’π‘‘π‘’)(π‘š1βˆ’πœ€)π‘šπ‘’π‘‘π‘’ξ‚Ό1/π‘Ÿξƒ―expβˆ’π‘‘π‘’+π‘“π‘™ξ€·π‘š1ξ€Έβˆ’πœ€π‘šπ‘’(𝑀2+πœ€)π‘Ÿ+ξ€·π‘š1ξ€Έξƒ°βˆ’πœ€def=𝑐2πœ€.(2.34) Set π‘š2πœ€ξ€½π‘=min1πœ€,𝑐2πœ€ξ€Ύ.(2.35) We claim that 𝑦(π‘˜)β‰₯π‘š2πœ€ for π‘˜β‰₯𝑛0. By a way of contradiction, assume that there exists a π‘ž0β‰₯𝑛0, such that 𝑦(π‘ž0)<π‘š2πœ€. Then π‘ž0β‰₯𝑛0+2. Let Μƒπ‘ž0β‰₯𝑛0+2 be the smallest integer such that 𝑦(Μƒπ‘ž0)<π‘š2πœ€. Then 𝑦(Μƒπ‘ž0)<𝑦(Μƒπ‘ž0βˆ’1), which implies that 𝑦(π‘ž0)β‰€π‘š2πœ€, a contradiction, this proves the claim.
Case (ii)
We assume that 𝑦(π‘˜+1)>𝑦(π‘˜) for all π‘˜>π‘˜3. According to (2.30), limπ‘˜β†’+βˆžπ‘¦(π‘˜) exists, denoted by 𝑦, then limπ‘˜β†’+βˆžπ‘¦(π‘˜)=𝑦.(2.36) We claim that 𝑦β‰₯π‘š2πœ€.(2.37) By the way of contradiction, assume that 𝑦<π‘š2πœ€. Taking limit in the second equation in system (1.4) gives limπ‘˜β†’+βˆžξ‚»βˆ’π‘‘(π‘˜)+𝑓(π‘˜)π‘₯(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿξ‚Ό(π‘˜)+π‘₯(π‘˜)=0,(2.38) which is a contradiction since for π‘˜>π‘˜3,βˆ’π‘‘(π‘˜)+𝑓(π‘˜)π‘₯(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ(π‘˜)+π‘₯(π‘˜)β‰₯βˆ’π‘‘π‘’+π‘“π‘™ξ€·π‘š1ξ€Έβˆ’πœ€π‘šπ‘’π‘¦π‘Ÿ+ξ€·π‘š1ξ€Έβˆ’πœ€>0.(2.39) The above analysis show that liminfπ‘˜β†’+βˆžπ‘¦(π‘˜)β‰₯π‘š2πœ€.(2.40) Letting πœ€β†’0, we have liminfπ‘˜β†’+βˆžπ‘¦(π‘˜)β‰₯π‘š2,(2.41) where π‘š2ξƒ―ξ‚»=min(π‘“π‘™βˆ’π‘‘π‘’)π‘š1π‘šπ‘’π‘‘π‘’ξ‚Ό1/π‘Ÿ,ξ‚»(π‘“π‘™βˆ’π‘‘π‘’)π‘š1π‘šπ‘’π‘‘π‘’ξ‚Ό1/π‘Ÿξ‚»expβˆ’π‘‘π‘’+π‘“π‘™π‘š1π‘šπ‘’π‘€π‘Ÿ2+π‘š1ξ‚Όξƒ°.(2.42) According to Claims 1–4, we can easily find that the result of Theorem 2.4 holds.

3. Global Attractivity

Theorem 3.1. Assume that (𝐻1) and (𝐻2) hold. Assume further that there exist positive constants 𝛼, 𝛽, and 𝛿 such that 𝑏𝛼min𝑙,2𝑀1βˆ’π‘π‘’ξ‚Όπ‘βˆ’π›Όπ‘’π‘€21βˆ’(π‘Ÿ/2)4π‘šπ‘™π‘š2π‘“βˆ’π›½π‘’π‘€11/24π‘š1π‘š2π‘Ÿ/2(𝐻>𝛿,3ξ€Έ)𝑓𝛽minπ‘™π‘šπ‘™π‘š1π‘Ÿ(π‘šπ‘’π‘€π‘Ÿ2+𝑀1)2𝑀21βˆ’π‘Ÿ,2𝑀2βˆ’π‘“π‘’π‘€11/2π‘Ÿ4π‘š2π‘š11/2ξƒ°π‘βˆ’π›Όπ‘’π‘€11/24π‘šπ‘™π‘šπ‘Ÿ2π‘š11/2π‘βˆ’π›Όπ‘’π‘€π‘Ÿ2(1βˆ’π‘Ÿ)4π‘š1π‘šπ‘Ÿ2𝐻>𝛿.(4ξ€Έ) Then system (1.4) with initial condition (1.6) is globally attractive, that is, for any two positive solutions (π‘₯1(π‘˜),𝑦1(π‘˜)) and (π‘₯2(π‘˜),𝑦2(π‘˜)) of system (1.4), one has limπ‘˜β†’+∞||π‘₯1(π‘˜)βˆ’π‘₯2||(π‘˜)=0,limπ‘˜β†’+∞||𝑦1(π‘˜)βˆ’π‘¦2||(π‘˜)=0.(3.1)

Proof. From conditions (𝐻3) and (𝐻4), there exists an enough small positive constant πœ€<min{π‘š1/2,π‘š2/2} such that 𝑏𝛼min𝑙,2𝑀1+πœ€βˆ’π‘π‘’ξ‚Όπ‘βˆ’π›Όπ‘’(𝑀2+πœ€)1βˆ’(π‘Ÿ/2)4π‘šπ‘™ξ€·π‘š2ξ€Έπ‘“βˆ’πœ€βˆ’π›½π‘’(𝑀1+πœ€)1/24ξ€·π‘š1ξ€Έβˆ’πœ€(π‘š2βˆ’πœ€)π‘Ÿ/2𝑓>𝛿,𝛽minπ‘™π‘šπ‘™ξ€·π‘š1ξ€Έπ‘Ÿβˆ’πœ€[π‘šπ‘’(𝑀2+πœ€)π‘Ÿ+(𝑀1+πœ€)]2(𝑀2+πœ€)1βˆ’π‘Ÿ,2𝑀2βˆ’π‘“+πœ€π‘’(𝑀1+πœ€)1/2π‘Ÿ4ξ€·π‘š2ξ€Έβˆ’πœ€(π‘š1βˆ’πœ€)1/2ξƒ°π‘βˆ’π›Όπ‘’(𝑀1+πœ€)1/24π‘šπ‘™(π‘š2βˆ’πœ€)π‘Ÿ(π‘š1βˆ’πœ€)1/2π‘βˆ’π›Όπ‘’(𝑀2+πœ€)π‘Ÿ(1βˆ’π‘Ÿ)4ξ€·π‘š1ξ€Έβˆ’πœ€(π‘š2βˆ’πœ€)π‘Ÿ>𝛿.(3.2) Since (𝐻1) and (𝐻2) hold, for any positive solutions (π‘₯1(π‘˜),𝑦1(π‘˜)) and (π‘₯2(π‘˜),𝑦2(π‘˜)) of system (1.4), it follows from Theorem 2.4 that π‘š1≀liminfπ‘˜β†’+∞π‘₯𝑖(π‘˜)≀limsupπ‘˜β†’+∞π‘₯𝑖(π‘˜)≀𝑀1,π‘š2≀liminfπ‘˜β†’+βˆžπ‘¦π‘–(π‘˜)≀limsupπ‘˜β†’+βˆžπ‘¦π‘–(π‘˜)≀𝑀2,𝑖=1,2.(3.3) For above πœ€ and (3.3), there exists a π‘˜4>0 such that for all π‘˜>π‘˜4, π‘š1βˆ’πœ€β‰€π‘₯𝑖(π‘˜)≀𝑀1+πœ€,π‘š2βˆ’πœ€β‰€π‘₯𝑖(π‘˜)≀𝑀2+πœ€,𝑖=1,2.(3.4) Let 𝑉1||(π‘˜)=lnπ‘₯1(π‘˜)βˆ’lnπ‘₯2||(π‘˜).(3.5) Then from the first equation of system (1.3), we have 𝑉1||(π‘˜+1)=lnπ‘₯1(π‘˜+1)βˆ’lnπ‘₯2||≀||(π‘˜+1)lnπ‘₯1(π‘˜)βˆ’lnπ‘₯2ξ€·π‘₯(π‘˜)βˆ’π‘(π‘˜)1(π‘˜)βˆ’π‘₯2ξ€Έ||||||𝑦(π‘˜)+𝑐(π‘˜)1(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1βˆ’π‘¦(π‘˜)2(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2||||.(π‘˜)(3.6) Using the Mean Value Theorem, we get π‘₯1(π‘˜)βˆ’π‘₯2ξ€·(π‘˜)=explnπ‘₯1ξ€Έξ€·(π‘˜)βˆ’explnπ‘₯2ξ€Έ(π‘˜)=πœ‰1ξ€·(π‘˜)lnπ‘₯1(π‘˜)βˆ’lnπ‘₯2ξ€Έ,𝑦(π‘˜)11βˆ’π‘Ÿ(π‘˜)βˆ’π‘¦21βˆ’π‘Ÿ(π‘˜)=(1βˆ’π‘Ÿ)πœ‰2βˆ’π‘Ÿξ€·π‘¦(π‘˜)1(π‘˜)βˆ’π‘¦2ξ€Έ,(π‘˜)(3.7) where πœ‰1(π‘˜) lies between π‘₯1(π‘˜) and π‘₯2(π‘˜), πœ‰2(π‘˜) lies between 𝑦1(π‘˜) and 𝑦2(π‘˜).
It follows from (3.6), (3.7) that 𝑉1||(π‘˜+1)≀lnπ‘₯1(π‘˜)βˆ’lnπ‘₯2||βˆ’ξƒ©1(π‘˜)πœ‰1βˆ’||||1(π‘˜)πœ‰1||||ξƒͺ||π‘₯(π‘˜)βˆ’π‘(π‘˜)1(π‘˜)βˆ’π‘₯2||+||||(π‘˜)𝑐(π‘˜)𝑦1(π‘˜)ξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1π‘š(π‘˜)ξ€Έξ€·(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2ξ€Έ||||||π‘₯(π‘˜)1(π‘˜)βˆ’π‘₯2||+||||(π‘˜)𝑐(π‘˜)π‘₯1(π‘˜)ξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1(π‘˜)ξ€Έξ€·π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2(ξ€Έ||||||π‘¦π‘˜)1(π‘˜)βˆ’π‘¦2||+||||(π‘˜)𝑐(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)π‘¦π‘Ÿ2(π‘˜)ξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1(π‘˜)ξ€Έξ€·π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2ξ€Έ(π‘˜)1βˆ’π‘Ÿπœ‰π‘Ÿ2(||||||π‘¦π‘˜)1(π‘˜)βˆ’π‘¦2||.(π‘˜)(3.8) And so, for π‘˜>π‘˜4Δ𝑉1ξ‚»π‘β‰€βˆ’min𝑙,2𝑀1+πœ€βˆ’π‘π‘’ξ‚Ό||π‘₯1(π‘˜)βˆ’π‘₯2||+𝑐(π‘˜)𝑒(𝑀2+πœ€)1βˆ’(π‘Ÿ/2)4π‘šπ‘™(π‘š2βˆ’πœ€)π‘Ÿ/2ξ€·π‘š1ξ€Έ||π‘₯βˆ’πœ€1(π‘˜)βˆ’π‘₯2(||+π‘π‘˜)𝑒(𝑀1+πœ€)1/24π‘šπ‘™(π‘š2βˆ’πœ€)π‘Ÿ(π‘š1βˆ’πœ€)1/2||𝑦1(π‘˜)βˆ’π‘¦2||+𝑐(π‘˜)𝑒(𝑀2+πœ€)π‘Ÿ(1βˆ’π‘Ÿ)4ξ€·π‘š1ξ€Έβˆ’πœ€(π‘š2βˆ’πœ€)π‘Ÿ||𝑦1(π‘˜)βˆ’π‘¦2||.(π‘˜)(3.9) Let 𝑉2||(π‘˜)=ln𝑦1(π‘˜)βˆ’ln𝑦2||(π‘˜).(3.10) Then from the second equation of system (1.4), we have 𝑉2||(π‘˜+1)=ln𝑦1(π‘˜+1)βˆ’ln𝑦2||=||||(π‘˜+1)ln𝑦1(π‘˜)βˆ’ln𝑦2ξ‚΅π‘₯(π‘˜)+𝑓(π‘˜)1(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1βˆ’π‘₯(π‘˜)2(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2ξ‚Ά||||≀||||(π‘˜)ln𝑦1(π‘˜)βˆ’ln𝑦2(π‘˜)βˆ’π‘“(π‘˜)π‘š(π‘˜)π‘₯1𝑦(π‘˜)π‘Ÿ1(π‘˜)βˆ’π‘¦π‘Ÿ2ξ€Έ(π‘˜)ξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1(π‘˜)ξ€Έξ€·π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2(ξ€Έ||||+||||π‘˜)𝑓(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ1ξ€·π‘₯(π‘˜)1(π‘˜)βˆ’π‘₯2ξ€Έ(π‘˜)ξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1(π‘˜)ξ€Έξ€·π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2ξ€Έ||||.(π‘˜)(3.11) Using the Mean Value Theorem, we get 𝑦1(π‘˜)βˆ’π‘¦2ξ€·(π‘˜)=expln𝑦1ξ€Έξ€·(π‘˜)βˆ’expln𝑦2ξ€Έ(π‘˜)=πœ‰3ξ€·(π‘˜)ln𝑦1(π‘˜)βˆ’ln𝑦2ξ€Έ,𝑦(𝑛)π‘Ÿ1(π‘˜)βˆ’π‘¦π‘Ÿ2(π‘˜)=π‘Ÿπœ‰4π‘Ÿβˆ’1𝑦(π‘˜)1(π‘˜)βˆ’π‘¦2ξ€Έ,(π‘˜)(3.12) where πœ‰3(π‘˜),πœ‰4(π‘˜) lie between 𝑦1(π‘˜) and 𝑦2(π‘˜), respectively. Then, it follows from (3.11), (3.12) that for π‘˜>π‘˜4,Δ𝑉21β‰€βˆ’πœ‰3βˆ’||||1(π‘˜)πœ‰3βˆ’(π‘˜)𝑓(π‘˜)π‘š(π‘˜)π‘₯1(π‘˜)π‘Ÿξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1(π‘˜)ξ€Έξ€·π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2ξ€Έπœ‰(π‘˜)41βˆ’π‘Ÿ||||ξƒͺΓ—||𝑦(π‘˜)1(π‘˜)βˆ’π‘¦2||+(π‘˜)𝑓(π‘˜)π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)ξ€·π‘š(π‘˜)π‘¦π‘Ÿ1(π‘˜)+π‘₯1(π‘˜)ξ€Έξ€·π‘š(π‘˜)π‘¦π‘Ÿ2(π‘˜)+π‘₯2ξ€Έ||π‘₯(π‘˜)1(π‘˜)βˆ’π‘₯2||𝑓(π‘˜)β‰€βˆ’minπ‘™π‘šπ‘™ξ€·π‘š1ξ€Έπ‘Ÿβˆ’πœ€[π‘šπ‘’(𝑀2+πœ€)π‘Ÿ+(𝑀1+πœ€)]2(𝑀2+πœ€)1βˆ’π‘Ÿ,2𝑀2βˆ’π‘“+πœ€π‘’(𝑀1+πœ€)1/2π‘Ÿ4ξ€·π‘š2ξ€Έβˆ’πœ€(π‘š1βˆ’πœ€)1/2ξƒ°Γ—||𝑦1(π‘˜)βˆ’π‘¦2||+𝑓(π‘˜)𝑒(𝑀1+πœ€)π‘Ÿ/24ξ€·π‘š1ξ€Έβˆ’πœ€(π‘š2βˆ’πœ€)π‘Ÿ/2||π‘₯1(π‘˜)βˆ’π‘₯2||.(π‘˜)(3.13) Now we define a Lyapunov function as follows: 𝑉(π‘˜)=𝛼𝑉1(π‘˜)+𝛽𝑉2(π‘˜).(3.14) Calculating the difference of 𝑉 along the solution of system (1.4), for π‘˜>π‘˜4, it follows from (3.9) and (3.13) that ξƒ¬ξ‚»π‘Ξ”π‘‰β‰€βˆ’π›Όmin𝑙,2𝑀1+πœ€βˆ’π‘π‘’ξ‚Όπ‘βˆ’π›Όπ‘’(𝑀2+πœ€)1βˆ’(π‘Ÿ/2)4π‘šπ‘™(π‘š2βˆ’πœ€)π‘Ÿ/2ξ€·π‘š1ξ€Έπ‘“βˆ’πœ€βˆ’π›½π‘’(𝑀1+πœ€)π‘Ÿ/24ξ€·π‘š1ξ€Έβˆ’πœ€(π‘š2βˆ’πœ€)π‘Ÿ/2ξƒ­Γ—||π‘₯1(π‘˜)βˆ’π‘₯2||βˆ’ξƒ¬ξƒ―π‘“(π‘˜)𝛽minπ‘™π‘šπ‘™ξ€·π‘š1ξ€Έπ‘Ÿβˆ’πœ€[π‘šπ‘’(𝑀2+πœ€)π‘Ÿ+(𝑀1+πœ€)]2(𝑀2+πœ€)1βˆ’π‘Ÿ,2𝑀2βˆ’π‘“+πœ€π‘’(𝑀1+πœ€)1/2π‘Ÿ4ξ€·π‘š2ξ€Έβˆ’πœ€(π‘š1βˆ’πœ€)1/2ξƒ°π‘βˆ’π›Όπ‘’(𝑀1+πœ€)1/24π‘šπ‘™(π‘š2βˆ’πœ€)π‘Ÿ(π‘š1βˆ’πœ€)1/2π‘βˆ’π›Όπ‘’(𝑀2+πœ€)π‘Ÿ(1βˆ’π‘Ÿ)4ξ€·π‘š1ξ€Έβˆ’πœ€(π‘š2βˆ’πœ€)π‘Ÿξƒ­Γ—||𝑦1(π‘˜)βˆ’π‘¦2(||ξ€·||π‘₯π‘˜)β‰€βˆ’π›Ώ1(π‘˜)βˆ’π‘₯2(||+||π‘¦π‘˜)1(π‘˜)βˆ’π‘¦2(||ξ€Έ.π‘˜)(3.15) Summating both sides of the above inequalities from π‘˜4 to π‘˜, we have π‘˜ξ“π‘=π‘˜40π‘₯0200𝑑(𝑉(𝑝+1)βˆ’π‘£(𝑝))β‰€βˆ’π›Ώπ‘˜ξ“π‘=π‘˜4ξ€·||π‘₯0π‘₯0200𝑑1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||ξ€Έ(𝑝),(3.16) which implies 𝑉(π‘˜+1)+π›Ώπ‘˜ξ“π‘=π‘˜4ξ€·||π‘₯0π‘₯0200𝑑1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||ξ€Έξ€·π‘˜(𝑝)≀𝑉4ξ€Έ.(3.17) It follows that π‘˜ξ“π‘=π‘˜4ξ€·||π‘₯0π‘₯0200𝑑1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||ξ€Έβ‰€π‘‰ξ€·π‘˜(𝑝)4𝛿.(3.18) Using the fundamental theorem of positive series, there exists small enough positive constant πœ€>0 such that +βˆžξ“π‘=π‘˜4ξ€·||π‘₯0π‘₯0200𝑑1(𝑝)βˆ’π‘₯2||+||𝑦(𝑝)1(𝑝)βˆ’π‘¦2||ξ€Έβ‰€π‘‰ξ€·π‘˜(𝑝)4𝛿,(3.19) which implies that limπ‘˜β†’+βˆžξ€·||π‘₯1(π‘˜)βˆ’π‘₯2||+||𝑦(π‘˜)1(π‘˜)βˆ’π‘¦2||ξ€Έ(π‘˜)=0,(3.20) that is limπ‘˜β†’+∞||π‘₯1(π‘˜)βˆ’π‘₯2||(π‘˜)=0,limπ‘˜β†’+∞||𝑦1(π‘˜)βˆ’π‘¦2||(π‘˜)=0.(3.21) 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 𝑦.

Theorem 4.1. Assume that βˆ’π‘‘π‘™+𝑓𝑒𝐻<0.(5ξ€Έ) Then, the species 𝑦 will be driven to extinction, and the species π‘₯ is permanent, that is, for any positive solution (π‘₯(π‘˜),𝑦(π‘˜)) of system (1.4), limπ‘˜β†’+βˆžπ‘šπ‘¦(π‘˜)=0,βˆ—β‰€liminfπ‘˜β†’+∞π‘₯(π‘˜)≀limsupπ‘˜β†’+∞π‘₯(π‘˜)≀𝑀1,(4.1) where π‘šβˆ—=π‘Žπ‘™π‘π‘’ξ€½π‘Žexpπ‘™βˆ’π‘π‘’π‘€1ξ€Ύ,𝑀1=1𝑏𝑙exp(π‘Žπ‘’βˆ’1).(4.2)

Proof. For condition (𝐻5), there exists small enough positive 𝛾>0, such that βˆ’π‘‘π‘™+𝑓𝑒<βˆ’π›Ύ<0(4.3) for all π‘˜βˆˆπ‘, from (4.3) and the second equation of the system (1.4), one can easily obtain that 𝑦(π‘˜+1)=𝑦(π‘˜)expβˆ’π‘‘(π‘˜)+𝑓(π‘˜)π‘₯(π‘˜)ξ‚Όξ€½π‘š(π‘˜)𝑦(π‘˜)+π‘₯(π‘˜)<𝑦(π‘˜)expβˆ’π‘‘π‘™+𝑓𝑒<𝑦(π‘˜)exp{βˆ’π›Ύ}.(4.4) Therefore, 𝑦(π‘˜+1)<𝑦(0)exp{βˆ’π‘˜π›Ύ},(4.5) which yields limπ‘˜β†’+βˆžπ‘¦(π‘˜)=0.(4.6) From the proof of Theorem 3.1, we have limsupπ‘˜β†’+∞π‘₯(π‘˜)≀𝑀1.(4.7) For enough small positive constant πœ€>0, π‘Žπ‘™βˆ’π‘π‘’πœ€1βˆ’π‘Ÿπ‘šπ‘™>0.(4.8) For above πœ€, from (2.9) and (4.6), there exists a π‘˜5>0 such that for all π‘˜>π‘˜5, π‘₯(π‘˜)<𝑀1+πœ€,𝑦(π‘˜)<πœ€.(4.9) From the first equation of (1.4), we have ξ‚»π‘Žπ‘₯(π‘˜+1)β‰₯π‘₯(π‘˜)expπ‘™βˆ’π‘π‘’πœ€1βˆ’π‘Ÿπ‘šπ‘™βˆ’π‘π‘’ξ‚Όπ‘₯(π‘˜).(4.10) By Lemma 2.3, we have liminfπ‘˜β†’+∞π‘₯π‘Ž(π‘˜)β‰₯π‘™βˆ’ξ€·π‘π‘’πœ€1βˆ’π‘Ÿ/π‘šπ‘™ξ€Έπ‘π‘’ξ‚»π‘Žexpπ‘™βˆ’π‘π‘’πœ€1βˆ’π‘Ÿπ‘šπ‘™βˆ’π‘π‘’ξ€·π‘€1ξ€Έξ‚Ό+πœ€.(4.11) Setting πœ€β†’0 in (4.11) leads to liminfπ‘˜β†’+βˆžπ‘Žπ‘₯(π‘˜)β‰₯π‘™π‘π‘’ξ€½π‘Žexpπ‘™βˆ’π‘π‘’π‘€1ξ€Ύdef=π‘šβˆ—.(4.12) 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: ξ‚»π‘₯(π‘˜+1)=π‘₯(π‘˜)exp1.41+0.12cos(π‘˜)βˆ’1.78π‘₯(π‘˜)βˆ’0.33𝑦(π‘˜)2.16𝑦1/2ξ‚Ό,ξ‚»(π‘˜)+π‘₯(π‘˜)𝑦(π‘˜+1)=𝑦(π‘˜)expβˆ’0.62+1.79π‘₯(π‘˜)2.16𝑦1/2ξ‚Ό.(π‘˜)+π‘₯(π‘˜)(5.1)
One could easily see that there exist positive constants 𝛼=0.01,𝛽=0.05,𝛿=0.001 such that π‘Žπ‘™βˆ’π‘π‘’π‘€21βˆ’π‘Ÿπ‘šπ‘™π‘“β‰ˆ2.3281>0,𝑙>π‘‘π‘’ξ‚»π‘β‰ˆ1.1700>0,𝛼min𝑙,2𝑀1βˆ’π‘π‘’ξ‚Όπ‘βˆ’π›Όπ‘’π‘€21βˆ’(π‘Ÿ/2)4π‘šπ‘™π‘š2π‘“βˆ’π›½π‘’π‘€11/24π‘š1π‘š2π‘Ÿ/2ξƒ―π‘“β‰ˆ0.0011>𝛿,𝛽minπ‘™π‘šπ‘™π‘š1π‘Ÿ(π‘šπ‘’π‘€π‘Ÿ2+𝑀1)2𝑀21βˆ’π‘Ÿ,2𝑀2βˆ’π‘“π‘’π‘€11/2π‘Ÿ4π‘š2π‘š11/2ξƒ°π‘βˆ’π›Όπ‘’π‘€11/24π‘šπ‘™π‘šπ‘Ÿ2π‘š11/2π‘βˆ’π›Όπ‘’π‘€π‘Ÿ2(1βˆ’π‘Ÿ)4π‘š1π‘šπ‘Ÿ2β‰ˆ0.0107>𝛿.(5.2) Clearly, conditions (𝐻1)–(𝐻4) 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 π‘Ÿβˆˆ(0,1). If π‘Ÿ=1 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 𝑦. 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 π‘Ÿ=1. If π‘Žπ‘™βˆ’π‘π‘’/π‘šπ‘™>0 holds, then the prey species π‘₯ is permanence. If π‘Ÿ=0 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 π‘Ÿ=0.

References

  1. A. A. Berryman, β€œThe origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992. View at: Publisher Site | Google Scholar
  2. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. View at: Zentralblatt MATH | MathSciNet
  3. R. Arditi and H. Saiah, β€œEmpirical evidence of the role of heterogeneity in ratio-dependent consumption,” Ecology, vol. 73, no. 5, pp. 1544–1551, 1992. View at: Publisher Site | Google Scholar
  4. R. Arditi, L. R. Ginzburg, and H. R. Akcakaya, β€œVariation in plankton densities among lakes: a case for ratio-dependent predation models,” The American Naturalist, vol. 138, no. 5, pp. 1287–1296, 1991. View at: Publisher Site | Google Scholar
  5. B. Dai, H. Su, and D. Hu, β€œPeriodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 126–134, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  6. F. Chen and J. Shi, β€œOn a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response and diffusion,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 358–369, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. A. P. Gutierrez, β€œPhysiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm,” Ecology, vol. 73, no. 5, pp. 1552–1563, 1992. View at: Publisher Site | Google Scholar
  8. R. Xu, F. A. Davidson, and M. A. J. Chaplain, β€œPersistence and stability for a two-species ratio-dependent predator-prey system with distributed time delay,” Journal of Mathematical Analysis and Applications, vol. 269, no. 1, pp. 256–277, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. R. Arditi and L. R. Ginzburg, β€œCoupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989. View at: Publisher Site | Google Scholar
  10. M. P. Hassell and G. C. Varley, β€œNew inductive population model for insect parasites and its bearing on biological control,” Nature, vol. 223, no. 5211, pp. 1133–1137, 1969. View at: Publisher Site | Google Scholar
  11. S.-B. Hsu, T.-W. Hwang, and Y. Kuang, β€œGlobal dynamics of a predator-prey model with Hassell-Varley type functional response,” Discrete and Continuous Dynamical Systems, vol. 10, no. 4, pp. 857–871, 2008. View at: Google Scholar | MathSciNet
  12. J. Yang, β€œDynamics behaviors of a discrete ratio-dependent predator-prey system with holling type III functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 186539, 19 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. F. Chen, β€œPermanence in a discrete Lotka-Volterra competition model with deviating arguments,” Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2150–2155, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. F. Chen, β€œPermanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. X. Li and W. Yang, β€œPermanence of a discrete predator-prey systems with Beddington-deAngelis functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 149267, 8 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. X. Chen and C. Fengde, β€œStable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1446–1454, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. Z. Zhou and X. Zou, β€œStable periodic solutions in a discrete periodic logistic equation,” Applied Mathematics Letters, vol. 16, no. 2, pp. 165–171, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  18. L. Chen, J. Xu, and Z. Li, β€œPermanence and global attractivity of a delayed discrete predator-prey system with general Holling-type functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 629620, 17 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. L. L. Wang and Y. H. Fan, β€œPermanence for a discrete Nicholson's blowflies model with feedback control and delay,” International Journal of Biomathematics, vol. 1, no. 4, pp. 433–442, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  20. F. Chen, β€œPermanence for the discrete mutualism model with time delays,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 431–435, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  21. F. Chen, L. Wu, and Z. Li, β€œPermanence and global attractivity of the discrete Gilpin-Ayala type population model,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1214–1227, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. F. Chen, β€œPermanence of a discrete N-species cooperation system with time delays and feedback controls,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 23–29, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  23. F. Chen, β€œPermanence and global stability of nonautonomous Lotka-Volterra system with predator-prey and deviating arguments,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 1082–1100, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  24. H.-F. Huo and W.-T. Li, β€œPermanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 135–144, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2009 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.


More related articles

524Β Views | 495Β Downloads | 5Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.