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]: 𝑑𝑥𝑥𝑑𝑡=ğ‘Žğ‘¥1−𝑘−𝑐𝑥𝑦,𝑚+𝑥𝑑𝑦𝑑𝑡=𝑦−𝑑+𝑓𝑥,𝑚+𝑥𝑥(0)>0,𝑦(0)>0,(1.1) where 𝑥, 𝑦 denote the density of prey and predator species at time 𝑡, respectively. The constants ğ‘Ž, 𝑘, 𝑐, 𝑚, 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 “ratio-dependent’’ theory.

Arditi and Ginzburg [9] proposed the following predator-prey model with ratio-dependent type functional response: 𝑑𝑥𝑥𝑑𝑡=ğ‘Žğ‘¥1−𝑘−𝑐𝑥𝑦,𝑚𝑦+𝑥𝑑𝑦𝑑𝑡=𝑦−𝑑+𝑓𝑥,𝑚𝑦+𝑥𝑥(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: 𝑑𝑥𝑥𝑑𝑡=ğ‘¥ğ‘Žâˆ’ğ‘˜î‚âˆ’ğ‘ğ‘¥ğ‘¦ğ‘šğ‘¦ğ‘Ÿ,+𝑥𝑑𝑦𝑑𝑡=𝑦−𝑑+𝑓𝑥𝑚𝑦𝑟+𝑥,𝑟∈(0,1),𝑥(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): 𝑥(𝑘+1)=𝑥(𝑘)expğ‘Ž(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑐(𝑘)𝑦(𝑘)𝑚(𝑘)𝑦𝑟,(𝑘)+𝑥(𝑘)𝑦(𝑘+1)=𝑦(𝑘)exp−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(,𝑘)+𝑥(𝑘)(1.4) where 𝑟∈(0,1); {ğ‘Ž(𝑘)}, {𝑏(𝑘)}, {𝑐(𝑘)}, {𝑑(𝑘)}, {𝑚(𝑘)}, {𝑓(𝑘)} are all bounded nonnegative sequences. For the rest of the paper, we use the following notations: for any bounded sequence {𝑔(𝑘)}, set 𝑔𝑢=sup𝑘∈𝑁𝑔(𝑘),𝑔𝑙=inf𝑘∈𝑁𝑔(𝑘).(1.5)

By the biological meaning, we will focus our discussion on the positive solution of system of (1.3). Thus, we require that 𝑥(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 𝑚 and 𝑀, which are independent of the solution of system (1.4), such that for any positive solution {𝑥(𝑘),𝑦(𝑘)} of system (1.4) satisfies 𝑚≤liminf𝑘→+∞{𝑥(𝑘),𝑦(𝑘)}≤limsup𝑘→+∞{𝑥(𝑘),𝑦(𝑘)}≤𝑀.(2.1)

Lemma 2.2 (see [23]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘)>0 and 𝑥(𝑘+1)≤𝑥(𝑘)exp{ğ‘Ž(𝑘)−𝑏(𝑘)𝑥(𝑘)}(2.2) for 𝑘∈𝑁, where ğ‘Ž(𝑘) and 𝑏(𝑘) are all nonnegative sequences bounded above and below by positive constants. Then limsup𝑘→+∞1𝑥(𝑘)≤𝑏𝑙exp(ğ‘Žğ‘¢âˆ’1).(2.3)

Lemma 2.3 (see [23]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘+1)≥𝑥(𝑘)exp{ğ‘Ž(𝑘)−𝑏(𝑘)𝑥(𝑘)},𝑘≥𝑁0,(2.4)limsup𝑘→+âˆžğ‘¥(𝑘)≤𝑥∗ and 𝑥(𝑁0)>0, where ğ‘Ž(𝑘) and 𝑏(𝑘) are all nonnegative sequences bounded above and below by positive constants and 𝑁0∈𝑁. Then liminf𝑘→+âˆžğ‘¥ğ‘Ž(𝑘)â‰¥ğ‘™î€½ğ‘Žexp𝑙−𝑏𝑢𝑥∗𝑏𝑢.(2.5)

Theorem 2.4. Assume that ğ‘Žğ‘™âˆ’ğ‘ğ‘¢ğ‘€21−𝑟𝑚𝑙(𝐻>0,1)𝑓𝑙>𝑑𝑢(𝐻2) hold, then system (1.4) is permanent, that is, for any positive solution {𝑥(𝑘),𝑦(𝑘)} of system (1.4), one has 𝑚1≤liminf𝑘→+âˆžğ‘¥(𝑘)≤limsup𝑘→+âˆžğ‘¥(𝑘)≤𝑀1,𝑚2≤liminf𝑘→+âˆžğ‘¥(𝑘)≤limsup𝑘→+âˆžğ‘¦(𝑘)≤𝑀2,(2.6) where 𝑚1=ğ‘Žğ‘™âˆ’î€·ğ‘ğ‘¢ğ‘€21−𝑟/ğ‘šğ‘™î€¸ğ‘ğ‘¢îƒ¯ğ‘Žexp𝑙−𝑐𝑢𝑀21−𝑟𝑚𝑙−𝑏𝑢𝑀1,𝑚2âŽ§âŽªâŽ¨âŽªâŽ©îƒ¯î€·ğ‘“=min𝑙−𝑑𝑢𝑚1𝑚𝑢𝑑𝑢1/𝑟,𝑓𝑙−𝑑𝑢𝑚1𝑚𝑢𝑑𝑢1/𝑟exp−𝑑𝑢+𝑓𝑙𝑚1𝑚𝑢𝑀𝑟2+𝑚1⎫⎪⎬⎪⎭,𝑀1=1𝑏𝑙exp(ğ‘Žğ‘¢ğ‘€âˆ’1),2=𝑓𝑢𝑀1𝑚𝑙𝑑𝑙1/𝑟exp−𝑑𝑙+𝑓𝑢.(2.7)

Proof. We divided the proof into four claims.Claim 1. From the first equation of (1.4), we have 𝑥(𝑘+1)≤𝑥(𝑘)exp{ğ‘Ž(𝑘)−𝑏(𝑘)𝑥(𝑘)}.(2.8) By Lemma 2.2, we have limsup𝑘→+∞1𝑥(𝑘)≤𝑏𝑙exp(ğ‘Žğ‘¢âˆ’1)def=𝑀1.(2.9) Above inequality shows that for any 𝜀>0, there exists a 𝑘1>0, such that 𝑥(𝑘+1)≤𝑀1+𝜀,∀𝑘≥𝑘1.(2.10)Claim 2. We divide it into two cases to prove that limsup𝑘→+âˆžğ‘¦(𝑘)≤𝑀2.(2.11)Case (i)
There exists an 𝑙0≥𝑘1, such that 𝑦(𝑙0+1)≥𝑦(𝑙0). Then by the second equation of system (1.4), we have 𝑙−𝑑0+𝑓𝑙0𝑥𝑙0𝑚𝑙0𝑦𝑟𝑙0𝑙+𝑥0≥0.(2.12) Hence, 𝑙−𝑑0+𝑓𝑙0𝑥𝑙0𝑚𝑙0𝑦𝑟𝑙0≥0,(2.13) therefore, 𝑦𝑟𝑙0≤𝑙0𝑥𝑙0𝑚𝑙0𝑑𝑙0≤𝑓𝑢𝑀1+𝜀𝑚𝑙𝑑𝑙,(2.14) and so, 𝑦𝑙0≤𝑓𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟.(2.15) It follows that 𝑦𝑙0𝑙+1=𝑦0𝑙exp−𝑑0+𝑓𝑙0𝑥𝑙0𝑚𝑙0𝑦𝑟𝑙0𝑙+𝑥0≤𝑓𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟exp−𝑑𝑙+𝑓𝑢def=𝑀2𝜀.(2.16) We claim that 𝑦(𝑘)≤𝑀2𝜀∀𝑘≥𝑙0.(2.17) By a way of contradiction, assume that there exists a 𝑝0>𝑙0 such that 𝑦(𝑝0)>𝑀2𝜀. Then 𝑝0≥𝑙0+2. Let 𝑦(̃𝑝0)≥𝑙0+2 be the smallest integer such that 𝑦(̃𝑝0)≥𝑀2𝜀. Then 𝑦(̃𝑝0)>𝑦(̃𝑝0−1). The above argument produces that 𝑦(̃𝑝0)≤𝑀2𝜀, a contradiction. This prove the claim.
Case (ii)
We assume that 𝑦(𝑘+1)<𝑦(𝑘) for all 𝐾≥𝐾1. Since 𝑦(𝑘) is nonincreasing and has a lower bound 0, we know that lim𝑘→+âˆžğ‘¦(𝑘) exists, denoted by 𝑦, then lim𝑘→+âˆžğ‘¦(𝑘)=𝑦.(2.18) We claim that 𝑓𝑦≤𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟.(2.19) By a way of contradiction, assume that 𝑦>{𝑓𝑢(𝑀1+𝜀)/𝑚𝑙𝑑𝑙}1/𝑟. Taking limit in the second equation in system (1.4) gives lim𝑘→+âˆžî‚»âˆ’ğ‘‘(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(𝑘)+𝑥(𝑘)=0,(2.20) which is a contradiction since for 𝐾>𝐾1−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(𝑘)+𝑥(𝑘)≤−𝑑𝑙+𝑓𝑢𝑀1+𝜀𝑚𝑙𝑦𝑟<0.(2.21) This prove the claim, then we have limsup𝑘→+âˆžğ‘¦(𝑘)=lim𝑘→+âˆžğ‘¦(𝑘)=𝑓𝑦≤𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟.(2.22) Combining Cases (i) and (ii), we see that limsup𝑘→+âˆžğ‘¦(𝑘)≤𝑀2𝜀.(2.23) Let 𝜀→0, we have limsup𝑘→+âˆžî‚»ğ‘“ğ‘¦(𝑘)≤𝑢𝑀1𝑚𝑙𝑑𝑙1/𝑟exp−𝑑𝑙+𝑓𝑢=𝑀2.(2.24)
Claim 3 (liminfğ‘˜â†’âˆžğ‘¥(𝑘)≥𝑚1). Conditions (𝐻1) imply that for enough small positive constant 𝜀, we have ğ‘Žğ‘™âˆ’ğ‘ğ‘¢(𝑀2+𝜀)1−𝑟𝑚𝑙>0.(2.25) For above 𝜀, it follows form Claims 1 and 2 that there exists a 𝑘2 such that for all 𝑘>𝑘2𝑥(𝑘)≤𝑀1+𝜀,𝑦(𝑘)≤𝑀2+𝜀.(2.26) From the first equation of (1.4), we have îƒ¯ğ‘Žğ‘¥(𝑘+1)≥𝑥(𝑘)exp𝑙−𝑐𝑢(𝑀2+𝜀)1−𝑟𝑚𝑙−𝑏𝑢𝑥(𝑘).(2.27) By applying Lemma 2.3 to above inequality, we have liminf𝑘→+âˆžğ‘Žğ‘¥(𝑘)≥𝑙−𝑐𝑢(𝑀2+𝜀)1−𝑟/ğ‘šğ‘™î€¸ğ‘ğ‘¢îƒ¯ğ‘Žexp𝑙−𝑐𝑢(𝑀2+𝜀)1−𝑟𝑚𝑙−𝑏𝑢𝑀1+𝜀.(2.28) Setting 𝜀→0 in (2.28) leads to liminf𝑘→+âˆžğ‘Žğ‘¥(𝑘)≥𝑙−𝑐𝑢𝑀21−𝑟/ğ‘šğ‘™î€¸ğ‘ğ‘¢îƒ¯ğ‘Žexp𝑙−𝑐𝑢𝑀21−𝑟𝑚𝑙−𝑏𝑢𝑀1def=𝑚1.(2.29) This ends the proof of Claim 3.Claim 4. For any small positive constant 𝜀<𝑚1/2, from Claims 1–3, it follows that there exists a 𝑘3>𝑘2 such that for all 𝑘>𝑘3𝑥(𝑘)≥𝑚1−𝜀,𝑥(𝑘)≤𝑀1+𝜀,𝑦(𝑘)≤𝑀2+𝜀.(2.30) We present two cases to prove that liminf𝑘→+âˆžğ‘¦(𝑘)≥𝑚2(2.31)Case (i)
There exists an 𝑛0≥𝑘3 such that 𝑦(𝑛0+1)≤𝑦(𝑛0), then 𝑛−𝑑0+𝑓𝑛0𝑥𝑛0𝑚𝑛0𝑦𝑟𝑛0𝑛+𝑥0≤0.(2.32) Hence 𝑦𝑛0≥(𝑓𝑙−𝑑𝑢)(𝑚1−𝜀)𝑚𝑢𝑑𝑢1/𝑟def=𝑐1𝜀,(2.33) and so, 𝑦𝑛0≥+1(𝑓𝑙−𝑑𝑢)(𝑚1−𝜀)𝑚𝑢𝑑𝑢1/𝑟exp−𝑑𝑢+𝑓𝑙𝑚1−𝜀𝑚𝑢(𝑀2+𝜀)𝑟+𝑚1−𝜀def=𝑐2𝜀.(2.34) Set 𝑚2𝜀𝑐=min1𝜀,𝑐2𝜀.(2.35) We claim that 𝑦(𝑘)≥𝑚2𝜀 for 𝑘≥𝑛0. By a way of contradiction, assume that there exists a ğ‘ž0≥𝑛0, such that 𝑦(ğ‘ž0)<𝑚2𝜀. Then ğ‘ž0≥𝑛0+2. Let Ìƒğ‘ž0≥𝑛0+2 be the smallest integer such that 𝑦(Ìƒğ‘ž0)<𝑚2𝜀. Then 𝑦(Ìƒğ‘ž0)<𝑦(Ìƒğ‘ž0−1), which implies that 𝑦(ğ‘ž0)≤𝑚2𝜀, a contradiction, this proves the claim.
Case (ii)
We assume that 𝑦(𝑘+1)>𝑦(𝑘) for all 𝑘>𝑘3. According to (2.30), lim𝑘→+âˆžğ‘¦(𝑘) exists, denoted by 𝑦, then lim𝑘→+âˆžğ‘¦(𝑘)=𝑦.(2.36) We claim that 𝑦≥𝑚2𝜀.(2.37) By the way of contradiction, assume that 𝑦<𝑚2𝜀. Taking limit in the second equation in system (1.4) gives lim𝑘→+âˆžî‚»âˆ’ğ‘‘(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(𝑘)+𝑥(𝑘)=0,(2.38) which is a contradiction since for 𝑘>𝑘3,−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(𝑘)+𝑥(𝑘)≥−𝑑𝑢+𝑓𝑙𝑚1−𝜀𝑚𝑢𝑦𝑟+𝑚1−𝜀>0.(2.39) The above analysis show that liminf𝑘→+âˆžğ‘¦(𝑘)≥𝑚2𝜀.(2.40) Letting 𝜀→0, we have liminf𝑘→+âˆžğ‘¦(𝑘)≥𝑚2,(2.41) where 𝑚2=min(𝑓𝑙−𝑑𝑢)𝑚1𝑚𝑢𝑑𝑢1/𝑟,(𝑓𝑙−𝑑𝑢)𝑚1𝑚𝑢𝑑𝑢1/𝑟exp−𝑑𝑢+𝑓𝑙𝑚1𝑚𝑢𝑀𝑟2+𝑚1.(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,2𝑀1−𝑏𝑢𝑐−𝛼𝑢𝑀21−(𝑟/2)4𝑚𝑙𝑚2𝑓−𝛽𝑢𝑀11/24𝑚1𝑚2𝑟/2(𝐻>𝛿,3)𝑓𝛽min𝑙𝑚𝑙𝑚1𝑟(𝑚𝑢𝑀𝑟2+𝑀1)2𝑀21−𝑟,2𝑀2−𝑓𝑢𝑀11/2𝑟4𝑚2𝑚11/2𝑐−𝛼𝑢𝑀11/24𝑚𝑙𝑚𝑟2𝑚11/2𝑐−𝛼𝑢𝑀𝑟2(1−𝑟)4𝑚1𝑚𝑟2𝐻>𝛿.(4) Then system (1.4) with initial condition (1.6) is globally attractive, that is, for any two positive solutions (𝑥1(𝑘),𝑦1(𝑘)) and (𝑥2(𝑘),𝑦2(𝑘)) of system (1.4), one has lim𝑘→+∞||𝑥1(𝑘)−𝑥2||(𝑘)=0,lim𝑘→+∞||𝑦1(𝑘)−𝑦2||(𝑘)=0.(3.1)

Proof. From conditions (𝐻3) and (𝐻4), there exists an enough small positive constant 𝜀<min{𝑚1/2,𝑚2/2} such that 𝑏𝛼min𝑙,2𝑀1+𝜀−𝑏𝑢𝑐−𝛼𝑢(𝑀2+𝜀)1−(𝑟/2)4𝑚𝑙𝑚2𝑓−𝜀−𝛽𝑢(𝑀1+𝜀)1/24𝑚1−𝜀(𝑚2−𝜀)𝑟/2𝑓>𝛿,𝛽min𝑙𝑚𝑙𝑚1𝑟−𝜀[𝑚𝑢(𝑀2+𝜀)𝑟+(𝑀1+𝜀)]2(𝑀2+𝜀)1−𝑟,2𝑀2−𝑓+𝜀𝑢(𝑀1+𝜀)1/2𝑟4𝑚2−𝜀(𝑚1−𝜀)1/2𝑐−𝛼𝑢(𝑀1+𝜀)1/24𝑚𝑙(𝑚2−𝜀)𝑟(𝑚1−𝜀)1/2𝑐−𝛼𝑢(𝑀2+𝜀)𝑟(1−𝑟)4𝑚1−𝜀(𝑚2−𝜀)𝑟>𝛿.(3.2) Since (𝐻1) and (𝐻2) hold, for any positive solutions (𝑥1(𝑘),𝑦1(𝑘)) and (𝑥2(𝑘),𝑦2(𝑘)) of system (1.4), it follows from Theorem 2.4 that 𝑚1≤liminf𝑘→+âˆžğ‘¥ğ‘–(𝑘)≤limsup𝑘→+âˆžğ‘¥ğ‘–(𝑘)≤𝑀1,𝑚2≤liminf𝑘→+âˆžğ‘¦ğ‘–(𝑘)≤limsup𝑘→+âˆžğ‘¦ğ‘–(𝑘)≤𝑀2,𝑖=1,2.(3.3) For above 𝜀 and (3.3), there exists a 𝑘4>0 such that for all 𝑘>𝑘4, 𝑚1−𝜀≤𝑥𝑖(𝑘)≤𝑀1+𝜀,𝑚2−𝜀≤𝑥𝑖(𝑘)≤𝑀2+𝜀,𝑖=1,2.(3.4) Let 𝑉1||(𝑘)=ln𝑥1(𝑘)−ln𝑥2||(𝑘).(3.5) Then from the first equation of system (1.3), we have 𝑉1||(𝑘+1)=ln𝑥1(𝑘+1)−ln𝑥2||≤||(𝑘+1)ln𝑥1(𝑘)−ln𝑥2𝑥(𝑘)−𝑏(𝑘)1(𝑘)−𝑥2||||||𝑦(𝑘)+𝑐(𝑘)1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1−𝑦(𝑘)2(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2||||.(𝑘)(3.6) Using the Mean Value Theorem, we get 𝑥1(𝑘)−𝑥2(𝑘)=expln𝑥1(𝑘)−expln𝑥2(𝑘)=𝜉1(𝑘)ln𝑥1(𝑘)−ln𝑥2,𝑦(𝑘)11−𝑟(𝑘)−𝑦21−𝑟(𝑘)=(1−𝑟)𝜉2−𝑟𝑦(𝑘)1(𝑘)−𝑦2,(𝑘)(3.7) where 𝜉1(𝑘) lies between 𝑥1(𝑘) and 𝑥2(𝑘), 𝜉2(𝑘) lies between 𝑦1(𝑘) and 𝑦2(𝑘).
It follows from (3.6), (3.7) that 𝑉1||(𝑘+1)≤ln𝑥1(𝑘)−ln𝑥2||−1(𝑘)𝜉1−||||1(𝑘)𝜉1||||||𝑥(𝑘)−𝑏(𝑘)1(𝑘)−𝑥2||+||||(𝑘)𝑐(𝑘)𝑦1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1𝑚(𝑘)(𝑘)𝑦𝑟2(𝑘)+𝑥2||||||𝑥(𝑘)1(𝑘)−𝑥2||+||||(𝑘)𝑐(𝑘)𝑥1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2(||||||𝑦𝑘)1(𝑘)−𝑦2||+||||(𝑘)𝑐(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)𝑦𝑟2(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2(𝑘)1−𝑟𝜉𝑟2(||||||𝑦𝑘)1(𝑘)−𝑦2||.(𝑘)(3.8) And so, for 𝑘>𝑘4Δ𝑉1𝑏≤−min𝑙,2𝑀1+𝜀−𝑏𝑢||𝑥1(𝑘)−𝑥2||+𝑐(𝑘)𝑢(𝑀2+𝜀)1−(𝑟/2)4𝑚𝑙(𝑚2−𝜀)𝑟/2𝑚1||𝑥−𝜀1(𝑘)−𝑥2(||+𝑐𝑘)𝑢(𝑀1+𝜀)1/24𝑚𝑙(𝑚2−𝜀)𝑟(𝑚1−𝜀)1/2||𝑦1(𝑘)−𝑦2||+𝑐(𝑘)𝑢(𝑀2+𝜀)𝑟(1−𝑟)4𝑚1−𝜀(𝑚2−𝜀)𝑟||𝑦1(𝑘)−𝑦2||.(𝑘)(3.9) Let 𝑉2||(𝑘)=ln𝑦1(𝑘)−ln𝑦2||(𝑘).(3.10) Then from the second equation of system (1.4), we have 𝑉2||(𝑘+1)=ln𝑦1(𝑘+1)−ln𝑦2||=||||(𝑘+1)ln𝑦1(𝑘)−ln𝑦2𝑥(𝑘)+𝑓(𝑘)1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1−𝑥(𝑘)2(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2||||≤||||(𝑘)ln𝑦1(𝑘)−ln𝑦2(𝑘)−𝑓(𝑘)𝑚(𝑘)𝑥1𝑦(𝑘)𝑟1(𝑘)−𝑦𝑟2(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2(||||+||||𝑘)𝑓(𝑘)𝑚(𝑘)𝑦𝑟1𝑥(𝑘)1(𝑘)−𝑥2(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2||||.(𝑘)(3.11) Using the Mean Value Theorem, we get 𝑦1(𝑘)−𝑦2(𝑘)=expln𝑦1(𝑘)−expln𝑦2(𝑘)=𝜉3(𝑘)ln𝑦1(𝑘)−ln𝑦2,𝑦(𝑛)𝑟1(𝑘)−𝑦𝑟2(𝑘)=𝑟𝜉4𝑟−1𝑦(𝑘)1(𝑘)−𝑦2,(𝑘)(3.12) where 𝜉3(𝑘),𝜉4(𝑘) lie between 𝑦1(𝑘) and 𝑦2(𝑘), respectively. Then, it follows from (3.11), (3.12) that for 𝑘>𝑘4,Δ𝑉21≤−𝜉3−||||1(𝑘)𝜉3−(𝑘)𝑓(𝑘)𝑚(𝑘)𝑥1(𝑘)𝑟𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2𝜉(𝑘)41−𝑟||||×||𝑦(𝑘)1(𝑘)−𝑦2||+(𝑘)𝑓(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2||𝑥(𝑘)1(𝑘)−𝑥2||𝑓(𝑘)≤−min𝑙𝑚𝑙𝑚1𝑟−𝜀[𝑚𝑢(𝑀2+𝜀)𝑟+(𝑀1+𝜀)]2(𝑀2+𝜀)1−𝑟,2𝑀2−𝑓+𝜀𝑢(𝑀1+𝜀)1/2𝑟4𝑚2−𝜀(𝑚1−𝜀)1/2×||𝑦1(𝑘)−𝑦2||+𝑓(𝑘)𝑢(𝑀1+𝜀)𝑟/24𝑚1−𝜀(𝑚2−𝜀)𝑟/2||𝑥1(𝑘)−𝑥2||.(𝑘)(3.13) Now we define a Lyapunov function as follows: 𝑉(𝑘)=𝛼𝑉1(𝑘)+𝛽𝑉2(𝑘).(3.14) Calculating the difference of 𝑉 along the solution of system (1.4), for 𝑘>𝑘4, it follows from (3.9) and (3.13) that 𝑏Δ𝑉≤−𝛼min𝑙,2𝑀1+𝜀−𝑏𝑢𝑐−𝛼𝑢(𝑀2+𝜀)1−(𝑟/2)4𝑚𝑙(𝑚2−𝜀)𝑟/2𝑚1𝑓−𝜀−𝛽𝑢(𝑀1+𝜀)𝑟/24𝑚1−𝜀(𝑚2−𝜀)𝑟/2×||𝑥1(𝑘)−𝑥2||−𝑓(𝑘)𝛽min𝑙𝑚𝑙𝑚1𝑟−𝜀[𝑚𝑢(𝑀2+𝜀)𝑟+(𝑀1+𝜀)]2(𝑀2+𝜀)1−𝑟,2𝑀2−𝑓+𝜀𝑢(𝑀1+𝜀)1/2𝑟4𝑚2−𝜀(𝑚1−𝜀)1/2𝑐−𝛼𝑢(𝑀1+𝜀)1/24𝑚𝑙(𝑚2−𝜀)𝑟(𝑚1−𝜀)1/2𝑐−𝛼𝑢(𝑀2+𝜀)𝑟(1−𝑟)4𝑚1−𝜀(𝑚2−𝜀)𝑟×||𝑦1(𝑘)−𝑦2(||||𝑥𝑘)≤−𝛿1(𝑘)−𝑥2(||+||𝑦𝑘)1(𝑘)−𝑦2(||.𝑘)(3.15) Summating both sides of the above inequalities from 𝑘4 to 𝑘, we have 𝑘𝑝=𝑘40𝑥0200𝑑(𝑉(𝑝+1)−𝑣(𝑝))≤−𝛿𝑘𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)−𝑥2||+||𝑦(𝑝)1(𝑝)−𝑦2||(𝑝),(3.16) which implies 𝑉(𝑘+1)+𝛿𝑘𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)−𝑥2||+||𝑦(𝑝)1(𝑝)−𝑦2||𝑘(𝑝)≤𝑉4.(3.17) It follows that 𝑘𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)−𝑥2||+||𝑦(𝑝)1(𝑝)−𝑦2||≤𝑉𝑘(𝑝)4𝛿.(3.18) Using the fundamental theorem of positive series, there exists small enough positive constant 𝜀>0 such that +âˆžî“ğ‘=𝑘4||𝑥0𝑥0200𝑑1(𝑝)−𝑥2||+||𝑦(𝑝)1(𝑝)−𝑦2||≤𝑉𝑘(𝑝)4𝛿,(3.19) which implies that lim𝑘→+∞||𝑥1(𝑘)−𝑥2||+||𝑦(𝑘)1(𝑘)−𝑦2||(𝑘)=0,(3.20) that is lim𝑘→+∞||𝑥1(𝑘)−𝑥2||(𝑘)=0,lim𝑘→+∞||𝑦1(𝑘)−𝑦2||(𝑘)=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 −𝑑𝑙+𝑓𝑢𝐻<0.(5) Then, the species 𝑦 will be driven to extinction, and the species 𝑥 is permanent, that is, for any positive solution (𝑥(𝑘),𝑦(𝑘)) of system (1.4), lim𝑘→+âˆžğ‘šğ‘¦(𝑘)=0,∗≤liminf𝑘→+âˆžğ‘¥(𝑘)≤limsup𝑘→+âˆžğ‘¥(𝑘)≤𝑀1,(4.1) where 𝑚∗=ğ‘Žğ‘™ğ‘ğ‘¢î€½ğ‘Žexp𝑙−𝑏𝑢𝑀1,𝑀1=1𝑏𝑙exp(ğ‘Žğ‘¢âˆ’1).(4.2)

Proof. For condition (𝐻5), there exists small enough positive 𝛾>0, such that −𝑑𝑙+𝑓𝑢<−𝛾<0(4.3) for all 𝑘∈𝑁, from (4.3) and the second equation of the system (1.4), one can easily obtain that 𝑦(𝑘+1)=𝑦(𝑘)exp−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦(𝑘)+𝑥(𝑘)<𝑦(𝑘)exp−𝑑𝑙+𝑓𝑢<𝑦(𝑘)exp{−𝛾}.(4.4) Therefore, 𝑦(𝑘+1)<𝑦(0)exp{−𝑘𝛾},(4.5) which yields lim𝑘→+âˆžğ‘¦(𝑘)=0.(4.6) From the proof of Theorem 3.1, we have limsup𝑘→+âˆžğ‘¥(𝑘)≤𝑀1.(4.7) For enough small positive constant 𝜀>0, ğ‘Žğ‘™âˆ’ğ‘ğ‘¢ğœ€1−𝑟𝑚𝑙>0.(4.8) For above 𝜀, from (2.9) and (4.6), there exists a 𝑘5>0 such that for all 𝑘>𝑘5, 𝑥(𝑘)<𝑀1+𝜀,𝑦(𝑘)<𝜀.(4.9) From the first equation of (1.4), we have î‚»ğ‘Žğ‘¥(𝑘+1)≥𝑥(𝑘)exp𝑙−𝑐𝑢𝜀1−𝑟𝑚𝑙−𝑏𝑢𝑥(𝑘).(4.10) By Lemma 2.3, we have liminf𝑘→+âˆžğ‘¥ğ‘Ž(𝑘)≥𝑙−𝑐𝑢𝜀1−𝑟/ğ‘šğ‘™î€¸ğ‘ğ‘¢î‚»ğ‘Žexp𝑙−𝑐𝑢𝜀1−𝑟𝑚𝑙−𝑏𝑢𝑀1+𝜀.(4.11) Setting 𝜀→0 in (4.11) leads to liminf𝑘→+âˆžğ‘Žğ‘¥(𝑘)â‰¥ğ‘™ğ‘ğ‘¢î€½ğ‘Žexp𝑙−𝑏𝑢𝑀1def=𝑚∗.(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: 𝑥(𝑘+1)=𝑥(𝑘)exp1.41+0.12cos(𝑘)−1.78𝑥(𝑘)−0.33𝑦(𝑘)2.16𝑦1/2,(𝑘)+𝑥(𝑘)𝑦(𝑘+1)=𝑦(𝑘)exp−0.62+1.79𝑥(𝑘)2.16𝑦1/2.(𝑘)+𝑥(𝑘)(5.1)
One could easily see that there exist positive constants 𝛼=0.01,𝛽=0.05,𝛿=0.001 such that ğ‘Žğ‘™âˆ’ğ‘ğ‘¢ğ‘€21−𝑟𝑚𝑙𝑓≈2.3281>0,𝑙>𝑑𝑢𝑏≈1.1700>0,𝛼min𝑙,2𝑀1−𝑏𝑢𝑐−𝛼𝑢𝑀21−(𝑟/2)4𝑚𝑙𝑚2𝑓−𝛽𝑢𝑀11/24𝑚1𝑚2𝑟/2𝑓≈0.0011>𝛿,𝛽min𝑙𝑚𝑙𝑚1𝑟(𝑚𝑢𝑀𝑟2+𝑀1)2𝑀21−𝑟,2𝑀2−𝑓𝑢𝑀11/2𝑟4𝑚2𝑚11/2𝑐−𝛼𝑢𝑀11/24𝑚𝑙𝑚𝑟2𝑚11/2𝑐−𝛼𝑢𝑀𝑟2(1−𝑟)4𝑚1𝑚𝑟2≈0.0107>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 𝑟∈(0,1). If 𝑟=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 𝑟=1. If ğ‘Žğ‘™âˆ’ğ‘ğ‘¢/𝑚𝑙>0 holds, then the prey species 𝑥 is permanence. If 𝑟=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 𝑟=0.