Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

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

Runxin Wu, Lin Li, "Permanence and Global Attractivity of Discrete Predator-Prey System with Hassell-Varley Type Functional Response", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 323065, 17 pages, 2009. 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]: 𝑑𝑥𝑥𝑑𝑡=𝑎𝑥1𝑘𝑐𝑥𝑦,𝑚+𝑥𝑑𝑦𝑑𝑡=𝑦𝑑+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 𝑎, 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 [38] 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: 𝑑𝑥𝑥𝑑𝑡=𝑎𝑥1𝑘𝑐𝑥𝑦,𝑚𝑦+𝑥𝑑𝑦𝑑𝑡=𝑦𝑑+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: 𝑑𝑥𝑥𝑑𝑡=𝑥𝑎𝑘𝑐𝑥𝑦𝑚𝑦𝑟,+𝑥𝑑𝑦𝑑𝑡=𝑦𝑑+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 [1224]. This motivated us to propose and study the discrete analogous of predator-prey system (1.3): 𝑥(𝑘+1)=𝑥(𝑘)exp𝑎(𝑘)𝑏(𝑘)𝑥(𝑘)𝑐(𝑘)𝑦(𝑘)𝑚(𝑘)𝑦𝑟,(𝑘)+𝑥(𝑘)𝑦(𝑘+1)=𝑦(𝑘)exp𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(,𝑘)+𝑥(𝑘)(1.4) where 𝑟(0,1); {𝑎(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=sup𝑘𝑁𝑔(𝑘),𝑔𝑙=inf𝑘𝑁𝑔(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 𝑚liminf𝑘+{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{𝑎(𝑘)𝑏(𝑘)𝑥(𝑘)}(2.2) for 𝑘𝑁, where 𝑎(𝑘) and 𝑏(𝑘) are all nonnegative sequences bounded above and below by positive constants. Then limsup𝑘+1𝑥(𝑘)𝑏𝑙exp(𝑎𝑢1).(2.3)

Lemma 2.3 (see [23]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘+1)𝑥(𝑘)exp{𝑎(𝑘)𝑏(𝑘)𝑥(𝑘)},𝑘𝑁0,(2.4)limsup𝑘+𝑥(𝑘)𝑥 and 𝑥(𝑁0)>0, where 𝑎(𝑘) and 𝑏(𝑘) are all nonnegative sequences bounded above and below by positive constants and 𝑁0𝑁. Then liminf𝑘+𝑥𝑎(𝑘)𝑙𝑎exp𝑙𝑏𝑢𝑥𝑏𝑢.(2.5)

Theorem 2.4. Assume that 𝑎𝑙𝑐𝑢𝑀21𝑟𝑚𝑙(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 𝑚1liminf𝑘+𝑥(𝑘)limsup𝑘+𝑥(𝑘)𝑀1,𝑚2liminf𝑘+𝑥(𝑘)limsup𝑘+𝑦(𝑘)𝑀2,(2.6) where 𝑚1=𝑎𝑙𝑐𝑢𝑀21𝑟/𝑚𝑙𝑏𝑢𝑎exp𝑙𝑐𝑢𝑀21𝑟𝑚𝑙𝑏𝑢𝑀1,𝑚2𝑓=min𝑙𝑑𝑢𝑚1𝑚𝑢𝑑𝑢1/𝑟,𝑓𝑙𝑑𝑢𝑚1𝑚𝑢𝑑𝑢1/𝑟exp𝑑𝑢+𝑓𝑙𝑚1𝑚𝑢𝑀𝑟2+𝑚1,𝑀1=1𝑏𝑙exp(𝑎𝑢𝑀1),2=𝑓𝑢𝑀1𝑚𝑙𝑑𝑙1/𝑟exp𝑑𝑙+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{𝑎(𝑘)𝑏(𝑘)𝑥(𝑘)}.(2.8) By Lemma 2.2, we have limsup𝑘+1𝑥(𝑘)𝑏𝑙exp(𝑎𝑢1)def=𝑀1.(2.9) Above inequality shows that for any 𝜀>0, there exists a 𝑘1>0, such that 𝑥(𝑘+1)𝑀1+𝜀,𝑘𝑘1.(2.10)Claim 2. We divide it into two cases to prove that limsup𝑘+𝑦(𝑘)𝑀2.(2.11)Case (i)
There exists an 𝑙0𝑘1, such that 𝑦(𝑙0+1)𝑦(𝑙0). Then by the second equation of system (1.4), we have 𝑙𝑑0+𝑓𝑙0𝑥𝑙0𝑚𝑙0𝑦𝑟𝑙0𝑙+𝑥00.(2.12) Hence, 𝑙𝑑0+𝑓𝑙0𝑥𝑙0𝑚𝑙0𝑦𝑟𝑙00,(2.13) therefore, 𝑦𝑟𝑙0𝑙0𝑥𝑙0𝑚𝑙0𝑑𝑙0𝑓𝑢𝑀1+𝜀𝑚𝑙𝑑𝑙,(2.14) and so, 𝑦𝑙0𝑓𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟.(2.15) It follows that 𝑦𝑙0𝑙+1=𝑦0𝑙exp𝑑0+𝑓𝑙0𝑥𝑙0𝑚𝑙0𝑦𝑟𝑙0𝑙+𝑥0𝑓𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟exp𝑑𝑙+𝑓𝑢def=𝑀2𝜀.(2.16) We claim that 𝑦(𝑘)𝑀2𝜀𝑘𝑙0.(2.17) By a way of contradiction, assume that there exists a 𝑝0>𝑙0 such that 𝑦(𝑝0)>𝑀2𝜀. Then 𝑝0𝑙0+2. Let 𝑦(̃𝑝0)𝑙0+2 be the smallest integer such that 𝑦(̃𝑝0)𝑀2𝜀. Then 𝑦(̃𝑝0)>𝑦(̃𝑝01). 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 𝐾𝐾1. Since 𝑦(𝑘) is nonincreasing and has a lower bound 0, we know that lim𝑘+𝑦(𝑘) exists, denoted by 𝑦, then lim𝑘+𝑦(𝑘)=𝑦.(2.18) We claim that 𝑓𝑦𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/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+𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(𝑘)+𝑥(𝑘)=0,(2.20) which is a contradiction since for 𝐾>𝐾1𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦𝑟(𝑘)+𝑥(𝑘)𝑑𝑙+𝑓𝑢𝑀1+𝜀𝑚𝑙𝑦𝑟<0.(2.21) This prove the claim, then we have limsup𝑘+𝑦(𝑘)=lim𝑘+𝑦(𝑘)=𝑓𝑦𝑢(𝑀1+𝜀)𝑚𝑙𝑑𝑙1/𝑟.(2.22) Combining Cases (i) and (ii), we see that limsup𝑘+𝑦(𝑘)𝑀2𝜀.(2.23) Let 𝜀0, we have limsup𝑘+𝑓𝑦(𝑘)𝑢𝑀1𝑚𝑙𝑑𝑙1/𝑟exp𝑑𝑙+𝑓𝑢=𝑀2.(2.24)
Claim 3 (liminf𝑘𝑥(𝑘)𝑚1). Conditions (𝐻1) imply that for enough small positive constant 𝜀, we have 𝑎𝑙𝑐𝑢(𝑀2+𝜀)1𝑟𝑚𝑙>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 𝑎𝑥(𝑘+1)𝑥(𝑘)exp𝑙𝑐𝑢(𝑀2+𝜀)1𝑟𝑚𝑙𝑏𝑢𝑥(𝑘).(2.27) By applying Lemma 2.3 to above inequality, we have liminf𝑘+𝑎𝑥(𝑘)𝑙𝑐𝑢(𝑀2+𝜀)1𝑟/𝑚𝑙𝑏𝑢𝑎exp𝑙𝑐𝑢(𝑀2+𝜀)1𝑟𝑚𝑙𝑏𝑢𝑀1+𝜀.(2.28) Setting 𝜀0 in (2.28) leads to liminf𝑘+𝑎𝑥(𝑘)𝑙𝑐𝑢𝑀21𝑟/𝑚𝑙𝑏𝑢𝑎exp𝑙𝑐𝑢𝑀21𝑟𝑚𝑙𝑏𝑢𝑀1def=𝑚1.(2.29) This ends the proof of Claim 3.Claim 4. For any small positive constant 𝜀<𝑚1/2, from Claims 13, it follows that there exists a 𝑘3>𝑘2 such that for all 𝑘>𝑘3𝑥(𝑘)𝑚1𝜀,𝑥(𝑘)𝑀1+𝜀,𝑦(𝑘)𝑀2+𝜀.(2.30) We present two cases to prove that liminf𝑘+𝑦(𝑘)𝑚2(2.31)Case (i)
There exists an 𝑛0𝑘3 such that 𝑦(𝑛0+1)𝑦(𝑛0), then 𝑛𝑑0+𝑓𝑛0𝑥𝑛0𝑚𝑛0𝑦𝑟𝑛0𝑛+𝑥00.(2.32) Hence 𝑦𝑛0(𝑓𝑙𝑑𝑢)(𝑚1𝜀)𝑚𝑢𝑑𝑢1/𝑟def=𝑐1𝜀,(2.33) and so, 𝑦𝑛0+1(𝑓𝑙𝑑𝑢)(𝑚1𝜀)𝑚𝑢𝑑𝑢1/𝑟exp𝑑𝑢+𝑓𝑙𝑚1𝜀𝑚𝑢(𝑀2+𝜀)𝑟+𝑚1𝜀def=𝑐2𝜀.(2.34) Set 𝑚2𝜀𝑐=min1𝜀,𝑐2𝜀.(2.35) We claim that 𝑦(𝑘)𝑚2𝜀 for 𝑘𝑛0. By a way of contradiction, assume that there exists a 𝑞0𝑛0, such that 𝑦(𝑞0)<𝑚2𝜀. Then 𝑞0𝑛0+2. Let ̃𝑞0𝑛0+2 be the smallest integer such that 𝑦(̃𝑞0)<𝑚2𝜀. Then 𝑦(̃𝑞0)<𝑦(̃𝑞01), which implies that 𝑦(𝑞0)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+𝑦(𝑘) exists, denoted by 𝑦, then lim𝑘+𝑦(𝑘)=𝑦.(2.36) We claim that 𝑦𝑚2𝜀.(2.37) By the way of contradiction, assume that 𝑦<𝑚2𝜀. Taking limit in the second equation in system (1.4) gives lim𝑘+𝑑(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+𝑓𝑙𝑚1𝜀𝑚𝑢𝑦𝑟+𝑚1𝜀>0.(2.39) The above analysis show that liminf𝑘+𝑦(𝑘)𝑚2𝜀.(2.40) Letting 𝜀0, we have liminf𝑘+𝑦(𝑘)𝑚2,(2.41) where 𝑚2=min(𝑓𝑙𝑑𝑢)𝑚1𝑚𝑢𝑑𝑢1/𝑟,(𝑓𝑙𝑑𝑢)𝑚1𝑚𝑢𝑑𝑢1/𝑟exp𝑑𝑢+𝑓𝑙𝑚1𝑚𝑢𝑀𝑟2+𝑚1.(2.42) According to Claims 14, 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,2𝑀1𝑏𝑢𝑐𝛼𝑢𝑀21(𝑟/2)4𝑚𝑙𝑚2𝑓𝛽𝑢𝑀11/24𝑚1𝑚2𝑟/2(𝐻>𝛿,3)𝑓𝛽min𝑙𝑚𝑙𝑚1𝑟(𝑚𝑢𝑀𝑟2+𝑀1)2𝑀21𝑟,2𝑀2𝑓𝑢𝑀11/2𝑟4𝑚2𝑚11/2𝑐𝛼𝑢𝑀11/24𝑚𝑙𝑚𝑟2𝑚11/2𝑐𝛼𝑢𝑀𝑟2(1𝑟)4𝑚1𝑚𝑟2𝐻>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+𝜀𝑏𝑢𝑐𝛼𝑢(𝑀2+𝜀)1(𝑟/2)4𝑚𝑙𝑚2𝑓𝜀𝛽𝑢(𝑀1+𝜀)1/24𝑚1𝜀(𝑚2𝜀)𝑟/2𝑓>𝛿,𝛽min𝑙𝑚𝑙𝑚1𝑟𝜀[𝑚𝑢(𝑀2+𝜀)𝑟+(𝑀1+𝜀)]2(𝑀2+𝜀)1𝑟,2𝑀2𝑓+𝜀𝑢(𝑀1+𝜀)1/2𝑟4𝑚2𝜀(𝑚1𝜀)1/2𝑐𝛼𝑢(𝑀1+𝜀)1/24𝑚𝑙(𝑚2𝜀)𝑟(𝑚1𝜀)1/2𝑐𝛼𝑢(𝑀2+𝜀)𝑟(1𝑟)4𝑚1𝜀(𝑚2𝜀)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 𝑚1liminf𝑘+𝑥𝑖(𝑘)limsup𝑘+𝑥𝑖(𝑘)𝑀1,𝑚2liminf𝑘+𝑦𝑖(𝑘)limsup𝑘+𝑦𝑖(𝑘)𝑀2,𝑖=1,2.(3.3) For above 𝜀 and (3.3), there exists a 𝑘4>0 such that for all 𝑘>𝑘4, 𝑚1𝜀𝑥𝑖(𝑘)𝑀1+𝜀,𝑚2𝜀𝑥𝑖(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)ln𝑥2𝑥(𝑘)𝑏(𝑘)1(𝑘)𝑥2||||||𝑦(𝑘)+𝑐(𝑘)1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1𝑦(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)11𝑟(𝑘)𝑦21𝑟(𝑘)=(1𝑟)𝜉2𝑟𝑦(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)1𝑟𝜉𝑟2(||||||𝑦𝑘)1(𝑘)𝑦2||.(𝑘)(3.8) And so, for 𝑘>𝑘4Δ𝑉1𝑏min𝑙,2𝑀1+𝜀𝑏𝑢||𝑥1(𝑘)𝑥2||+𝑐(𝑘)𝑢(𝑀2+𝜀)1(𝑟/2)4𝑚𝑙(𝑚2𝜀)𝑟/2𝑚1||𝑥𝜀1(𝑘)𝑥2(||+𝑐𝑘)𝑢(𝑀1+𝜀)1/24𝑚𝑙(𝑚2𝜀)𝑟(𝑚1𝜀)1/2||𝑦1(𝑘)𝑦2||+𝑐(𝑘)𝑢(𝑀2+𝜀)𝑟(1𝑟)4𝑚1𝜀(𝑚2𝜀)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)ln𝑦2𝑥(𝑘)+𝑓(𝑘)1(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1𝑥(𝑘)2(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2||||||||(𝑘)ln𝑦1(𝑘)ln𝑦2(𝑘)𝑓(𝑘)𝑚(𝑘)𝑥1𝑦(𝑘)𝑟1(𝑘)𝑦𝑟2(𝑘)𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2(||||+||||𝑘)𝑓(𝑘)𝑚(𝑘)𝑦𝑟1𝑥(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)=𝑟𝜉4𝑟1𝑦(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,Δ𝑉21𝜉3||||1(𝑘)𝜉3(𝑘)𝑓(𝑘)𝑚(𝑘)𝑥1(𝑘)𝑟𝑚(𝑘)𝑦𝑟1(𝑘)+𝑥1(𝑘)𝑚(𝑘)𝑦𝑟2(𝑘)+𝑥2𝜉(𝑘)41𝑟||||×||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)min𝑙𝑚𝑙𝑚1𝑟𝜀[𝑚𝑢(𝑀2+𝜀)𝑟+(𝑀1+𝜀)]2(𝑀2+𝜀)1𝑟,2𝑀2𝑓+𝜀𝑢(𝑀1+𝜀)1/2𝑟4𝑚2𝜀(𝑚1𝜀)1/2×||𝑦1(𝑘)𝑦2||+𝑓(𝑘)𝑢(𝑀1+𝜀)𝑟/24𝑚1𝜀(𝑚2𝜀)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 𝑏Δ𝑉𝛼min𝑙,2𝑀1+𝜀𝑏𝑢𝑐𝛼𝑢(𝑀2+𝜀)1(𝑟/2)4𝑚𝑙(𝑚2𝜀)𝑟/2𝑚1𝑓𝜀𝛽𝑢(𝑀1+𝜀)𝑟/24𝑚1𝜀(𝑚2𝜀)𝑟/2×||𝑥1(𝑘)𝑥2||𝑓(𝑘)𝛽min𝑙𝑚𝑙𝑚1𝑟𝜀[𝑚𝑢(𝑀2+𝜀)𝑟+(𝑀1+𝜀)]2(𝑀2+𝜀)1𝑟,2𝑀2𝑓+𝜀𝑢(𝑀1+𝜀)1/2𝑟4𝑚2𝜀(𝑚1𝜀)1/2𝑐𝛼𝑢(𝑀1+𝜀)1/24𝑚𝑙(𝑚2𝜀)𝑟(𝑚1𝜀)1/2𝑐𝛼𝑢(𝑀2+𝜀)𝑟(1𝑟)4𝑚1𝜀(𝑚2𝜀)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 𝑘𝑝=𝑘40𝑥0200𝑑(𝑉(𝑝+1)𝑣(𝑝))𝛿𝑘𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||(𝑝),(3.16) which implies 𝑉(𝑘+1)+𝛿𝑘𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||𝑘(𝑝)𝑉4.(3.17) It follows that 𝑘𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||𝑉𝑘(𝑝)4𝛿.(3.18) Using the fundamental theorem of positive series, there exists small enough positive constant 𝜀>0 such that +𝑝=𝑘4||𝑥0𝑥0200𝑑1(𝑝)𝑥2||+||𝑦(𝑝)1(𝑝)𝑦2||𝑉𝑘(𝑝)4𝛿,(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+𝑓𝑢𝐻<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+𝑚𝑦(𝑘)=0,liminf𝑘+𝑥(𝑘)limsup𝑘+𝑥(𝑘)𝑀1,(4.1) where 𝑚=𝑎𝑙𝑏𝑢𝑎exp𝑙𝑏𝑢𝑀1,𝑀1=1𝑏𝑙exp(𝑎𝑢1).(4.2)

Proof. For condition (𝐻5), there exists small enough positive 𝛾>0, such that 𝑑𝑙+𝑓𝑢<𝛾<0(4.3) for all 𝑘𝑁, from (4.3) and the second equation of the system (1.4), one can easily obtain that 𝑦(𝑘+1)=𝑦(𝑘)exp𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦(𝑘)+𝑥(𝑘)<𝑦(𝑘)exp𝑑𝑙+𝑓𝑢<𝑦(𝑘)exp{𝛾}.(4.4) Therefore, 𝑦(𝑘+1)<𝑦(0)exp{𝑘𝛾},(4.5) which yields lim𝑘+𝑦(𝑘)=0.(4.6) From the proof of Theorem 3.1, we have limsup𝑘+𝑥(𝑘)𝑀1.(4.7) For enough small positive constant 𝜀>0, 𝑎𝑙𝑐𝑢𝜀1𝑟𝑚𝑙>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 𝑎𝑥(𝑘+1)𝑥(𝑘)exp𝑙𝑐𝑢𝜀1𝑟𝑚𝑙𝑏𝑢𝑥(𝑘).(4.10) By Lemma 2.3, we have liminf𝑘+𝑥𝑎(𝑘)𝑙𝑐𝑢𝜀1𝑟/𝑚𝑙𝑏𝑢𝑎exp𝑙𝑐𝑢𝜀1𝑟𝑚𝑙𝑏𝑢𝑀1+𝜀.(4.11) Setting 𝜀0 in (4.11) leads to liminf𝑘+𝑎𝑥(𝑘)𝑙𝑏𝑢𝑎exp𝑙𝑏𝑢𝑀1def=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)exp0.62+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 𝑎𝑙𝑐𝑢𝑀21𝑟𝑚𝑙𝑓2.3281>0,𝑙>𝑑𝑢𝑏1.1700>0,𝛼min𝑙,2𝑀1𝑏𝑢𝑐𝛼𝑢𝑀21(𝑟/2)4𝑚𝑙𝑚2𝑓𝛽𝑢𝑀11/24𝑚1𝑚2𝑟/2𝑓0.0011>𝛿,𝛽min𝑙𝑚𝑙𝑚1𝑟(𝑚𝑢𝑀𝑟2+𝑀1)2𝑀21𝑟,2𝑀2𝑓𝑢𝑀11/2𝑟4𝑚2𝑚11/2𝑐𝛼𝑢𝑀11/24𝑚𝑙𝑚𝑟2𝑚11/2𝑐𝛼𝑢𝑀𝑟2(1𝑟)4𝑚1𝑚𝑟20.0107>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 𝑎𝑙𝑐𝑢/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.

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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.

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