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

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]: 𝑑𝑥𝑑𝑑𝑦𝑡=𝑎𝑥1−𝑥𝑘−𝑚𝑐𝑥+𝑦𝑥,𝑥(𝑑0𝑡)>=0𝑦,−𝑑𝑦+(0𝑚)𝑓>+𝑥0𝑥,,(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 [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: 𝑑𝑥𝑑𝑑𝑦𝑡=𝑎𝑥1−𝑥𝑘−𝑚𝑦𝑐𝑥+𝑦𝑥,𝑥(𝑑0𝑡)>=𝑦0,−𝑑𝑦+(0𝑚)𝑦>𝑓+𝑥0.𝑥,(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: 𝑑𝑥𝑑𝑑𝑦𝑡=𝑥𝑎−𝑥𝑘−𝑚𝑦𝑐^𝑟𝑥+𝑦𝑥,𝑥(𝑑0𝑡)>=0𝑦,−𝑑𝑦+(0𝑚)𝑦>𝑓^𝑟𝑥0+.𝑥,𝑟∈(0,1),(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)exp𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑐(𝑘)𝑦(𝑘)𝑦(𝑘+1)=𝑦(𝑘)exp−𝑑(𝑘)+𝑓(𝑘)𝑥𝑚(𝑘()𝑘)𝑦^𝑟(𝑘)+𝑥(𝑘),𝑚(𝑘)𝑦^𝑟(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 𝑔^𝑢=sup_𝑘∈𝑁𝑔(𝑘),𝑔^𝑙=inf_𝑘∈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≤liminf_𝑘→+∞{𝑥(𝑘),𝑦(𝑘)}≤limsup_𝑘→+∞{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 𝑘∈𝑁, where 𝑎(𝑘) and 𝑏(𝑘) are all nonnegative sequences bounded above and below by positive constants. Then limsup_𝑘→+∞𝑥(𝑘)≤1𝑏^𝑙exp(𝑎^𝑢−1).(2.3)

Lemma 2.3 (see [23]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘+1)≥𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)},𝑘≥𝑁₀,(2.4)limsup_𝑘→+∞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 𝑁₀∈𝑁. Then liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙exp𝑎^𝑙−𝑏^𝑢𝑥^∗𝑏^𝑢.(2.5)

Theorem 2.4. Assume that 𝑎^𝑙−𝑐^𝑢𝑀^21−𝑟𝑚^𝑙𝑓_𝑙>0,(𝐻₁)>𝑑^𝑢(𝐻₂) hold, then system (1.4) is permanent, that is, for any positive solution {𝑥(𝑘),𝑦(𝑘)} of system (1.4), one has 𝑚𝑚₁₂≤liminf_𝑘→+∞𝑥(𝑘)≤limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁,≤liminf_𝑘→+∞𝑥(𝑘)≤limsup_𝑘→+∞𝑦(𝑘)≤𝑀₂,(2.6) where 𝑚₁=𝑎^𝑙−𝑐^𝑢𝑀^21−𝑟/𝑚^𝑙𝑏^𝑢exp𝑎^𝑙−𝑐^𝑢𝑀^21−𝑟𝑚^𝑙−𝑏^𝑢𝑀₁𝑚₂,=min⎧⎪⎨⎪⎩𝑓^𝑙−𝑑^𝑢𝑚₁𝑚^𝑢𝑑^𝑢^1/𝑟,𝑓^𝑙−𝑑^𝑢𝑚₁𝑚^𝑢𝑑^𝑢^1/𝑟exp−𝑑^𝑢+𝑓^𝑙𝑚₁𝑀₁𝑚^𝑢𝑀^𝑟2+𝑚₁⎫⎪⎬⎪⎭,=1𝑏^𝑙𝑀exp(𝑎^𝑢−1),₂=𝑓^𝑢𝑀₁𝑚^𝑙𝑑^𝑙^1/𝑟exp−𝑑^𝑙+𝑓^𝑢.(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 limsup_𝑘→+∞𝑥(𝑘)≤1𝑏^𝑙exp(𝑎^𝑢−1)=^def𝑀₁.(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 limsup_𝑘→+∞𝑦(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 −𝑑𝑙₀+𝑓𝑙⁰𝑥𝑙₀𝑚𝑙₀𝑦^𝑟𝑙₀+𝑥𝑙₀≥0.(2.12) Hence, −𝑑𝑙₀+𝑓𝑙⁰𝑥𝑙₀𝑚𝑙₀𝑦^𝑟𝑙₀≥0,(2.13) therefore, 𝑦^𝑟𝑙₀≤𝑙⁰𝑥𝑙₀𝑚𝑙₀𝑑𝑙₀≤𝑓^𝑢𝑀₁+𝜀𝑚^𝑙𝑑^𝑙,(2.14) and so, 𝑦𝑙₀≤𝑓^𝑢(𝑀₁+𝜀)𝑚^𝑙𝑑^𝑙^1/𝑟.(2.15) It follows that 𝑦𝑙₀+1=𝑦𝑙₀exp−𝑑𝑙₀+𝑓𝑙⁰𝑥𝑙₀≤𝑓_𝑢𝑚𝑙₀𝑦^𝑟𝑙₀+𝑥𝑙₀(𝑀₁+𝜀)𝑚^𝑙𝑑^𝑙^1/𝑟exp−𝑑^𝑙+𝑓^𝑢=^def𝑀_2𝜀.(2.16) We claim that 𝑦(𝑘)≤𝑀_2𝜀∀𝑘≥𝑙₀.(2.17) By a way of contradiction, assume that there exists a 𝑝₀>𝑙₀ such that 𝑦(𝑝₀)>𝑀_2𝜀. Then 𝑝₀≥𝑙₀+2. Let 𝑦(𝑝₀̃)≥𝑙₀+2 be the smallest integer such that 𝑦(𝑝₀̃)≥𝑀_2𝜀. Then 𝑦(𝑝₀̃)>𝑦(𝑝₀̃−1). The above argument produces that 𝑦(𝑝₀̃)≤𝑀_2𝜀, 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 lim_𝑘→+∞𝑦(𝑘) exists, denoted by 𝑦, then lim_𝑘→+∞𝑦(𝑘)=𝑦.(2.18) We claim that 𝑦≤𝑓^𝑢(𝑀₁+𝜀)𝑚^𝑙𝑑^𝑙^1/𝑟.(2.19) By a way of contradiction, assume that 𝑦>{𝑓^𝑢(𝑀₁+𝜀)/𝑚^𝑙𝑑^𝑙}^1/𝑟. Taking limit in the second equation in system (1.4) gives lim_𝑘→+∞−𝑑(𝑘)+𝑚(𝑘𝑓)𝑦(^𝑟𝑘()𝑘𝑥)(+𝑘)𝑥(𝑘)=0,(2.20) which is a contradiction since for 𝐾>𝐾₁−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦^𝑟(𝑘)+𝑥(𝑘)≤−𝑑^𝑙+𝑓^𝑢𝑀₁+𝜀𝑚^𝑙𝑦^𝑟<0.(2.21) This prove the claim, then we have limsup_𝑘→+∞𝑦(𝑘)=lim_𝑘→+∞𝑦(𝑘)=𝑦≤𝑓^𝑢(𝑀₁+𝜀)𝑚^𝑙𝑑^𝑙^1/𝑟.(2.22) Combining Cases (i) and (ii), we see that limsup_𝑘→+∞𝑦(𝑘)≤𝑀_2𝜀.(2.23) Let 𝜀→0, we have limsup_𝑘→+∞𝑦(𝑘)≤𝑓^𝑢𝑀₁𝑚^𝑙𝑑^𝑙^1/𝑟exp−𝑑^𝑙+𝑓^𝑢=𝑀₂.(2.24)Claim 3 (liminf_𝑘→∞𝑥(𝑘)≥𝑚₁). Conditions (𝐻₁) imply that for enough small positive constant 𝜀, we have 𝑎^𝑙−𝑐^𝑢(𝑀₂+𝜀)^1−𝑟𝑚^𝑙>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 𝑥(𝑘+1)≥𝑥(𝑘)exp𝑎^𝑙−𝑐^𝑢(𝑀₂+𝜀)^1−𝑟𝑚^𝑙−𝑏^𝑢𝑥(𝑘).(2.27) By applying Lemma 2.3 to above inequality, we have liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙−𝑐^𝑢(𝑀₂+𝜀)^1−𝑟/𝑚^𝑙𝑏^𝑢exp𝑎^𝑙−𝑐^𝑢(𝑀₂+𝜀)^1−𝑟𝑚^𝑙−𝑏^𝑢𝑀₁+𝜀.(2.28) Setting 𝜀→0 in (2.28) leads to liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙−𝑐^𝑢𝑀^21−𝑟/𝑚^𝑙𝑏^𝑢exp𝑎^𝑙−𝑐^𝑢𝑀^21−𝑟𝑚^𝑙−𝑏^𝑢𝑀₁^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 liminf_𝑘→+∞𝑦(𝑘)≥𝑚₂(2.31)Case (i)
There exists an 𝑛₀≥𝑘₃ such that 𝑦(𝑛₀+1)≤𝑦(𝑛₀), then −𝑑𝑛₀+𝑓𝑛⁰𝑥𝑛₀𝑚𝑛₀𝑦^𝑟𝑛₀+𝑥𝑛₀≤0.(2.32) Hence 𝑦𝑛₀≥(𝑓^𝑙−𝑑^𝑢)(𝑚₁−𝜀)𝑚^𝑢𝑑^𝑢^1/𝑟def=𝑐_1𝜀,(2.33) and so, 𝑦𝑛₀+1≥(𝑓^𝑙−𝑑^𝑢)(𝑚₁−𝜀)𝑚^𝑢𝑑^𝑢^1/𝑟exp−𝑑^𝑢+𝑓^𝑙𝑚₁−𝜀𝑚^𝑢(𝑀₂+𝜀)^𝑟+𝑚₁−𝜀^def=𝑐_2𝜀.(2.34) Set 𝑚_2𝜀=min𝑐_1𝜀,𝑐_2𝜀.(2.35) We claim that 𝑦(𝑘)≥𝑚_2𝜀 for 𝑘≥𝑛₀. By a way of contradiction, assume that there exists a 𝑞₀≥𝑛₀, such that 𝑦(𝑞₀)<𝑚_2𝜀. Then 𝑞₀≥𝑛₀+2. Let 𝑞̃₀≥𝑛₀+2 be the smallest integer such that 𝑦(𝑞̃₀)<𝑚_2𝜀. Then 𝑦(𝑞̃₀)<𝑦(𝑞̃₀−1), which implies that 𝑦(𝑞₀)≤𝑚_2𝜀, a contradiction, this proves the claim.Case (ii)
We assume that 𝑦(𝑘+1)>𝑦(𝑘) for all 𝑘>𝑘₃. According to (2.30), lim_𝑘→+∞𝑦(𝑘) exists, denoted by 𝑦, then lim_𝑘→+∞𝑦(𝑘)=𝑦.(2.36) We claim that 𝑦≥𝑚_2𝜀.(2.37) By the way of contradiction, assume that 𝑦<𝑚_2𝜀. Taking limit in the second equation in system (1.4) gives lim_𝑘→+∞−𝑑(𝑘)+𝑚(𝑘𝑓)𝑦(^𝑟𝑘()𝑘𝑥)(+𝑘)𝑥(𝑘)=0,(2.38) which is a contradiction since for 𝑘>𝑘₃,−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝑚(𝑘)𝑦^𝑟(𝑘)+𝑥(𝑘)≥−𝑑^𝑢+𝑓^𝑙𝑚₁−𝜀𝑚^𝑢𝑦^𝑟+𝑚₁−𝜀>0.(2.39) The above analysis show that liminf_𝑘→+∞𝑦(𝑘)≥𝑚_2𝜀.(2.40) Letting 𝜀→0, we have liminf_𝑘→+∞𝑦(𝑘)≥𝑚₂,(2.41) where 𝑚₂=min(𝑓^𝑙−𝑑^𝑢)𝑚₁𝑚^𝑢𝑑^𝑢^1/𝑟,(𝑓^𝑙−𝑑^𝑢)𝑚₁𝑚^𝑢𝑑^𝑢^1/𝑟exp−𝑑^𝑢+𝑓^𝑙𝑚₁𝑚^𝑢𝑀^𝑟2+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 𝛼min𝑏^𝑙,2𝑀₁−𝑏^𝑢−𝛼𝑐^𝑢𝑀^{21−(𝑟/2)}4𝑚^𝑙𝑚₂−𝛽𝑓^𝑢𝑀^11/24𝑚₁𝑚^2𝑟/2𝛽min𝑓_𝑙>𝛿,(𝐻₃)𝑚^𝑙𝑚₁𝑟(𝑚^𝑢𝑀^𝑟2+𝑀₁)²𝑀^21−𝑟,2𝑀₂−𝑓^𝑢𝑀^11/2𝑟4𝑚₂𝑚^11/2−𝛼𝑐^𝑢𝑀^11/24𝑚^𝑙𝑚^𝑟2𝑚^11/2−𝛼𝑐^𝑢𝑀^𝑟2(1−𝑟)4𝑚₁𝑚^𝑟2>𝛿.(𝐻₄) 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 lim_𝑘→+∞||𝑥₁(𝑘)−𝑥₂(𝑘)||=0,lim_𝑘→+∞||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 𝛼min𝑏^𝑙,2𝑀₁+𝜀−𝑏^𝑢−𝛼𝑐^𝑢(𝑀₂+𝜀)^1−(𝑟/2)4𝑚^𝑙𝑚₂−𝜀−𝛽𝑓^𝑢(𝑀₁+𝜀)^1/24𝑚₁−𝜀(𝑚₂−𝜀)^𝑟/2𝛽min𝑓_𝑙>𝛿,𝑚^𝑙𝑚₁−𝜀𝑟[𝑚^𝑢(𝑀₂+𝜀)^𝑟+(𝑀₁+𝜀)]²(𝑀₂+𝜀)^1−𝑟,2𝑀₂+𝜀−𝑓^𝑢(𝑀₁+𝜀)^1/2𝑟−𝛼𝑐_𝑢4𝑚₂−𝜀(𝑚₁−𝜀)^1/2(𝑀₁+𝜀)^1/24𝑚^𝑙(𝑚₂−𝜀)^𝑟(𝑚₁−𝜀)^1/2−𝛼𝑐^𝑢(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 𝑚𝑚₁₂≤liminf_𝑘→+∞𝑥_𝑖(𝑘)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀₁,≤liminf_𝑘→+∞𝑦_𝑖(𝑘)≤limsup_𝑘→+∞𝑦_𝑖(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≤𝑥_𝑖(𝑘)≤𝑀₁+𝜀,𝑚₂−𝜀≤𝑥_𝑖(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)=≤||||llnn𝑥𝑥₁₁(𝑘+1)−ln𝑥₂(𝑘+1)||+𝑐(𝑘)(||||𝑘)−ln𝑥𝑦₂₁(𝑘)−𝑏(𝑘)𝑥₁(𝑘)−𝑥₂(𝑘)||(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)−𝑦²(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||||.(3.6) Using the Mean Value Theorem, we get 𝑥₁(𝑘)−𝑥₂(𝑘)𝑦=expln𝑥₁(𝑘)−expln𝑥₂(𝑘)=𝜉₁(𝑘)ln𝑥₁(𝑘)−ln𝑥₂(𝑘)_11−𝑟,(𝑘)−𝑦^21−𝑟(𝑘)=(1−𝑟)𝜉^2−𝑟(𝑘)𝑦₁(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(𝑘)||−1𝜉₁(𝑘)−||||𝜉₁1(𝑘)−𝑏(𝑘)||||||𝑥¹+||||𝑐(𝑘)𝑦₁(𝑘)−𝑥₂(𝑘)||(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||||||𝑥¹+||||𝑐(𝑘)𝑥₁(𝑘)−𝑥₂(𝑘)||(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||||||𝑦¹+||||𝑐(𝑘)𝑚(𝑘)𝑦_𝑟1(𝑘)−𝑦₂(𝑘)||(𝑘)𝑦^𝑟2(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)𝜉1^𝑟2−(𝑘𝑟)||||||𝑦¹(𝑘)−𝑦₂(𝑘)||.(3.8) And so, for 𝑘>𝑘₄Δ𝑉₁≤−min𝑏^𝑙,2𝑀₁+𝜀−𝑏^𝑢||𝑥¹+𝑐(𝑘)−𝑥₂(𝑘)_𝑢||(𝑀₂+𝜀)^1−(𝑟/2)4𝑚^𝑙(𝑚₂−𝜀)^𝑟/2𝑚₁−𝜀||𝑥¹+𝑐(𝑘)−𝑥₂(𝑘)_𝑢||(𝑀₁+𝜀)^1/24𝑚^𝑙(𝑚₂−𝜀)^𝑟(𝑚₁−𝜀)^1/2+𝑐||𝑦₁(𝑘)−𝑦₂(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)==||||||llnn𝑦𝑦₁₁(𝑘+1)−ln𝑦₂(𝑘+1)||(𝑘)−ln𝑦₂(𝑘)+𝑓(𝑘)𝑥¹(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)−𝑥²(𝑘)≤||||ln𝑦₁𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||||(𝑘)−ln𝑦₂(𝑘)−𝑓(𝑘)𝑚(𝑘)𝑥¹(𝑘)𝑦^𝑟1(𝑘)−𝑦^𝑟2(𝑘)+||||𝑓(𝑘)𝑚(𝑘)𝑦_𝑟1𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||||(𝑘)𝑥₁(𝑘)−𝑥₂(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||||.(3.11) Using the Mean Value Theorem, we get 𝑦₁(𝑘)−𝑦₂(𝑘)=ex𝑦pln𝑦₁(𝑘)−expln𝑦₂(𝑘)=𝜉₃(𝑘)ln𝑦₁(𝑘)−ln𝑦₂(𝑛)_𝑟1,(𝑘)−𝑦^𝑟2(𝑘)=𝑟𝜉^4𝑟−1(𝑘)𝑦₁(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≤−1𝜉₃(𝑘)−||||𝜉₃1(𝑘)−𝑓(𝑘)𝑚(𝑘)𝑥¹(𝑘)𝑟×||𝑦𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)𝜉^41−𝑟(𝑘)₁||||+(𝑘)−𝑦₂(𝑘)𝑓||(𝑘)𝑚(𝑘)𝑦_𝑟1(𝑘)𝑚(𝑘)𝑦^𝑟1(𝑘)+𝑥₁(𝑘)𝑚(𝑘)𝑦^𝑟2(𝑘)+𝑥₂(𝑘)||𝑥¹≤−min𝑓_𝑙(𝑘)−𝑥₂(𝑘)||𝑚^𝑙𝑚₁−𝜀𝑟[𝑚^𝑢(𝑀₂+𝜀)^𝑟+(𝑀₁+𝜀)]²(𝑀₂+𝜀)^1−𝑟,2𝑀₂+𝜀−𝑓^𝑢(𝑀₁+𝜀)^1/2𝑟×||𝑦₁4𝑚₂−𝜀(𝑚₁−𝜀)^1/2+(𝑓𝑘)−𝑦₂(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 Δ𝑉≤−𝛼min𝑏^𝑙,2𝑀₁+𝜀−𝑏^𝑢−𝛼𝑐^𝑢(𝑀₂+𝜀)^1−(𝑟/2)4𝑚^𝑙(𝑚₂−𝜀)^𝑟/2𝑚₁−𝜀−𝛽𝑓^𝑢(𝑀₁+𝜀)^𝑟/2×||𝑥₁4𝑚₁−𝜀(𝑚₂−𝜀)^𝑟/2−𝛽m(𝑘i)n−𝑥₂(𝑘)||𝑓_𝑙𝑚^𝑙𝑚₁−𝜀𝑟[𝑚^𝑢(𝑀₂+𝜀)^𝑟+(𝑀₁+𝜀)]²(𝑀₂+𝜀)^1−𝑟,2𝑀₂+𝜀−𝑓^𝑢(𝑀₁+𝜀)^1/2𝑟−𝛼𝑐_𝑢4𝑚₂−𝜀(𝑚₁−𝜀)^1/2(𝑀₁+𝜀)^1/24𝑚^𝑙(𝑚₂−𝜀)^𝑟(𝑚₁−𝜀)^1/2−𝛼𝑐^𝑢(𝑀₂+𝜀)^𝑟(1−𝑟)4𝑚₁−𝜀(𝑚₂−𝜀)^𝑟×||𝑦¹≤−𝛿||𝑥₁(𝑘)−𝑦₂(𝑘)||(𝑘)−𝑥₂(𝑘)||+||𝑦₁(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))≤−𝛿^𝑘_𝑝_=𝑘40𝑥0200𝑑||𝑥₁(𝑝)−𝑥₂(𝑝)||+||𝑦₁(𝑝)−𝑦₂(𝑝)||,(3.16) which implies 𝑉(𝑘+1)+𝛿^𝑘_𝑝_=𝑘40𝑥0200𝑑||𝑥₁(𝑝)−𝑥₂(𝑝)||+||𝑦₁(𝑝)−𝑦₂(𝑝)||≤𝑉𝑘₄.(3.17) It follows that ^𝑘_𝑝=𝑘40𝑥0200𝑑||𝑥₁(𝑝)−𝑥₂(𝑝)||+||𝑦₁(𝑝)−𝑦₂(𝑝)||≤𝑉𝑘⁴𝛿.(3.18) Using the fundamental theorem of positive series, there exists small enough positive constant 𝜀>0 such that ⁺^∞_𝑝=𝑘40𝑥0200𝑑||𝑥₁(𝑝)−𝑥₂(𝑝)||+||𝑦₁(𝑝)−𝑦₂(𝑝)||≤𝑉𝑘⁴𝛿,(3.19) which implies that lim_𝑘→+∞||𝑥₁(𝑘)−𝑥₂(𝑘)||+||𝑦₁(𝑘)−𝑦₂(𝑘)||=0,(3.20) that is lim_𝑘→+∞||𝑥₁(𝑘)−𝑥₂(𝑘)||=0,lim_𝑘→+∞||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 −𝑑^𝑙+𝑓^𝑢<0.(𝐻₅) 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), 𝑚lim_𝑘→+∞𝑦(𝑘)=0,_∗≤liminf_𝑘→+∞𝑥(𝑘)≤limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁,(4.1) where 𝑚_∗=𝑎^𝑙𝑏^𝑢𝑀₁exp𝑎^𝑙−𝑏^𝑢𝑀₁,=1𝑏^𝑙exp(𝑎^𝑢−1).(4.2)

Proof. For condition (𝐻₅), there exists small enough positive 𝛾>0, such that −𝑑^𝑙+𝑓^𝑢<−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)<𝑦(𝑘)exp−𝑑_𝑙𝑚(𝑘)𝑦(𝑘)+𝑥(𝑘)<𝑦(𝑘)exp{−𝛾}.+𝑓^𝑢(4.4) Therefore, 𝑦(𝑘+1)<𝑦(0)exp{−𝑘𝛾},(4.5) which yields lim_𝑘→+∞𝑦(𝑘)=0.(4.6) From the proof of Theorem 3.1, we have limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁.(4.7) For enough small positive constant 𝜀>0, 𝑎^𝑙−𝑐^𝑢𝜀^1−𝑟𝑚^𝑙>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 𝑥(𝑘+1)≥𝑥(𝑘)exp𝑎^𝑙−𝑐^𝑢𝜀^1−𝑟𝑚^𝑙−𝑏^𝑢𝑥(𝑘).(4.10) By Lemma 2.3, we have liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙−𝑐^𝑢𝜀^1−𝑟/𝑚^𝑙𝑏^𝑢exp𝑎^𝑙−𝑐^𝑢𝜀^1−𝑟𝑚^𝑙−𝑏^𝑢𝑀₁+𝜀.(4.11) Setting 𝜀→0 in (4.11) leads to liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑀₁=^def𝑚_∗.(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)exp1.41+0.12cos(𝑘)−1.78𝑥(𝑘)−0.33𝑦(𝑘)𝑦(𝑘+1)=𝑦(𝑘)exp−0.62+1.79𝑥(𝑘)2.16𝑦^1/2(𝑘)+𝑥(𝑘),2.16𝑦^1/2(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 𝑎^𝑙−𝑐^𝑢𝑀^21−𝑟𝑚^𝑙𝑓≈2.3281>0,_𝑙𝛼min𝑏_𝑙>𝑑^𝑢≈1.1700>0,,2𝑀₁−𝑏^𝑢−𝛼𝑐^𝑢𝑀^{21−(𝑟/2)}4𝑚^𝑙𝑚₂−𝛽𝑓^𝑢𝑀^11/24𝑚₁𝑚^2𝑟/2𝛽min𝑓_𝑙≈0.0011>𝛿,𝑚^𝑙𝑚₁𝑟(𝑚^𝑢𝑀^𝑟2+𝑀₁)²𝑀^21−𝑟,2𝑀₂−𝑓^𝑢𝑀^11/2𝑟4𝑚₂𝑚^11/2−𝛼𝑐^𝑢𝑀^11/24𝑚^𝑙𝑚^𝑟2𝑚^11/2−𝛼𝑐^𝑢𝑀^𝑟2(1−𝑟)4𝑚₁𝑚^𝑟2≈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.

Figure 1 

Dynamics behavior of system (1.4) with initial conditions (𝑥(0),𝑦(0))=(0.9,0.6),(0.7,0.4),(0.5,0.5), respectively.

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 𝑎^𝑙−𝑐^𝑢/𝑚^𝑙>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.