Permanence and Global Attractivity of the Discrete Predator-Prey System with Hassell-Varley-Holling III 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-Holling III type functional response are obtained. An example together with its numerical simulation shows that the main results are verifiable.

1. Introduction

Recently, there were many works on predator-prey system done by scholars [1–6]. In particular, since Hassell-Varley [7] proposed a general predator-prey model with Hassell-Varley type functional response in 1969, many excellent works have been conducted for the Hassell-Varley type system [1, 7–13].

Liu and Huang [8] studied the following discrete predator-prey system with Hassell-Varley-Holling III type functional response: 𝑥()(𝑘)𝑎(𝑘)(𝑘)𝑥(𝑘)−𝐴(𝑘)𝑥(𝑘)𝑦(𝑘)𝑘+1=𝑥exp−𝑏𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘),𝑦()(𝑘)(𝑘)+𝐵(𝑘)𝑥𝑘+1=𝑦exp−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘),𝑅∈(0,1), (1) where 𝑥(𝑘), 𝑦(𝑘) denote the density of prey and predator species at the 𝑘th generation, respectively. 𝑎, 𝑏, 𝐴, 𝑟, 𝑑, 𝐵 are all periodic positive sequences with common period 𝑋. Here 𝑎(𝑘) represents the intrinsic growth rate of prey species at the 𝑘th generation, and 𝑏(𝑘) measures the intraspecific effects of the 𝑘th generation of prey species on their own population; 𝑑(𝑘) is the death rate of the predator; 𝐴(𝑘) is the capturing rate; 𝐵(𝑘) is the maximal growth rate of the predator. Liu and Huang obtained the necessary and sufficient conditions for the existences of positive periodic solutions by applying a new estimation technique of solutions and the invariance property of homotopy. As we know, the persistent property is one of the most important topics in the study of population dynamics. For more papers on permanence and extinction of population dynamics, one could refer to [2–5, 14–17] and the references cited therein. The purpose of this paper is to investigate permanence and global attractivity of this system.

We argue that a general nonautonomous nonperiodic system is more appropriate, and thus, we assume that the coefficients of system (1) satisfy the following:

(A) 𝑎, 𝑏, 𝐴, 𝑟, 𝑑, 𝐵 are nonnegative sequences bounded above and below by positive constants.

By the biological meaning, we consider (1) together with the following initial conditions as 𝑥(0)>0,𝑦(0)>0. (2)

For the rest of the paper, we use the following notations: for any bounded sequence {ℎ(𝑘)}, set ℎ^𝑢=sup_𝑘∈𝑁{ℎ(𝑘)} and ℎ^𝑙=inf_𝑘∈𝑁{ℎ(𝑘)}.

2. Permanence

Now, let us state several lemmas which will be useful to prove our main conclusion.

Definition 1 (see [5]). System (1) said to be permanent if there exist positive constants 𝑚 and 𝑀, which are independent of the solution of system (1), such that for any positive solution {𝑥(𝑘),𝑦(𝑘)} of system (1) satisfies 𝑚≤liminf_𝑘→+∞𝑥(𝑘),𝑦(𝑘)≤limsup_𝑘→+∞𝑥(𝑘),𝑦(𝑘)≤𝑀. (3)

Lemma 2 (see [14]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘)>0 and 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)}, (4) 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^𝑙exp𝑎^𝑢−1. (5)

Lemma 3 (see [14]). Assume that {𝑥(𝑘)} satisfies 𝑥(𝑘+1)≥𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)},𝑘≥𝑁₀, (6)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≥min^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑥^∗,𝑎^𝑙𝑏^𝑢. (7)

Theorem 4. Assume that 𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅2𝑟^𝑙>0,𝐵^𝑙−𝑑^𝑢𝑟^𝑢>0 𝐻₁𝐻₂ hold, then system (1) is permanent, that is, for any positive solution {𝑥(𝑘),𝑦(𝑘)} of system (1), one has 𝑚₁≤liminf_𝑘→+∞𝑥(𝑘)≤limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁,𝑚₂≤liminf_𝑘→+∞𝑦(𝑘)≤limsup_𝑘→+∞𝑦(𝑘)≤𝑀₂, (8) where 𝑚₁=𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅/2𝑟^𝑙𝑏^𝑢×exp𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅2𝑟^𝑙−𝑏^𝑢𝑀₁,𝑚₂𝐵=min^𝑙−𝑟^𝑢𝑑^𝑢𝑚²₁𝑑^𝑢1/2𝑅,𝐵^𝑙−𝑟^𝑢𝑑^𝑢𝑚²₁𝑑^𝑢1/2𝑅×exp−𝑑^𝑢+𝐵^𝑙𝑚²₁𝑟^𝑢𝑚²₁+𝑀₂^2𝑅,𝑀₁=1𝑏^𝑙exp𝑎^𝑢,𝑀−1₂=𝐵^𝑢𝑀²₁𝑑^𝑙1/2𝑅exp−𝑑^𝑙+𝐵^𝑢𝑟^𝑙. (9)

Proof. We divided the proof into four steps.
Step 1. We show limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁. (10) From the first equation of (1), we have 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)}. (11) By Lemma 2, we have limsup_𝑘→+∞𝑥(𝑘)≤1𝑏^𝑙exp𝑎^𝑢−1=𝑀₁. (12) Previous inequality shows that for any 𝜀>0, there exists a 𝑘₁>0, such that 𝑥(𝑘)≤𝑀₁+𝜀,∀𝑘≥𝑘₁. (13)
Step 2. We prove limsup_𝑘→+∞𝑦(𝑘)≤𝑀₂ by distinguishing two cases.
Case 1. There exists a 𝑙₀≥𝑘₁, such that 𝑦(𝑙₀+1)≥𝑦(𝑙₀).
By the second equation of system (1), we have −𝑑𝑙₀+𝐵𝑙₀𝑥²𝑙₀𝑟𝑙₀𝑥²𝑙₀+𝑦^2𝑅𝑙₀≥0, (14) which implies −𝑑𝑙₀+𝐵𝑙₀𝑥²𝑙₀𝑦^2𝑅𝑙₀≥0. (15) The previous inequality combined with (13) leads to 𝑦(𝑙₀)≤{𝐵^𝑢(𝑀₁+𝜀)²/𝑑^𝑙}^1/2𝑅. Thus, from the second equation of system (1), again we have 𝑦𝑙₀𝑙+1=𝑦₀exp−𝑑𝑙₀+𝐵𝑙₀𝑥²𝑙₀𝑟𝑙₀𝑥²𝑙₀+𝑦^2𝑅𝑙₀≤𝐵^𝑢(𝑀₁+𝜀)²𝑑^𝑙1/2𝑅exp−𝑑^𝑙+𝐵^𝑢𝑙_de𝑟_f=𝑀_2𝜀. (16) We claim that 𝑦(𝑘)≤𝑀_2𝜀∀𝑘≥𝑙₀. (17) By a way of contradiction, assume that there exists a 𝑝₀≥𝑙₀ such that 𝑦(𝑝₀)>𝑀_2𝜀. Then 𝑝₀≥𝑙₀+2. Let 𝑝₀≥𝑙₀+2 be the smallest integer such that 𝑦(𝑝₀)>𝑀_2𝜀. Then 𝑦(𝑝₀)>𝑦(𝑝₀−1). The previous argument produces that 𝑦(𝑝₀)≤𝑀_2𝜀, a contradiction. This proves the claim. Therefore, limsup_𝑘→+∞𝑦(𝑘)≤𝑀_2𝜀. Setting 𝜀→0 in it leads to limsup_𝑘→+∞𝑦(𝑘)≤𝑀₂.
Case 2. Suppose 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, we claim that 𝐵𝑦≤^𝑢𝑀²₁𝑑^𝑙1/2𝑅. (18) By a way of contradiction, assume that 𝑦>{𝐵^𝑢𝑀²₁/𝑑^𝑙}^1/2𝑅.
Taking limit in the second equation in system (1) gives lim_𝑘→+∞(𝑘)+𝐵(𝑘)𝑥−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)=0, (19) however, lim_𝑘→+∞(𝑘)+𝐵(𝑘)𝑥−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)≤limsup_𝑘→+∞(𝑘)+𝐵(𝑘)𝑥−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)≤−𝑑^𝑙+𝐵^𝑢𝑀²₁𝑦^2𝑅<0, (20) which is a contradiction. It implies that 𝑦≤{𝐵^𝑢𝑀²₁/𝑑^𝑙}^1/2𝑅. By the fact 𝐵^𝑢>𝑑^𝑙𝑟^𝑙, we obtain that 𝐵𝑦≤^𝑢𝑀²₁𝑑^𝑙1/2𝑅≤𝐵^𝑢𝑀²₁𝑑^𝑙1/2𝑅exp−𝑑^𝑙+𝐵^𝑢𝑟^𝑙=𝑀₂. (21) Therefore, we have lim_𝑘→+∞𝑦(𝑘)=𝑦≤𝑀₂. (22) Then, limsup_𝑘→+∞𝑦(𝑘)≤𝑀₂. (23)
Step 3. We verify liminf_𝑘→+∞𝑥(𝑘)≥𝑚₁. (24) Conditions 𝐻₁ imply that for enough small positive constant 𝜀, we have 𝑎^𝑙−𝐴^𝑢𝑀₂+𝜀^1−𝑅2𝑟^𝑙>0. (25) For the previous 𝜀, it follows from Steps 1 and 2 that there exists a 𝑘₂ such that for all 𝑘≥𝑘₂𝑥(𝑘)≤𝑀₁+𝜀,𝑦(𝑘)≤𝑀₂+𝜀. (26) Then, for 𝑘≥𝑘₂, it follows from (26) and the first equation of system (1) that 𝑥(𝑘+1)≥𝑥(𝑘)exp𝑎^𝑙−𝐴^𝑢𝑀₂+𝜀^1−𝑅2𝑟^𝑙−𝑏^𝑢𝑥(𝑘). (27) According to Lemma 3, one has liminf_𝑘→+∞𝑥(𝑘)≥min𝑚_1∗,𝑎^𝑙−𝐴^𝑢𝑀₂+𝜀^1−𝑅/2𝑟^𝑙𝑏^𝑢=𝑚_1∗, (28) where 𝑚_1∗=𝑎^𝑙−𝐴^𝑢𝑀₂+𝜀^1−𝑅/2𝑟^𝑙𝑏^𝑢×exp𝑎^𝑙−𝐴^𝑢𝑀₂+𝜀^1−𝑅2𝑟^𝑙−𝑏^𝑢𝑀₁+𝜀. (29)
Setting 𝜀→0 in (28) leads to liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅/2𝑟^𝑙𝑏^𝑢×exp𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅2𝑟^𝑙−𝑏^𝑢𝑀₁=𝑚₁. (30) By the fact that min_𝑥∈𝑅⁺{[exp(𝑥−1)]/𝑥}=1, we see that 𝑀₁=exp(𝑎^𝑢−1)/𝑏^𝑙≥𝑎^𝑢/𝑏^𝑙≥𝑎^𝑙/𝑏^𝑢≥(𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅/2𝑟^𝑙)/𝑏^𝑙≥𝑚₁.
This ends the proof of Step 3.
Step 4. We present two cases to prove that liminf_𝑘→+∞𝑦(𝑘)≥𝑚₂. (31) For any small positive constant 𝜀<𝑚₁/2, from Step 1 to Step 3, it follows that there exists a 𝑘₃≥𝑘₂ such that for all 𝑘≥𝑘₃𝑥(𝑘)≥𝑚₁−𝜀,𝑥(𝑘)≤𝑀₁𝑦(𝑘)+𝜀,≤𝑀₂+𝜀. (32)
Case 1. There exists a 𝑛₀≥𝑘₃ such that 𝑦(𝑛₀+1)≤𝑦(𝑛₀), then −𝑑𝑛₀+𝐵𝑛₀𝑥²𝑛₀𝑟𝑛₀𝑥²𝑛₀+𝑦^2𝑅𝑛₀≤0. (33) Hence, 𝑦𝑛₀≥𝐵^𝑙−𝑟^𝑢𝑑^𝑢𝑚₁−𝜀²𝑑^{𝑢1/2𝑅def}=𝑐_1𝜀, (34) and so, 𝑦𝑛₀≥𝐵+1_𝑙−𝑟^𝑢𝑑^𝑢𝑚₁−𝜀²𝑑^𝑢1/2𝑅×exp−𝑑^𝑢+𝐵^𝑙𝑚₁−𝜀²𝑟^𝑢𝑚₁−𝜀²+𝑀₂+𝜀^2𝑅_def=𝑐_2𝜀. (35) Set 𝑚_2𝜀=min𝑐_1𝜀,𝑐_2𝜀. (36) We claim that 𝑦(𝑘)≥𝑚_2𝜀∀𝑘≥𝑛₀. (37) By a way of contradiction, assume that there exists a 𝑞₀≥𝑛₀, such that 𝑦(𝑞₀)<𝑚_2𝜀. Then 𝑞₀≥𝑛₀+2. Let 𝑞₀≥𝑛₀+2 be the smallest integer such that 𝑦(𝑞₀)<𝑚_2𝜀. Then 𝑦(𝑞₀)<𝑦(𝑞₀−1), which implies that 𝑦(𝑞₀)≥𝑚_2𝜀, a contradiction, this proves the claim. Therefore, liminf_𝑘→+∞𝑦(𝑘)≥𝑚_2𝜀, setting 𝜀→0 in it leads to liminf_𝑘→+∞𝑦(𝑘)≥𝑚₂.
Case 2. Assume that 𝑦(𝑘+1)>𝑦(𝑘) for all 𝑘≥𝑘₃, then, lim_𝑘→+∞𝑦(𝑘) exists, denoted by 𝑦, then lim_𝑘→+∞𝑦(𝑘)=𝑦. We claim that 𝑦≥𝑚₂. (38) By a way of contradiction, assume that 𝑦<𝑚₂. Taking limit in the second equation in system (1) gives lim_𝑘→+∞(𝑘)+𝐵(𝑘)𝑥−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)=0, (39) which is a contradiction since lim_𝑘→+∞(𝑘)+𝐵(𝑘)𝑥−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)≥liminf_𝑘→+∞(𝑘)+𝐵(𝑘)𝑥−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)≥−𝑑^𝑢+𝐵^𝑙𝑚²₁𝑟^𝑢𝑚²₁+𝑦^2𝑅>0. (40) This proves the claim, then we have lim_𝑘→+∞𝑦(𝑘)=𝑦≥𝑚₂. (41) So, liminf_𝑘→+∞𝑦(𝑘)≥𝑚₂. (42) Obviously, 𝑀₂={𝐵^𝑢𝑀²₁/𝑑^𝑙}^1/2𝑅exp{−𝑑^𝑙+𝐵^𝑢/𝑟^𝑙}≥{(𝐵^𝑙−𝑟^𝑢𝑑^𝑢)𝑚²₁/𝑑^𝑢}^1/2𝑅≥𝑚₂. This completes the proof of the theorem.

3. Global Attractivity

Definition 5 (see [18]). System (1) is said to be globally attractive if any two positive solutions (𝑥₁(𝑘),𝑦₁(𝑘)) and (𝑥₂(𝑘),𝑦₂(𝑘)) of system (1) satisfy lim_𝑘→+∞𝑥₁(𝑘)−𝑥₂(𝑘)=0,lim_𝑘→+∞𝑦₁(𝑘)−𝑦₂(𝑘)=0. (43)

Theorem 6. Assume that 𝐻₁ and 𝐻₂ hold. Assume further that there exist positive constants 𝛼, 𝛽, and 𝛿 such that 𝛼min𝑏^𝑙,2𝑀₁−𝑏^𝑢𝐴−𝛼^𝑢𝑀₂^1−𝑅4𝑚^𝑅₂𝐴−𝛼^𝑢𝑀₂4𝑟^𝑙𝑚²₁𝐵−𝛽^𝑢𝑀^𝑅₂2𝑟^𝑙𝑚^𝑅₁>𝛿,𝛽min𝐺₁,𝐺₂,𝐺₃,𝐺₄𝐴−𝛼^𝑢𝑀₁4𝑚₂^2𝑅𝐴−𝛼^𝑢𝑀₂+𝜀^𝑅4𝑟^𝑙𝑚₁𝑚₂+𝜀^𝑅𝐴−𝛼^𝑢𝑅2𝑟^𝑙𝑚₁𝑀max²𝑚₂^1−𝑅,𝑀₂𝑚₂^𝑅>𝛿, 𝐻₃𝐻₄ where 𝐺₁=2𝑅𝐵^𝑙𝑚²₁𝑚₂^2𝑅−1𝑟^𝑢𝑀²₁+𝑀₂^2𝑅2,𝐺₂=2𝑅𝐵^𝑙𝑚²₁𝑀₂^2𝑅−1𝑟^𝑢𝑀²₁+𝑀₂^2𝑅2,𝐺₃=2𝑀₂−2𝑅𝐵^𝑢𝑀²₁𝑀₂^2𝑅−1𝑟^𝑙𝑚²₁+𝑚₂^2𝑅2,𝐺₄=2𝑀₂−2𝑅𝐵^𝑢𝑀²₁𝑚₂^2𝑅−1𝑟^𝑙𝑚²₁+𝑚₂^2𝑅2. (44)

Then, system (1), with initial condition (2), is globally attractive, that is, for any two positive solutions (𝑥₁(𝑘),𝑦₁(𝑘)) and (𝑥₂(𝑘),𝑦₂(𝑘)) of system (1), we have lim_𝑘→+∞𝑥₁(𝑘)−𝑥₂(𝑘)=0,lim_𝑘→+∞𝑦₁(𝑘)−𝑦₂(𝑘)=0. (45)

Proof. From conditions 𝐻₃ and 𝐻₄, there exists an enough small positive constant 𝜀<min{𝑚₁/2,𝑚₂/2} such that 𝛼min𝑏^𝑙,2𝑀₁+𝜀−𝑏^𝑢𝐴−𝛼^𝑢𝑀₂+𝜀^1−𝑅4𝑚₂−𝜀^𝑅𝐴−𝛼^𝑢𝑀₂+𝜀4𝑟^𝑙𝑚₁−𝜀²𝐵−𝛽^𝑢𝑀₂+𝜀^𝑅2𝑟^𝑙𝑚₂−𝜀^𝑅𝑚₁𝐺−𝜀>𝛿,𝛽min_1𝜀,𝐺_2𝜀,𝐺_3𝜀,𝐺_4𝜀𝐴−𝛼^𝑢𝑀₁+𝜀4𝑚₂−𝜀^2𝑅𝐴−𝛼^𝑢4𝑟^𝑙𝑚₁𝑀−𝜀²+𝜀𝑚₂−𝜀^𝑅𝐴−𝛼^𝑢𝑅2𝑟^𝑙𝑚₁𝑀−𝜀×max²+𝜀𝑚₂−𝜀^1−𝑅,𝑀₂+𝜀𝑚₂−𝜀^𝑅>𝛿, (46) where 𝐺_1𝜀=2𝑅𝐵^𝑙𝑚₁−𝜀²𝑚₂−𝜀^2𝑅−1𝑟^𝑢𝑀₁+𝜀²+𝑀₂+𝜀^2𝑅2,𝐺_2𝜀=2𝑅𝐵^𝑙𝑚₁−𝜀²𝑀₂+𝜀^2𝑅−1𝑟^𝑢𝑀₁+𝜀²+𝑀₂+𝜀^2𝑅2,𝐺_3𝜀=2𝑀₂−+𝜀2𝑅𝐵^𝑢𝑀₁+𝜀²𝑀₂+𝜀^2𝑅−1𝑟^𝑙𝑚₁−𝜀²+𝑚₂−𝜀^2𝑅2,𝐺_4𝜀=2𝑀₂−+𝜀2𝑅𝐵^𝑢𝑀₁+𝜀²𝑚₂−𝜀^2𝑅−1𝑟^𝑙𝑚₁−𝜀²+𝑚₂−𝜀^2𝑅2. (47)
Since 𝐻₁ and 𝐻₂ hold, for any positive solutions (𝑥₁(𝑘),𝑦₁(𝑘)) and (𝑥₂(𝑘),𝑦₂(𝑘)) of system (1), it follows from Theorem 4 that 𝑚₁≤liminf_𝑘→+∞𝑥_𝑖(𝑘)≤limsup_𝑘→+∞𝑥_𝑖(𝑘)≤𝑀₁,𝑚₂≤liminf_𝑘→+∞𝑦_𝑖(𝑘)≤limsup_𝑘→+∞𝑦_𝑖(𝑘)≤𝑀₂,𝑖=1,2. (48) For the previous 𝜀 and (48), there exists a 𝑘₄>0 such that for all 𝑘>𝑘₄, 𝑚₁−𝜀≤𝑥_𝑖(𝑘)≤𝑀₁𝑚+𝜀,₂−𝜀≤𝑥_𝑖(𝑘)≤𝑀₂+𝜀,𝑖=1,2. (49) Let 𝑉₁(𝑘)=ln𝑥₁(𝑘)−ln𝑥₂(𝑘). (50) Then from the first equation of system (1), we have 𝑉₁()=𝑘+1ln𝑥₁(𝑘+1)−ln𝑥₂(≤𝑘+1)ln𝑥₁(𝑘)−ln𝑥₂(𝑘)−𝑏(𝑘)𝑥₁(𝑘)−𝑥₂(𝑘)(𝑘𝑥+𝐴)¹(𝑘)𝑦₁(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)−𝑥₂(𝑘)𝑦₂(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘). (51) Using the mean value theorem, we get 𝑥₁(𝑘)−𝑥₂(𝑘)=expln𝑥₁(𝑘)−expln𝑥₂(𝑘)=𝜉₁(𝑘)ln𝑥₁(𝑘)−ln𝑥₂(𝑘),𝑦₁^2𝑅(𝑘)−𝑦₂^2𝑅(𝑘)=2𝑅𝜉₂^2𝑅−1(𝑘)𝑦₁(𝑘)−𝑦₂(𝑘), (52) where 𝜉₁(𝑘) lies between 𝑥₁(𝑘) and 𝑥₂(𝑘), 𝜉₂(𝑘) lies between 𝑦₁(𝑘) and 𝑦₂(𝑘).
It follows from (51) and (52) that 𝑉₁()≤𝑘+1ln𝑥₁(𝑘)−ln𝑥₂(𝑘−1)𝜉₁(𝑘)−1𝜉₁(𝑘)−𝑏(𝑘)𝑥₁(𝑘)−𝑥₂(𝑘+𝐴(𝑘)𝑟(𝑘)𝑥₁)(𝑘)𝑥₂(𝑘)𝑦₁(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑥₁(𝑘)−𝑥₂(𝑘+𝐴(𝑘)𝑦)_12𝑅(𝑘)𝑦₂(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑥₁(𝑘)−𝑥₂(𝑘+𝐴(𝑘)𝑟(𝑘)𝑥)₂₁(𝑘)𝑥₂(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑦₁(𝑘)−𝑦₂(𝑘+𝐴(𝑘)𝑥)₁(𝑘)𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑦₁(𝑘)−𝑦₂(𝑘+𝐴(𝑘)𝑥)₁(𝑘)𝑦₁(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×2𝑅𝜉₂^2𝑅−1(𝑘)𝑦₁(𝑘)−𝑦₂(𝑘). (53) And so, for 𝑘>𝑘₄, Δ𝑉₁≤−min𝑏^𝑙,2𝑀₁+𝜀−𝑏^𝑢𝑥₁(𝑘)−𝑥₂(𝑘+𝐴_𝑢)𝑀₂+𝜀^1−𝑅4𝑚₂−𝜀^𝑅𝑥₁(𝑘)−𝑥₂(𝑘+𝐴_𝑢)𝑀₁+𝜀4𝑟^𝑙𝑚₁−𝜀²𝑥₁(𝑘)−𝑥₂(𝑘+𝐴_𝑢)𝑀₁+𝜀4𝑚₂−𝜀^2𝑅𝑦₁(𝑘)−𝑦₂(𝑘+𝐴_𝑢)4𝑟^𝑙𝑚₁𝑀−𝜀²+𝜀𝑚₂−𝜀^𝑅𝑦₁(𝑘)−𝑦₂(𝑘+)𝑅𝐴^𝑢2𝑟^𝑙𝑚₁𝑀−𝜀max²+𝜀𝑚₂−𝜀^1−𝑅,𝑀₂+𝜀𝑚₂−𝜀^𝑅×𝑦₁(𝑘)−𝑦₂(𝑘). (54) Let 𝑉₂(𝑘)=ln𝑦₁(𝑘)−ln𝑦₂(𝑘). (55) Then, from the second equation of system (1), we have 𝑉₂()=𝑘+1ln𝑦₁(𝑘+1)−ln𝑦₂(=𝑘+1)ln𝑦₁(𝑘)−ln𝑦₂(𝑘)(𝑘)×𝑥+𝐵₂₁(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)−𝑥²₂(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)≤ln𝑦₁(𝑘)−ln𝑦₂(𝑘)−𝐵(𝑘)𝑥²₂(𝑘)𝑦₁^2𝑅(𝑘)−𝑦₂^2𝑅(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)+𝐵(𝑘)𝑦₂^2𝑅(𝑘)𝑥₁(𝑘)+𝑥₂(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑥₁(𝑘)−𝑥₂(𝑘). (56) Using the mean value theorem, we get 𝑦₁(𝑘)−𝑦₂(𝑘)=expln𝑦₁(𝑘)−expln𝑦₂(𝑘)=𝜉₃(𝑘)ln𝑦₁(𝑘)−ln𝑦₂(𝑛),𝑦₁^2𝑅(𝑘)−𝑦₂^2𝑅(𝑘)=2𝑅𝜉₂^2𝑅−1(𝑘)𝑦₁(𝑘)−𝑦₂(𝑘), (57) where 𝜉₃(𝑘),𝜉₂(𝑘) lies between 𝑦₁(𝑘) and 𝑦₂(𝑘), respectively. Then, it follows from (56) and (57) that for 𝑘>𝑘₄, Δ𝑉₂1≤−𝜉₃(𝑘)−1𝜉₃(𝑘)−𝐵(𝑘)𝑥²₂(𝑘)2𝑅𝜉₂^2𝑅−1(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑦₁(𝑘)−𝑦₂(𝑘+𝐵(𝑘)𝑦)_22𝑅(𝑘)𝑥₁(𝑘)+𝑥₂(𝑘)𝑟(𝑘)𝑥²₁(𝑘)+𝑦₁^2𝑅(𝑘)𝑟(𝑘)𝑥²₂(𝑘)+𝑦₂^2𝑅(𝑘)×𝑥₁(𝑘)−𝑥₂(𝑘𝐺)≤−min_1𝜀,𝐺_2𝜀,𝐺_3𝜀,𝐺_4𝜀×𝑦₁(𝑘)−𝑦₂(𝑘+𝐵_𝑢)𝑀₂+𝜀^𝑅2𝑟^𝑙𝑚₁−𝜀𝑚₂−𝜀^𝑅𝑥₁(𝑘)−𝑥₂(𝑘). (58)
Now, we define a Lyapunov function as follows: 𝑉(𝑘)=𝛼𝑉₁(𝑘)+𝛽𝑉₂(𝑘). (59) Calculating the difference of 𝑉 along the solution of system (1), for 𝑘>𝑘₄, it follows from (54) and (58) that 𝑏Δ𝑉≤−𝛼min^𝑙,2𝑀₁+𝜀−𝑏^𝑢𝐴−𝛼^𝑢𝑀₂+𝜀^1−𝑅4𝑚₂−𝜀^𝑅𝐴−𝛼^𝑢𝑀₂+𝜀4𝑟^𝑙𝑚₁−𝜀²𝐵−𝛽^𝑢𝑀₂+𝜀^𝑅2𝑟^𝑙𝑚₂−𝜀^𝑅𝑚₁×𝑥−𝜀₁(𝑘)−𝑥₂(𝑘−𝐺)𝛽min_1𝜀,𝐺_2𝜀,𝐺_3𝜀,𝐺_4𝜀𝐴−𝛼^𝑢𝑀₁+𝜀4𝑚₂−𝜀^2𝑅𝐴−𝛼^𝑢𝑀₁+𝜀𝑀₂+𝜀^𝑅4𝑟^𝑙𝑚₁−𝜀𝑚₂−𝜀^𝑅𝐴−𝛼^𝑢2𝑅2𝑟^𝑙𝑚₁𝑀−𝜀×max²+𝜀𝑚₂−𝜀^1−𝑅,𝑀₂+𝜀𝑚₂−𝜀^𝑅×𝑦₁(𝑘)−𝑦₂(𝑘𝑥)≤−𝛿₁(𝑘)−𝑥₂(𝑘)+𝑦₁(𝑘)−𝑦₂(𝑘). (60) Summating both sides of the previous inequalities from 𝑘₄ to 𝑘, we have ^𝑘_𝑝=𝑘4𝑉𝑝+1−𝑣𝑝≤−𝛿^𝑘_𝑝=𝑘4𝑥₁𝑝−𝑥₂𝑝+𝑦₁𝑝−𝑦₂𝑝, (61) which implies 𝑉(𝑘+1)+𝛿^𝑘_𝑝=𝑘4𝑥₁𝑝−𝑥₂𝑝+𝑦₁𝑝−𝑦₂𝑝≤𝑉𝑘₄. (62) It follows that ^𝑘_𝑝=𝑘4𝑥₁𝑝−𝑥₂𝑝+𝑦₁𝑝−𝑦₂𝑝≤𝑉𝑘₄𝛿. (63) Using the fundamental theorem of positive series, there exists small enough positive constant 𝜀>0 such that ^+∞_𝑝=𝑘4𝑥₁𝑝−𝑥₂𝑝+𝑦₁𝑝−𝑦₂𝑝≤𝑉𝑘₄𝛿+𝜀, (64) which implies that lim_𝑘→+∞𝑥₁(𝑘)−𝑥₂(𝑘)+𝑦₁(𝑘)−𝑦₂(𝑘)=0, (65) that is, lim_𝑘→+∞𝑥₁(𝑘)−𝑥₂(𝑘)=0,lim_𝑘→+∞𝑦₁(𝑘)−𝑦₂(𝑘)=0. (66) This completes the proof of Theorem 6.

4. Extinction of the Predator Species

This section is devoted to study the extinction of the predator species 𝑦.

Theorem 7. Assume that −𝑑^𝑙+𝐵^𝑢𝑟^𝑙<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), lim_𝑘→+∞𝑦(𝑘)𝑚=0,_∗≤liminf_𝑘→+∞𝑥(𝑘)≤limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁, (67) where 𝑚_∗=𝑎^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑀₁,𝑀₁=1𝑏^𝑙exp𝑎^𝑢−1. (68)

Proof. For condition 𝐻₅, there exists small enough positive 𝛾>0, such that −𝑑^𝑙+𝐵^𝑢𝑟^𝑙<0, (69) for all 𝑘∈𝑁, from (69) and the second equation of the system (1), one can easily obtain that 𝑦()(𝑘)(𝑘)+𝐵(𝑘)𝑥𝑘+1=𝑦exp−𝑑²(𝑘)𝑟(𝑘)𝑥²(𝑘)+𝑦^2𝑅(𝑘)<𝑦(𝑘)exp−𝑑^𝑙+𝐵^𝑢𝑟^𝑙<𝑦(𝑘)exp−𝛾. (70) Therefore, 𝑦(𝑘+1)<𝑦(0)exp−𝑘𝛾, (71) which yields lim_𝑘→+∞𝑦(𝑘)=0. (72) From the proof of Theorem 4, we have limsup_𝑘→+∞𝑥(𝑘)≤𝑀₁. (73) For enough small positive constant 𝜀>0, 𝑎^𝑙−𝐴^𝑢𝜀^1−𝑅/2𝑟^𝑙𝑏^𝑢>0. (74) For the previous 𝜀, from (72) and (73) there exists a 𝑘₅>0 such that for all 𝑘>𝑘₅, 𝑥(𝑘)<𝑀₁+𝜀,𝑦(𝑘)<𝜀. (75) From the first equation of (1), we have 𝑥(𝑘+1)≥𝑥(𝑘)exp𝑎^𝑙−𝑎^𝑙−𝐴^𝑢𝜀^1−𝑅/2𝑟^𝑙𝑏^𝑢−𝑏^𝑢𝑥(𝑘). (76) By Lemma 3, we have liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙−𝐴^𝑢𝜀^1−𝑅/2𝑟^𝑙𝑏^𝑢×exp𝑎^𝑙−𝐴^𝑢𝜀^1−𝑅2𝑟^𝑙−𝑏^𝑢𝑀₁+𝜀. (77) Setting 𝜀→0 in (72) leads to liminf_𝑘→+∞𝑥(𝑘)≥𝑎^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑀=₁^def𝑚_∗. (78) The proof of Theorem 7 is completed.

5. Example

The following example shows the feasibility of the main results.

Example 8. Consider the following system: 𝑥()(𝑘)(𝑘)(𝑘)−(𝑘)𝑦(𝑘)𝑘+1=𝑥exp0.85+0.05cos−2.4𝑥1.7𝑥0.3𝑥(𝑘)²(𝑘),𝑦()(𝑘)(𝑘)+𝑦𝑘+1=𝑦exp−4.1+1.6𝑥²0.3𝑥(𝑘)²(𝑘).+𝑦 (79)
One could easily see that 𝑎^𝑙−𝐴^𝑢𝑀₂^1−𝑅2𝑟^𝑙=0.1228>0,𝐵^𝑙−𝑑^𝑢𝑟^𝑢=0.37>0. 𝐻₆𝐻₇ Clearly, conditions 𝐻₆ and 𝐻₇ are satisfied. It follows from Theorem 4 that the system is permanent. Numerical simulation from Figure 1 shows that solutions do converge and system is permanent and globally attractive.

Figure 1 

Dynamics behaviors of system (1) with initial conditions (𝑥(0),𝑦(0))=(0.3,0.3),(0.4,0.2),(0.2,0.4), respectively.

6. Conclusion

In this paper, a discrete predator-prey model with Hassell-Varley-Holling III type functional response is discussed. The main topics are focused on permanence, global attractivity, and extinction of predator species. The numerical simulation shows that the main results are verifiable.

The investigation in this paper suggests the following biological implications. Theorem 4 shows that the coefficients, such as the death rate of the predator, the capturing rate, and the intraspecific effects of prey species, influence permanence. Conditions 𝐻₁ and 𝐻₂ imply that the higher the intraspecific effects of prey species are, the more favourable permanence is. Those results have further application on predator-prey population dynamics. However, the conditions for global attractivity in Theorem 4 is so complicated that its application is very difficult. A further study is required to simplify the application.

Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (JA11193).