Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

Abstract

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

1. Introduction

Impulsive equations are found in almost every domain of applied science, such as population dynamics, ecology, biological systems, and optimal control. In recent years, the theory of impulsive differential equations has been an object of active research (see [1–4] and reference therein) since it is much richer than the corresponding theory of differential equations without impulsive effects.

It is well known that continuous-time dynamic systems play an important role in control theory, population dynamics, and so on. But in applications of continuous-time dynamic systems to some practical problems, such as computer simulation, experimental, or computational purposes, it is usual to formulate a discrete-time system which is a version of the continuous-time system. In some sense, the discrete time model inherits the dynamical characteristics of the continuous-time systems. We refer to [4–16] for related discussions of the importance and the need for discrete-time analogs to reflect the dynamics of their continuous-time counterparts. Nevertheless, the discrete-time version can but not always preserve the dynamics of its initial version because the theory of difference equations is a lot richer than the corresponding theory of differential equations as pointed out in [17, 18]. Therefore, it is important to study the dynamics of its initial version alone.

Due to the above facts, we construct the following discrete impulsive Lotka-Volterra predator-prey model concerning integrated pest management by piecewise Euler method: 𝑥_{𝑘𝑇+𝑙+1}=𝑥_{𝑘𝑇+𝑙}exp𝑟−𝑎𝑥_{𝑘𝑇+𝑙}−𝑏𝑦_{𝑘𝑇+𝑙},𝑦_{𝑘𝑇+𝑙+1}=𝑦_{𝑘𝑇+𝑙}exp−𝑑+𝑐𝑥_{𝑘𝑇+𝑙},𝑥_𝑘𝑇+0=1−𝑝₁𝑥_𝑘𝑇,𝑦_𝑘𝑇+0=1−𝑝₂𝑦_𝑘𝑇+𝜇,𝑙=0,1,2,…,𝑇−1,𝑘=0,1,2,…, (1) where 𝑟>0 is the intrinsic growth rate of pest, 𝑎>0 is the coefficient of intraspecific competition, 𝑏>0 is the per-capita rate of predation of the predator, 𝑑>0 is the death rate of predator, 𝑐>0 denotes the product of the per-capita rate of predation and the rate of conversing pest into predator, and 𝑇 is the period of the impulsive effect. 0≤𝑝₁<1(0≤𝑝₂<1) represents the fraction of pest (predator) which dies due to the pesticide, and 𝜇>0 is the release amount of predator at 𝑘𝑇, 𝑘∈ℤ₊. That is, we can use a combination of biological (periodic releasing natural enemies) and chemical (spraying pesticide) tactics that eradicates the pest to extinction and show the efficiency of integrated pest management strategy.

System (1) can be regarded as a discrete analogy of the following impulsive Lotka-Volterra predator-prey model concerning integrated pest management: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑟−𝑎𝑥₁−𝑏𝑥₂,𝑑𝑥₂𝑑𝑡=𝑥₂−𝑑+𝑐𝑥₁,Δ𝑥₁=1−𝑝₁𝑥₁,Δ𝑥₂=1−𝑝₂𝑥₂+𝜇, (2) where Δ𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡⁺)−𝑥_𝑖(𝑡), 𝑖=1,2. Liu et al. [19] discussed the dynamical behavior of model (2).

Recently, the studies of discrete impulsive model have received great attention from more scholar (see [5, 15, 16, 20–22]). The main difficulty for dynamical analysis of such equations comes from impulsive effect on the equations since the corresponding theory for impulsive difference equations have not yet been fully developed. The discrete impulsive model (1) gives a new form of describing the impulsive moment. In some papers, authors use (𝑘𝑇)⁺ to denote impulsive moment (see [20]). It is obvious that describing the impulsive moment of model (1) is easily realized at computer. In addition, some authors use 𝑘𝑇−1 to denote impulsive moment (see [16, 21]). Compared with it, model (1) is a better analogue of the continuous-time dynamic system.

The main aim of this paper is to construct the discrete impulsive model (1) and discuss the dynamical behaviors of the discrete impulsive model (1). We investigate the globally asymptotical stability of pest-eradication periodic solution system (1) and the permanence of system (1).

2. Global Qualitative Analysis for Model (1)

Before our main results, we will give some lemmas which will be useful for our main results. First, we present the Floquent theory for the linear 𝑇-periodic difference equation 𝑢_𝑛+1=𝐴_𝑛𝑢_𝑛, (3) where 𝑢_𝑛∈𝑅^𝑠, 𝑛=0,1,2,…. 𝐴_𝑛 is a real 𝑠×𝑠 matrix whose entries are functions of 𝑛 satisfying 𝐴_𝑛=𝐴_𝑛+𝑇 for a positive integer 𝑇 and is nonsingular; that is, det 𝐴_𝑛≠0 for all 𝑛=0,1,2,… . As usual 𝑀=𝐴_𝑇−1𝐴_𝑇−2⋯𝐴₀ denotes the monodromy of (3).

Lemma 1 (see [5, 6]). Let 𝐵 be an 𝑠×𝑠 nonsingular matrix and let 𝑇 be any positive integer. Then there exists some 𝑠×𝑠 matrix 𝐶, such that 𝐶^𝑇=𝐵.

Lemma 2 (see [5, 6]). If we write 𝑀=𝐷^𝑇, then there exists a change of variable 𝑢_𝑛=𝑆_𝑛𝑦_𝑛, with 𝑆_𝑛+𝑇=𝑆_𝑛 for all 𝑛=0,1,2,…, such that (3) becomes 𝑦_𝑛+1=𝐷𝑦_𝑛. (4)

Lemma 3 (see [5]). Let 𝑎_{𝑘𝑇+𝑙} be a solution of the following impulsive inequality: 𝑎_{𝑘𝑇+𝑙+1}≥𝑎_{𝑘𝑇+𝑙}(),𝑎exp−𝑑_𝑘𝑇+0=1−𝑝₂𝑎_𝑘𝑇𝑎+𝜇,𝑙=0,1,2,…,𝑇−1,₀₊₀=𝑎₀,𝑘=0,1,2,… (5) and let 𝑏_{𝑘𝑇+𝑙} be a solution of the following impulsive inequality 𝑏_{𝑘𝑇+𝑙+1}=𝑏_{𝑘𝑇+𝑙}(),𝑏exp−𝑑_𝑘𝑇+0=1−𝑝₂𝑏_𝑘𝑇𝑏+𝜇,𝑙=0,1,2,…,𝑇−1,₀₊₀=𝑎₀,𝑘=0,1,2,…. (6) Then 𝑎_{𝑘𝑇+𝑙}≥𝑏_{𝑘𝑇+𝑙}, 𝑙=0,1,2,…,𝑇−1.

Proof. First, we know that 𝑎₀₊₀=𝑎₀=𝑏₀₊₀. By induction, we assume that 𝑎_{𝑘𝑇+𝑙}≥𝑏_{𝑘𝑇+𝑙}, 𝑘=0,1,2,…, 𝑙=1,2,…,𝑇−1.
There are two cases.
Case  1. Considering 𝑙≠0, 𝑘=0,1,2,… or 𝑘=0, 𝑙=0,1,2,…,𝑇−1, we have 𝑎_{𝑘𝑇+𝑙+1}≥𝑎_{𝑘𝑇+𝑙}exp(−𝑑)≥𝑏_{𝑘𝑇+𝑙}exp(−𝑑)=𝑏_{𝑘𝑇+𝑙+1}. (7)Case  2. Considering 𝑙=0,  𝑘=1,2,…, we have 𝑎_𝑘𝑇+1≥𝑎_𝑘𝑇+0()=exp−𝑑1−𝑝₂𝑎_𝑘𝑇()≥+𝜇exp−𝑑1−𝑝₂𝑏_𝑘𝑇+𝜇exp(−𝑑)=𝑏_𝑘𝑇+1. (8)
Therefore, for both cases we have 𝑎_{𝑘𝑇+𝑙}≥𝑏_{𝑘𝑇+𝑙}. The proof is complete.

Lemma 4. Suppose (𝑥_{𝑘𝑇+𝑙},𝑦_{𝑘𝑇+𝑙}) is a solution of (1) subject to 𝑥₀>0, 𝑦₀>0; then 𝑥_{𝑘𝑇+𝑙}>0,𝑦_{𝑘𝑇+𝑙}>0 for all 𝑘𝑇+𝑙≥0, 𝑘=0,1,2,…, 𝑙=0,1,2,…,𝑇−1.

2.1. Global Stability of the Pest-Eradication Periodic Solution

Consider the following system: 𝑦_{𝑘𝑇+𝑙+1}=𝑦_{𝑘𝑇+𝑙}(),𝑦exp−𝑑_𝑘𝑇+0=1−𝑝₂𝑦_𝑘𝑇𝑦+𝜇,𝑙=0,1,2,…,𝑇−1,₀₊₀=𝑦₀>0,𝑘=0,1,2,…. (9)

Clearly, 𝑦_{𝑘𝑇+𝑙}=(𝜇/(1−(1−𝑝₂)exp(−𝑑𝑇)))exp(−𝑑𝑙),  𝑙=0,1,2,…,𝑇−1, with 𝑦₀₊₀=(𝜇/(1−(1−𝑝₂)exp(−𝑑𝑇))), is a positive periodic solution of (9). Since the solution of (9) is 𝑦_{𝑘𝑇+𝑙}=(1−𝑝₂)^𝑘(𝑦₀₊₀−(𝜇/(1−(1−𝑝₂)exp(−𝑑𝑇))))exp(−(𝑘𝑇+𝑙)𝑑)+𝑦_{𝑘𝑇+𝑙}, 𝑙=0,1,2,…,𝑇−1, 𝑘=0,1,2,3,…, we have the following lemma.

Lemma 5. Equation (9) has a positive periodic solution 𝑦_{𝑘𝑇+𝑙}, 𝑘=0,1,2,…, 𝑙=0,1,2,…,𝑇−1, and, for every solution 𝑦_{𝑘𝑇+𝑙} of (9), we have 𝑦_{𝑘𝑇+𝑙}→𝑦_{𝑘𝑇+𝑙} as 𝑘→∞.

Therefore, system (1) has a pest-eradication periodic solution: 0,𝑦_{𝑘𝑇+𝑙}=𝜇0,1−1−𝑝₂()()exp−𝑑𝑇exp−𝑑𝑙 (10) and 𝑦₀₊₀=𝜇/(1−(1−𝑝₂)exp(−𝑑𝑇)).

Now we give the conditions which assure the globally stability of the pest-eradication periodic solution (0,𝑦_{𝑘𝑇+𝑙}).

Theorem 6. Let (𝑥_{𝑘𝑇+𝑙},𝑦_{𝑘𝑇+𝑙}) be any solution of (1); then (0,𝑦_{𝑘𝑇+𝑙}) is globally asymptotically stable provided ()𝑟𝑇−𝑏𝜇1−exp−𝑑𝑇1−1−𝑝₂()()1exp−𝑑𝑇1−exp−𝑑<ln1−𝑝₁. (11)

Proof. Firstly, we proved the local stability. The local stability of periodic solution (0,𝑦_{𝑘𝑇+𝑙}) may be determined by considering the behavior of small amplitude perturbations of the solution. Defining 𝑥_{𝑘𝑇+𝑙}=𝑢_{𝑘𝑇+𝑙}, 𝑦_{𝑘𝑇+𝑙}=𝑣_{𝑘𝑇+𝑙}+𝑦_{𝑘𝑇+𝑙}, there may be written 𝑢_{𝑘𝑇+𝑙+1}𝑣_{𝑘𝑇+𝑙+1}=𝑢_{𝑘𝑇+𝑙}exp𝑟−𝑎𝑢_{𝑘𝑇+𝑙}−𝑏𝑣_{𝑘𝑇+𝑙}+𝑦_{𝑘𝑇+𝑙}𝑣_{𝑘𝑇+𝑙}+𝑦_{𝑘𝑇+𝑙}exp−𝑑+𝑐𝑢_{𝑘𝑇+𝑙}. (12) Equation (12) can be expanded in a Taylor series: after neglecting higher-order terms, the linearized equations read as 𝑢_{𝑘𝑇+𝑙+1}=exp𝑟−𝑏𝑦_{𝑘𝑇+𝑙}𝑢_{𝑘𝑇+𝑙},𝑣_{𝑘𝑇+𝑙+1}=𝑦_{𝑘𝑇+𝑙}𝑐exp(−𝑑)𝑢_{𝑘𝑇+𝑙}+exp(−𝑑)𝑣_{𝑘𝑇+𝑙},𝑢_𝑘𝑇+0=1−𝑝₁𝑢_𝑘𝑇,𝑣_𝑘𝑇+0=1−𝑝₂𝑣_𝑘𝑇,𝑙=0,1,2,…,𝑇−1,𝑘=0,1,2,…. (13) Hence 𝐴_{𝑘𝑇+𝑙}=exp𝑟−𝑏𝑦_{𝑘𝑇+𝑙}0𝑦_{𝑘𝑇+𝑙}𝑐exp(−𝑑)exp(−𝑑𝑇); (14) the fundamental solution matrix is𝑀=𝐴_𝑘𝑇+0𝐴_𝑘𝑇+1𝐴_𝑘𝑇+2⋯𝐴_{𝑘𝑇+𝑇−1}1−𝑝₁001−𝑝₂=exp𝑟𝑇−𝑏𝜇1−1−𝑝₂()()exp−𝑑𝑇1−exp−𝑑𝑇()0()1−exp−𝑑∗exp−𝑑𝑇1−𝑝₁001−𝑝₂=1−𝑝₁exp𝑟𝑇−𝑏𝜇1−1−𝑝₂()()exp−𝑑𝑇1−exp−𝑑𝑇()0∗1−exp−𝑑1−𝑝₂().exp−𝑑𝑇 (15)
There is no need to calculate the exact form of (∗) as it is not required in the analysis that follows.
The stability of the periodic solution (0,𝑦_{𝑘𝑇+𝑙}) is determined by the eigenvalues of 𝑀.
Let 𝜆₁,𝜆₂ be eigenvalues of matrix 𝑀. Then 𝜆₁=1−𝑝₁×exp𝑟𝑇−𝑏𝜇1−1−𝑝₂()()exp−𝑑𝑇1−exp−𝑑𝑇(),𝜆1−exp−𝑑₂=1−𝑝₂exp(−𝑑𝑇), (16) according to Lemmas 1 and 2, (0,𝑦_{𝑘𝑇+𝑙}) is locally stable if |𝜆₁|<1 and |𝜆₂|<1. Obviously, 𝜆₂<1. So, (0,𝑦_{𝑘𝑇+𝑙}) is locally stable if |𝜆₁|<1. That is, ()𝑟𝑇−𝑏𝜇1−exp−𝑑𝑇1−1−𝑝₂()()1exp−𝑑𝑇1−exp−𝑑<ln1−𝑝₁. (17)
In the following we prove the global attractivity. Choose 𝜀>0 such that 𝛿≜1−𝑝₁−()×exp𝑟𝑇+𝑏𝜀𝑇𝑏𝜇1−exp−𝑑𝑇1−1−𝑝₂exp(−𝑑𝑇)1−exp(−𝑑)<1. (18)
Noting that 𝑦_{𝑘𝑇+𝑙+1}≥𝑦_{𝑘𝑇+𝑙}exp(−𝑑), consider the following impulsive equation: 𝑧_{𝑘𝑇+𝑙+1}=𝑧_{𝑘𝑇+𝑙}{},𝑧exp−𝑑_𝑘𝑇+0=1−𝑝₂𝑧_𝑘𝑇𝑧+𝜇,𝑙=0,1,2,…,𝑇−1,𝑘=0,1,2,…,₀₊₀=𝑦₀≥0. (19)
By Lemmas 3 and 5, we have 𝑦_{𝑘𝑇+𝑙}≥𝑧_{𝑘𝑇+𝑙}>𝑦_{𝑘𝑇+𝑙}−𝜀, (20) for 𝑘 large enough, so 𝑥_{𝑘𝑇+𝑙+1}≤𝑥_{𝑘𝑇+𝑙}exp𝑟−𝑏𝑦_{𝑘𝑇+𝑙}≤𝑥_{𝑘𝑇+𝑙}exp𝑟−𝑏𝑦_{𝑘𝑇+𝑙}−𝜀, (21) which leads to 𝑥_𝑘𝑇+0=1−𝑝₁𝑥_𝑘𝑇≤1−𝑝₁𝑥_𝑘𝑇−1exp𝑟−𝑏𝑦_𝑘𝑇−1≤−𝜀1−𝑝₁𝑥_𝑘𝑇−2exp𝑟−𝑏𝑦_𝑘𝑇−2𝑦−𝜀×exp𝑟−𝑏_𝑘𝑇−1⋮≤−𝜀1−𝑝₁𝑥_{(𝑘−1)𝑇+0}𝑦×exp𝑇𝑟+𝑏𝑇𝜀−𝑏_{(𝑘−1)𝑇+0}+𝑦_{(𝑘−1)𝑇−1}+⋯+𝑦_𝑘𝑇−1≤1−𝑝₁𝑥_{(𝑘−1)𝑇+0}−(×exp𝑇𝑟+𝑏𝑇𝜀𝑏𝜇1−exp−𝑑𝑇)1−1−𝑝₂()().exp−𝑑𝑇1−exp−𝑑 (22)
Therefore, 𝑥_𝑘𝑇+0≤𝑥₀₊₀𝛿^𝑘 and 𝑥_𝑘𝑇+0→0 as 𝑘→+∞. Moreover, 0<𝑥_{𝑘𝑇+𝑙}≤𝑥_𝑘𝑇+0exp(𝑟𝑇+𝑏𝑇𝜀), so we have 𝑥_{𝑘𝑇+𝑙}→0 as 𝑘→∞.
Next, we prove that 𝑦_{𝑘𝑇+𝑙}→𝑦_{𝑘𝑇+𝑙} as 𝑘→∞ if lim_𝑘→∞𝑥_{𝑘𝑇+𝑙}=0. For 0<𝜀₁<𝑑/𝑐, there exists an 𝐾₀>0 such that 0<𝑥_{𝑘𝑇+𝑙}<𝜀₁ for all 𝑙=0,1,2,…,𝑇−1, and 𝑘>𝐾₀; then from (1) we have 𝑦_{𝑘𝑇+𝑙}exp(−𝑑)<𝑦_{𝑘𝑇+𝑙+1}<𝑦_{𝑘𝑇+𝑙}exp−𝑑+𝑐𝜀₁. (23)
By Lemmas 3 and 5, we obtain 𝑦¹_{𝑘𝑇+𝑙}≤𝑦_{𝑘𝑇+𝑙}≤𝑦²_{𝑘𝑇+𝑙},𝑦¹_{𝑘𝑇+𝑙}⟶𝑦_{𝑘𝑇+𝑙},𝑦²_{𝑘𝑇+𝑙}⟶𝑦²_{𝑘𝑇+𝑙}as𝑘⟶∞, (24) where 𝑦¹_{𝑘𝑇+𝑙} and 𝑦²_{𝑘𝑇+𝑙} are solutions of 𝑦¹_{𝑘𝑇+𝑙+1}=𝑦¹_{𝑘𝑇+𝑙}{},𝑦exp−𝑑_1𝑘𝑇+0=1−𝑝₂𝑦¹_𝑘𝑇𝑦+𝜇,𝑙=0,1,2,…,𝑇−1,𝑘=0,1,2,…,₁₀₊₀=𝑦₀𝑦≥0,_{2𝑘𝑇+𝑙+1}=𝑦²_{𝑘𝑇+𝑙}exp−𝑑+𝑐𝜀₁,𝑦²_𝑘𝑇+0=1−𝑝₂𝑦²_𝑘𝑇𝑦+𝜇,𝑙=0,1,2,…,𝑇−1,𝑘=0,1,2,…,₂₀₊₀=𝑦₀≥0, (25)(26) respectively, and 𝑦²_{𝑘𝑇+𝑙}=(𝜇/(1−(1−𝑝₂)exp((−𝑑+𝑐𝜀₁)𝑇)))exp((−𝑑+𝑐𝜀₁)𝑙),  𝑙=0,1,2,…,𝑇−1.
Therefore, for any 𝜀₂>0 there exists a 𝐾>0 such that 𝑦_{𝑘𝑇+𝑙}−𝜀₂<𝑦_{𝑘𝑇+𝑙}<𝑦²_{𝑘𝑇+𝑙}+𝜀₂. (27)
Let 𝜀₁→0; we have 𝑦²_{𝑘𝑇+𝑙}⟶𝑦_{𝑘𝑇+𝑙}as𝑘⟶∞, (28) which implies 𝑦_{𝑘𝑇+𝑙}→𝑦_{𝑘𝑇+𝑙} as 𝑘→∞. This completes the proof.

2.2. Permanence

Now we investigate the permanence of system (1). For convenience, let 𝑛=𝑘𝑇+𝑙. Firstly, we give the following definition.

Definition 7. System (1) is said to be permanent if there are constants 𝑚,𝑀>0 (independent of initial value) and a finite time 𝑇₀ such that for all solutions (𝑥_𝑛,𝑦_𝑛) with all initial values (𝑥₀₊₀>0,𝑦₀₊₀>0), 𝑚<𝑥_𝑛<𝑀,𝑚<𝑦_𝑛<𝑀 (29) hold for all 𝑛>𝑇₀. Here 𝑇₀ may depend on the initial values (𝑥₀₊₀,𝑦₀₊₀).

Lemma 8. There exist two positive constants 𝑀 and 𝑁, such that for every solution (𝑥_𝑛,𝑦_𝑛) of system (1), we have 𝑥_𝑛<𝑀𝑓𝑜𝑟𝑛>𝑁. (30)

Proof. To prove (30), we have two cases.
Case I. 𝑛≠𝑘𝑇+0, 𝑘=0,1,2,3,…. For any 𝑥₀>0, we have 𝑥_𝑛+1=𝑥_𝑛exp𝑟−𝑎𝑥_𝑛−𝑏𝑦_𝑛≤𝑥_𝑛exp𝑟−𝑎𝑥_𝑛≤exp{𝑟−1}𝑎≜𝑀; (31) here we used max_𝑥≥0{={}𝑥exp𝑏−𝑎𝑥}exp𝑏−1𝑎. (32)
Case II. 𝑛=𝑘𝑇+0, 𝑘=0,1,2,3,…, we have 𝑥_𝑘𝑇+0=1−𝑝₁𝑥_𝑘𝑇=1−𝑝₁𝑥_𝑘𝑇−1exp𝑟−𝑎𝑥_𝑘𝑇−1−𝑏𝑦_𝑘𝑇−1≤1−𝑝₁𝑥_𝑘𝑇−1exp𝑟−𝑎𝑥_𝑘𝑇−1≤1−𝑝₁𝑀<𝑀. (33)
This completes the proof.

In the following, without loss of generality, we assume 𝑀>𝑑/𝑐 is large enough.

Lemma 9. There exists a constant 𝑀₁ such that, for every solution (𝑥_𝑛,𝑦_𝑛) of (1), we have limsup_𝑛→∞𝑦_𝑛<𝑀₁. (34)

Proof. We first prove that there exists a 𝐾>0 such that liminf_𝑛→∞𝑦_𝑛<𝐾. (35) There exist 𝐾, 𝜀₀, 𝑀₂>𝜇/𝑝₂>0 such that 𝜀₀<𝑑𝑐;𝑀exp(𝑟−𝑏𝐾)<𝜀₀𝜇;𝐾>𝑒^𝐵−1+𝑝₂,𝜇1−𝑝₂()1exp𝐴𝑇−1<𝐾+1,𝐴𝑀ln²2(𝐾+1)>3, (36)(37) where 𝐴=−𝑑+𝑐𝑀, 𝐵=𝑑−𝑐𝜀₀. Now we claim that (35) holds. Otherwise, there would exist 𝑁₁∈ℕ and 𝜀>0, such that 𝑦_𝑛>𝐾+𝜀 for 𝑛>𝑁₁. By Lemma 8, there exists an 𝑁₂>𝑁₁ such that 𝑥_𝑛<𝑀,for𝑛>𝑁₂. (38) Then if 𝑛>𝑁₂𝑥_𝑛+1≤𝑥_𝑛((()exp𝑟−𝑏𝐾+𝜀))≤𝑀exp𝑟−𝑏𝐾<𝜀₀. (39)
There are two cases as follows.(1)If 𝑛=𝑘𝑇+0>𝑁₂+1, 𝑦_𝑛+1=𝑦_𝑘𝑇+1≤𝑦_𝑘𝑇+0exp−𝑑+𝑐𝑥_𝑘𝑇+0≤1−𝑝₂𝑦_𝑘𝑇+𝜇exp−𝑑+𝑐𝜀₀≤1−𝑝₂𝑦_𝑛+𝜇exp(−𝐵). (40) We claim that 𝑦_𝑛+1<𝑦_𝑛. Otherwise, 𝑦_𝑛+1≥𝑦_𝑛, so 1−𝑝₂𝑦_𝑛+𝜇exp(−𝐵)≥𝑦_𝑛; (41) hence 𝑦_𝑛≤𝜇𝑒^𝐵−1−𝑝₂<𝐾, (42) which is a contradiction.(2)If 𝑛≠𝑘𝑇+0>𝑁₂+1, 𝑦_𝑛+1≤𝑦_𝑛exp−𝑑+𝑐𝑥_𝑘𝑇+0≤𝑦_𝑛exp−𝑑+𝑐𝜀₀≤𝑦_𝑛exp(−𝐵)<𝑦_𝑛. (43)
Therefore, for both cases, we have 𝑦_𝑛+1<𝑦_𝑛, so lim_𝑛→∞𝑦_𝑛=0, which is a contradiction.
For 𝑀₂ given by (37), in the following we will prove that (34) holds.
Otherwise, limsup_𝑛→∞𝑦_𝑛>𝑀₂. (44)
By (35) and (44), there exist 𝑛₂>𝑛₁>𝑁₂, (45) such that 𝑦_𝑛1≤𝐾+1,𝑦_𝑛2≥𝑀₂,𝐾+1<𝑦_𝑛<𝑀₂for𝑛∈𝑛₁,𝑛₂. (46)(i)If 𝑛₁,𝑛₂ are at the same region ∈[𝑘𝑇,(𝑘+1)𝑇), then 𝑦_𝑛2≤𝑦_𝑛2−1exp(𝐴)≤𝑦_𝑛2−2()⋮exp2𝐴≤𝑦_𝑛1exp𝑛₂−𝑛₁𝐴 (47) so 𝑛₂−𝑛₁≥1𝐴𝑀ln²𝐾+1>3. (48)(ii)If 𝑛₁,𝑛₂ are not at the same region, we assume 𝑛₂∈[𝑘𝑇,(𝑘+1)𝑇) and 𝑛₁∈[𝑘𝑇,(𝑘+1)𝑇). Then 𝑦_𝑛2≤𝑦_𝑛2−1exp(𝐴)≤𝑦_𝑛2−2()⋮≤exp2𝐴1−𝑝₂𝑦_𝑘𝑇exp𝑛₂𝐴𝑛−𝑘𝑇+𝜇exp₂𝐴⋮≤−𝑘𝑇1−𝑝₂²𝑦_{(𝑘−1)𝑇}exp𝑛₂−(𝑘−1)𝑇𝐴+𝜇1−𝑝₂exp𝑛₂−()𝑇𝐴𝑛𝑘−1+𝜇exp₂𝐴⋮≤−𝑘𝑇1−𝑝₂^𝑘𝑘−𝑦_(𝑘+1)𝑇exp𝑛₂−𝑘+1𝑇𝐴+𝜇1−𝑝₂^{𝑘−𝑘−1}exp𝑛₂−𝑘+1𝑇𝐴+𝜇1−𝑝₂^{𝑘−𝑘−2}exp𝑛₂−𝑘+2𝑇𝐴+⋯+𝜇1−𝑝₂²exp𝑛₂−(𝑘−2)𝑇𝐴+𝜇1−𝑝₂exp𝑛₂−()𝑇𝐴𝑛𝑘−1+𝜇exp₂𝐴≤−𝑘𝑇1−𝑝₂^𝑘𝑘−𝑦_𝑛1exp𝑛₂−𝑛₁𝐴+𝜇exp𝑛₂−𝑘𝑇𝐴1−1−𝑝₂exp𝑘−𝑘𝐴𝑇1−1−𝑝₂exp(𝐴𝑇)≤𝑦_𝑛1exp𝑛₂−𝑛₁𝐴+𝜇1−𝑝₂exp𝑛₂−𝑘𝑇𝐴−exp𝑛₂−𝑘𝑇𝐴1−𝑝₂()()𝑛exp𝐴𝑇−1≤2𝐾+1exp₂−𝑛₁𝐴. (49)
So 𝑛₂−𝑛₁≥1𝐴𝑀ln²2(𝐾+1)>3. (50)
Therefore, the set {𝑛∣𝑛₁<𝑛<𝑛₂} is nonempty. So if 𝑛₂−1≠𝑘𝑇+0>𝑁₂+1, then 𝑥_𝑛2−1=𝑥_𝑛2−2exp𝑟−𝑎𝑥_𝑛2−2−𝑏𝑦_𝑛2−2≤𝑥_𝑛2−2exp𝑟−𝑏𝑦_𝑛2−2≤𝑀exp(𝑟−𝑏(𝐾+1))≤𝜀₀. (51)
If 𝑛₂−1=𝑘𝑇+0>𝑁₂+1, then 𝑥_𝑛2−1=𝑥_𝑘𝑇+0=1−𝑝₁𝑥_𝑘𝑇≤1−𝑝₁𝑥_𝑘𝑇−1((((exp𝑟−𝑏𝐾+1))≤𝑀exp𝑟−𝑏𝐾+1))≤𝜀₀. (52)
There are three case as follows.(1)If 𝑛₂=𝑘𝑇+1, 𝑘=1,2,3,… and 𝑀₂>𝜇/𝑝₂, 𝑦_𝑛2=𝑦_𝑘𝑇+1=𝑦_𝑘𝑇+0exp−𝑑+𝑐𝑥_𝑘𝑇+0=1−𝑝₂𝑦_𝑘𝑇+𝜇exp−𝑑+𝑐𝑥_𝑘𝑇+0≤1−𝑝₂𝑦_𝑘𝑇+𝜇exp−𝑑+𝑐𝜀₀≤1−𝑝₂𝑀₂+𝜇exp−𝑑+𝑐𝜀₀≤1−𝑝₂𝑀₂+𝜇≤𝑀₂. (53)(2)If 𝑛₂=𝑘𝑇+0,𝑘=1,2,3,… and 𝑀₂>𝜇/𝑝₂, 𝑦_𝑛2=𝑦_𝑘𝑇+0=1−𝑝₂𝑦_𝑘𝑇=+𝜇1−𝑝₂𝑦_𝑘𝑇−1exp−𝑑+𝑐𝑥_𝑘𝑇−1≤+𝜇1−𝑝₂𝑦_𝑘𝑇−1exp−𝑑+𝑐𝜀₀≤+𝜇1−𝑝₂𝑀₂exp−𝑑+𝑐𝜀₀+𝜇≤𝑀₂. (54)(3)Other cases are as follows: 𝑦_𝑛2=𝑦_𝑛2−1exp−𝑑+𝑐𝑥_𝑛2−1=𝑦_𝑛2−1exp−𝑑+𝑐𝑥_𝑛2−1≤𝑀₂exp−𝑑+𝑐𝜀₀≤𝑀₂, (55) which is a contradiction. This completes the proof.

Theorem 10. Equation (1) is permanence provided that ()𝑟𝑇−𝑏𝜇1−exp−𝑑𝑇1−1−𝑝₂()()1exp−𝑑𝑇1−exp−𝑑>ln1−𝑝₁ (56) holds true.

Proof. Suppose (𝑥_𝑛,𝑦_𝑛) is a solution of (1) with (𝑥₀>0,𝑦₀>0). By Lemmas 8 and 9, we have proved there exists a constant 𝑀>0 such that (𝑥_𝑛<𝑀,𝑦_𝑛<𝑀) for 𝑛 large enough.
From (20), we know 𝑦_𝑛>𝑦_𝑛−𝜀 for all 𝑛 large enough and some 𝜀>0, so 𝑦_𝑛≥(𝜇/(1−(1−𝑝₂)exp(−𝑑𝑇)))exp(−𝑑𝑇)−𝜀≜𝑚₂ for 𝑛 large enough.
Thus, we only need to find 𝑚₁>0 such that 𝑥_𝑛≥𝑚₁ for 𝑛 large enough.
We will do it in the following two steps.
Step I. From condition (56), let 0<𝑚₃<𝑑/𝑐, 𝜀₁>0 be small enough such that 𝜂≜(1−𝑝₁)exp(𝑇(𝑟−𝑎𝑚₃−𝑏𝜀₁)−(𝜇/(1−(1−𝑝₂)exp((−𝑑+𝑐𝑚₃)𝑇)))(𝑏(1−exp((−𝑑+𝑐𝑚₃)𝑇))/(1−exp(−𝑑+𝑐𝑚₃))))>1; we will prove that 𝑥_𝑛<𝑚₃ cannot hold for all 𝑛>0. Otherwise, 𝑦_𝑛+1≤𝑦_𝑛exp−𝑑+𝑐𝑚₃,𝑦_𝑘𝑇+0=1−𝑝₂𝑦_𝑘𝑇+𝜇. (57)
So we have 𝑦_𝑛≤𝑧_𝑛 and 𝑧_𝑛→𝑧_𝑛,  𝑛→∞, where 𝑧_𝑛 is the solution of 𝑧_𝑛+1=𝑧_𝑛exp−𝑑+𝑐𝑚₃,𝑧_𝑘𝑇+0=1−𝑝₂𝑧_𝑘𝑇𝑧+𝜇,₀₊₀=𝑦₀₊₀, (58) and 𝑧_𝑛=(𝜇/(1−(1−𝑝₂)exp((−𝑑+𝑐𝑚₃)𝑇)))exp((−𝑑+𝑐𝑚₃)𝑙),  𝑙=0,1,2,…,𝑇−1, with 𝑦₀₊₀=𝜇/(1−(1−𝑝₂)exp((−𝑑+𝑐𝑚₃)𝑇)), is a positive periodic solution of (58).
Therefore, there exists a 𝑁₁>0 such that 𝑦_𝑛≤𝑧_𝑛<𝑧_𝑛+𝜀₁,𝑥_𝑛+1≥𝑥_𝑛exp𝑟−𝑎𝑚₃−𝑏𝑧_𝑛+𝜀₁,𝑥_𝑘𝑇+0=1−𝑝₁𝑥_𝑘𝑇 (59)(60) for 𝑛>𝑁₁. Therefore there exists a 𝐾₁, such that if 𝑘>𝐾₁𝑥_{𝑘𝑇+𝑇}≥𝑥_{𝑘𝑇+𝑇−1}exp𝑟−𝑎𝑚₃−𝑏𝑧_{𝑘𝑇+𝑇−1}+𝜀₁≥𝑥_{𝑘𝑇+𝑇−1}exp𝑟−𝑎𝑚₃−𝑏𝜀₁exp−𝑏𝑧_{𝑘𝑇+𝑇−1}≥𝑥_{𝑘𝑇+𝑇−2}exp2𝑟−𝑎𝑚₃−𝑏𝜀₁×exp−𝑏𝑧_{𝑘𝑇+𝑇−1}+𝑧_{𝑘𝑇+𝑇−2}≥⋮≥𝑥_𝑘𝑇+0exp𝑇𝑟−𝑎𝑚₃−𝑏𝜀₁×exp−𝑏𝑧_{𝑘𝑇+𝑇−1}+𝑧_{𝑘𝑇+𝑇−2}+⋯+𝑧_𝑘𝑇+0)≥1−𝑝₁𝑥_𝑘𝑇×exp𝑇𝑟−𝑎𝑚₃−𝑏𝜀₁−𝜇1−1−𝑝₂exp−𝑑+𝑐𝑚₃𝑇×𝑏1−exp−𝑑+𝑐𝑚₃𝑇1−exp−𝑑+𝑐𝑚₃=𝑥_𝑘𝑇𝜂, (61) then 𝑥_{(𝑘+𝐾1)𝑇}≥𝑥_𝐾1𝑇𝜂^𝑘→∞ as 𝑘→∞, which is a contradiction to the boundedness of 𝑥_{𝑘𝑇+𝑙}. Hence there exists a 𝑛₁>0 such that 𝑥_𝑛1≥𝑚₃.
Step II. If 𝑥_𝑛≥𝑚₃ for all 𝑛≥𝑛₁, then our aim is achieved. Otherwise, 𝑥_𝑛<𝑚₃ for some 𝑛>𝑛₁. Set 𝑛^∗=min_𝑛>𝑛1{𝑛∣𝑥_𝑛<𝑚₃}. We have 𝑥_𝑛≥𝑚₃ for 𝑛∈[𝑛₁,𝑛^∗).
Assume that 𝑛^∗∈[𝑘𝑇,(𝑘+1)𝑇). It is easy to see that 𝑥_𝑛∗<𝑚₃, 𝑥_𝑛∗−1≥𝑚₃.
There are two possible cases for 𝑛∈[𝑛^∗,(𝑘+1)𝑇). Here, for 𝑀 large enough, we have 𝜂₁=𝑟−𝑎𝑚₃−𝑏𝑀<0 and 𝜂₂=𝑟−𝑎𝑀−𝑏𝑀<0.
Case (a). There exists a 𝑛^′′∈[𝑛^∗,(𝑘+1)𝑇), such that 𝑥_𝑛′′≥𝑚₃. Let 𝑛=min_𝑛>𝑛∗{𝑛∣𝑥_𝑛≥𝑚₃}; then 𝑥_𝑛<𝑚₃ for 𝑛∈[𝑛^∗,𝑛) and 𝑥_𝑛≥𝑚₃.
For 𝑛∈[𝑛^∗,𝑛−1), 𝑥_𝑛+1≥𝑥_𝑛exp𝜂₁⋮≥𝑥_𝑛∗exp𝑛+1−𝑛^∗𝜂₁. (62)
If 𝑛^∗≠𝑘𝑇+0, then 𝑥_𝑛∗=𝑥_𝑛∗−1exp𝑟−𝑎𝑥_𝑛∗−1−𝑏𝑦_𝑛∗−1≥𝑚₃exp𝜂₂,𝑥_𝑛+1≥𝑥_𝑛∗exp𝑛+1−𝑛^∗𝜂₁≥𝑚₃exp𝜂₂exp𝜂₁≥1−𝑝₁𝑚₃exp𝜂₂exp𝜂₁. (63)
If 𝑛^∗=𝑘𝑇+0, then 𝑘𝑇+0>𝑛₁.
Case (1). Consider 𝑘𝑇=𝑛₁, 𝑥_𝑛∗=1−𝑝₁𝑥_𝑘𝑇≥1−𝑝₁𝑚₃≥1−𝑝₁𝑚₃exp𝜂₂. (64)
Case (2). Consider 𝑘𝑇>𝑛₁, 𝑥_𝑛∗=1−𝑝₁𝑥_𝑘𝑇≥1−𝑝₁𝑥_𝑘𝑇−1exp𝑟−𝑎𝑥_𝑘𝑇−1−𝑏𝑦_𝑘𝑇−1≥1−𝑝₁𝑚₃exp𝜂₂,𝑥_𝑛+1≥𝑥_𝑛∗exp𝑛+1−𝑛^∗𝜂₁≥1−𝑝₁𝑚₃exp𝜂₂exp𝜂₁≥1−𝑝₁𝑚₃exp𝜂₂exp𝜂₁. (65)
Let 𝑚₁=(1−𝑝₁)𝑚₃exp(𝜂₂)exp(𝜂₁). So we have 𝑥_𝑛≥𝑚₁ for 𝑛∈[𝑛^∗,𝑛−1). For 𝑛≥𝑛 the same arguments can be continued since 𝑥_𝑛≥𝑚₃.
Case (b). There does not exist a 𝑛^′′∈[𝑛^∗,(𝑘+1)𝑇), such that 𝑥_𝑛′′≥𝑚₃; namely, 𝑥_𝑛<𝑚₃ for 𝑛∈[𝑛^∗,(𝑘+1)𝑇).
Choose 𝑛₂,𝑛₃∈ℤ⁺ such that 𝑛₂1𝑇>−𝑑+𝑐𝑚₃𝜀ln^′1,𝜂𝑀+𝜇exp₂1−𝑝₁^𝑛2+1exp𝜂₁𝑛₂+1𝑇𝜂^𝑛3₁>1. (66)
Let 𝑇^′=(𝑛₂+𝑛₃)𝑇. We claim that there must be a 𝑛∈[(𝑘+1)𝑇,(𝑘+1)𝑇+𝑇^′), such that 𝑥_𝑛≥𝑚₃. Otherwise, 𝑥_𝑛<𝑚₃, 𝑛∈[(𝑘+1)𝑇,(𝑘+1)𝑇+𝑇^′). Consider (58) with 𝑧_{(𝑘+1)𝑇+0}=𝑦_{(𝑘+1)𝑇+0}; we have 𝑧_{𝑚𝑇+𝑙}=𝑧_{(𝑘+1)𝑇+0}−𝜇1−1−𝑝₁exp−𝑑+𝑐𝑚₃𝑇×exp−𝑑+𝑐𝑚₃(𝑚𝑇+𝑙−(𝑘+1)𝑇)+𝑧_{𝑚𝑇+𝑙} (67) for 𝑚𝑇+𝑙∈(𝑛𝑇,(𝑛+1)𝑇) and 𝑘+1≤𝑛≤𝑘+1+𝑛₂+𝑛₃. Then for 𝑚∈(𝑘+1+𝑛₂,𝑘+1+𝑛₂+𝑛₃), 𝑧_{𝑚𝑇+𝑙}−𝑧_{𝑚𝑇+𝑙}<𝑀+𝜇exp−𝑑+𝑐𝑚₃(()𝑇)≤𝑚𝑇+𝑙−𝑘+1𝑀+𝜇exp−𝑑+𝑐𝑚₃(()𝑇)≤𝑚𝑇−𝑘+1𝑀+𝜇exp−𝑑+𝑐𝑚₃𝑛₂𝑇<𝜀^′₁,𝑦_{𝑚𝑇+𝑙}≤𝑧_{𝑚𝑇+𝑙}≤𝑧_{𝑚𝑇+𝑙}+𝜀^′₁, (68) which implies that (60) holds for 𝑚∈(𝑘+1+𝑛₂,𝑘+1+𝑛₂+𝑛₃).
So as in Step I, we have 𝑥_{(𝑘+1+𝑛2+𝑛3)𝑇}≥𝑥_{(𝑘+1+𝑛2)𝑇}𝜂^𝑛3. (69)
Since 𝑥_𝑛<𝑚₃,𝑛∈[𝑛^∗,(𝑘+1)𝑇), then 𝑥_𝑛<𝑚₃ for all 𝑛∈[𝑛^∗,((𝑘+1)+𝑛₂)𝑇].
So, 𝑥_𝑛+1≥𝑥_𝑛exp𝜂₁,𝑥_{((𝑘+1)+𝑛2)𝑇}≥𝑥_{((𝑘+1)+𝑛2)𝑇−1}exp𝜂₁≥⋮≥𝑥_𝑛∗1−𝑝₁^𝑛2exp𝜂₁𝑘+1+𝑛₂𝑇−𝑛^∗. (70)(71)
If 𝑛^∗≠𝑘𝑇+0, then 𝑥_{((𝑘+1)+𝑛2)𝑇}=𝑥_𝑛∗−1exp𝑟−𝑎𝑥_𝑛∗−1−𝑏𝑦_𝑛∗−1×1−𝑝₁^𝑛2exp𝜂₁𝑘+1+𝑛₂𝑇−𝑛^∗≥𝑚₃exp𝜂₂1−𝑝₁^𝑛2exp𝜂₁𝑘+1+𝑛₂𝑇−𝑛^∗≥𝑚₃exp𝜂₂1−𝑝₁^𝑛2exp𝜂₁𝑛₂+1𝑇. (72)
If 𝑛^∗=𝑘𝑇+0, then 𝑘𝑇+0>𝑛₁𝑥_{((𝑘+1)+𝑛2)𝑇}=𝑥_𝑛∗1−𝑝₁^𝑛2+1exp𝜂₁𝑘+1+𝑛₂𝑇−𝑛^∗=𝑥_𝑛∗−1exp𝑟−𝑎𝑥_𝑛∗−1−𝑏𝑦_𝑛∗−1×1−𝑝₁^𝑛2+1exp𝜂₁𝑘+1+𝑛₂𝑇−𝑛^∗≥𝑚₃exp𝜂₂1−𝑝₁^𝑛2+1exp𝜂₁𝑘+1+𝑛₂𝑇−𝑛^∗≥𝑚₃exp𝜂₂1−𝑝₁^𝑛2+1exp𝜂₁𝑛₂+1𝑇. (73)
So, 𝑥_{(𝑘+1)+𝑛2+𝑛3}≥𝑚₃exp𝜂₂1−𝑝₁^𝑛2+1×exp𝜂₁𝑛₂+1𝑇𝜂^𝑛3₁>𝑚₃, (74) which is a contradiction.
Let 𝑛=min_𝑛>𝑛∗{𝑛∣𝑥_𝑛≥𝑚₃}. Then 𝑥_𝑛≥𝑚₃, 𝑥_𝑛−1<𝑚₃. For 𝑛∈[𝑛^∗,𝑛−1], (70) holds. Suppose 𝑛∈[(𝑘+1)𝑇+(𝑘^′−1)𝑇,(𝑘+1)𝑇+𝑘^′𝑇)], 𝑘^′≤𝑛₂+𝑛₃.
We have 𝑥_𝑛≥𝑥_𝑛1exp𝜂₁⋮≥𝑥_𝑛∗1−𝑝₁^𝑘′exp𝑛−𝑛^∗𝜂₁. (75)
If 𝑛^∗≠𝑘𝑇+0, then 𝑥_𝑛≥𝑚₃exp𝜂₂1−𝑝₁^𝑘′exp𝑛−𝑛^∗𝜂₁≥𝑚₃exp𝜂₂1−𝑝₁^𝑘′exp𝑛₂+𝑛₃+1𝑇𝜂₁. (76)
If 𝑛^∗=𝑘𝑇+0, then 𝑘𝑇+0>𝑛₁𝑥_𝑛≥𝑚₃exp𝜂₂1−𝑝₁^𝑘′+1exp𝑛−𝑛^∗𝜂₁≥𝑚₃exp𝜂₂1−𝑝₁^𝑘′+1exp𝑛₂+𝑛₃+1𝑇𝜂₁. (77) Let 𝑚^′₁=𝑚₃exp(𝜂₂)(1−𝑝₁)^{𝑛2+𝑛3+1}exp((𝑛₂+𝑛₃+1)𝜂₁𝑇)<𝑚₁. So 𝑥_𝑛>𝑚^′₁ for [𝑛^∗,𝑛−1]. For 𝑛>𝑛−1, the same arguments can be continued since 𝑥_𝑛≥𝑚₃.
The proof is completed.

Remark 11. Let 𝑓(𝑇)()=𝑟𝑇−𝑏𝜇1−exp−𝑑𝑇1−1−𝑝₂()()1exp−𝑑𝑇1−exp−𝑑−ln1−𝑝₁. (78) Since 𝑓(0)=−ln(1/(1−𝑝₁))<0, 𝑓(𝑇)→∞ as 𝑇→∞ and 𝑓^′′(𝑇)=𝑏𝜇𝑝₂𝑑²()(1−exp−𝑑exp−𝑑𝑇)1+1−𝑝₂exp(−𝑑𝑇)1−1−𝑝₂exp(−𝑑𝑇)³>0, (79) so 𝑓(𝑇)=0 has a unique positive root, denoted by 𝑇_max. From Theorem 6 we know that the pest-eradication periodic solution is globally stable when 𝑇<𝑇_max. From Theorem 10, system (1) is permanent if 𝑇>𝑇_max.

3. Numerical Experiments

In Figure 1, we choose parameters of system (1) as 𝑑=0.4, 𝑎=0.4, 𝑏=0.9, 𝑐=0.6, 𝑟=0.2, 𝑝₁=0.8, 𝑝₂=0.4, 𝜇=0.2, 𝑇=10, and initial value 𝑥₀=1, 𝑦₀=1. It is easy to verify that condition (11) holds, and, by Theorem 6, the pest-eradication periodic solution of (1) is the global stability.

(a) Time-series of the pest population

(b) Time-series of the predator population

Figure 1 

Dynamical behavior of the pest-predator system of (1).

In Figure 2, we choose parameters of system (1) as 𝑑=0.4, 𝑎=0.4, 𝑏=1, 𝑐=0.6, 𝑟=0.3, 𝑝₁=0.2, 𝑝₂=0.01, 𝜇=0.2, 𝑇=10, and initial value 𝑥₀=1, 𝑦₀=1. It is easy to verify that condition (56) holds, and, by Theorem 10, (1) is permanence.

(a) Time-series of the pest population

(b) Time-series of the predator population

Figure 2 

Dynamical behavior of the pest-predator system of (1).

4. Conclusion

In this paper, by piecewise Euler method, we construct a discrete impulsive Lotka-Volterra predator-prey model concerning integrated pest management. The discrete impulsive model gives a new form of describing the impulsive moment. On the other hand, model (1) is a better analogue of the continuous-time impulsive dynamic system. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value and the discrete system is permanence if the impulsive period is larger than some critical value. By (78), the impulsive period critical value 𝑇_max can be obtained. Since 𝑇_max is a direct function with respect to 𝑝₁, 𝑝₂, and 𝜇, in order to obtain the object of integrated pest management, we can determine the impulsive period 𝑇 according to effect of the chemical pesticides on the populations and cost of the releasing natural enemies.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (DL12BB23) and the China Postdoctoral Science Foundation (415220).