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

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.


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][2][3][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][5][6][7][8][9][10][11][12][13][14][15][16] for related discussions of the importance and the need for discretetime 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: +0 = (1 −  2 )   + ,  = 0, 1, 2, . . .,  − 1,  = 0, 1, 2, . . ., (1) where  > 0 is the intrinsic growth rate of pest,  > 0 is the coefficient of intraspecific competition,  > 0 is the per-capita rate of predation of the predator,  > 0 is the death rate of predator,  > 0 denotes the product of the per-capita rate of predation and the rate of conversing pest into predator, and  is the period of the impulsive effect.0 ≤  1 < 1 (0 ≤  2 < 1) represents the fraction of pest (predator) which dies due to the pesticide, and  > 0 is the release amount of predator at ,  ∈ 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.
Recently, the studies of discrete impulsive model have received great attention from more scholar (see [5,15,16,[20][21][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 () + 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  − 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).(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 -periodic difference equation
Lemma 1 (see [5,6]).Let  be an × nonsingular matrix and let  be any positive integer.Then there exists some  ×  matrix , such that   = .
There are two cases.
Therefore, system (1) has a pest-eradication periodic solution: and ).Now we give the conditions which assure the globally stability of the pest-eradication periodic solution (0, ỹ+ ).Theorem 6.Let ( + ,  + ) be any solution of (1); then (0, ỹ+ ) is globally asymptotically stable provided ) . ( Proof.Firstly, we proved the local stability.The local stability of periodic solution (0, ỹ+ ) may be determined by considering the behavior of small amplitude perturbations of the solution.Defining  + =  + ,  + = V + + ỹ+ , there may be written Equation ( 12) can be expanded in a Taylor series: after neglecting higher-order terms, the linearized equations read as Hence the fundamental solution matrix is 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, ỹ+ ) is determined by the eigenvalues of .
Let  1 ,  2 be eigenvalues of matrix .Then according to Lemmas 1 and 2, (0, ỹ+ ) is locally stable if ) . ( In the following we prove the global attractivity.Choose  > 0 such that Noting that  ++1 ≥  + exp(−), consider the following impulsive equation: By Lemmas 3 and 5, we have for  large enough, so which leads to . . .
Lemma 8.There exist two positive constants  and , such that for every solution (  ,   ) of system (1), we have Proof.To prove (30), we have two cases.
In the following, without loss of generality, we assume  > / is large enough.Lemma 9.There exists a constant  1 such that, for every solution (  ,   ) of (1), we have Proof.We first prove that there exists a  > 0 such that lim inf There exist ,  0 ,  2 > / 2 > 0 such that where  = − + ,  =  −  0 .Now we claim that (35) holds.Otherwise, there would exist  1 ∈ N and  > 0, such that   >  +  for  >  1 .By Lemma 8, there exists an  2 >  1 such that There are two cases as follows. ( We claim that  +1 <   .Otherwise,  +1 ≥   , so hence which is a contradiction. ( Therefore, for both cases, we have  +1 <   , so lim  → ∞   = 0, which is a contradiction. For  2 given by (37), in the following we will prove that (34) holds.
holds true.
Thus, we only need to find  1 > 0 such that   ≥  1 for  large enough.
We will do it in the following two steps.

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  max can be obtained.Since  max is a direct function with respect to  1 ,  2 , and , in order to obtain the object of integrated pest management, we can determine the impulsive period  according to effect of the chemical pesticides on the populations and cost of the releasing natural enemies.