Dynamic Analysis of General Integrated Pest Management Model with Double Impulsive Control

A general predator-prey model with disease in the prey and double impulsive control is proposed and investigated for the purpose of integrated pest management. By using the Floquet theory, the comparison theorem of impulsive differential equations, and the persistence theory of dynamical systems, we obtain that if threshold value R 0 < 1, then the susceptible pest eradication periodic solution is globally asymptotically stable and if R 0 > 1, then the model is permanent. The numerical examples not only illustrate the theoretical results, but also show that when the model is permanent, then it may possess a unique globally attractive T-periodic solution.


Introduction
Integrated pest management (IPM) is a long term management tactic that uses a combination of chemical, biological strategies to reduce pests to tolerable level or below the threshold, with less cost to the farmers and minimal effect on the environment (see [1,2]).Such techniques include mechanical methods (erecting pest barriers or using pest traps) and biological methods (breeding natural predators of the pest or using biological insecticides).Some successful biological control examples contain the use of the predatory arthropod Orius sauteri against the pest Thrips palmi Karny to protect eggplant crops in greenhouses (see [3]) and the use of the predatory mites Phytoseiulus persimilis and Neoseiulus californicus to regulate the red spider mite Tetranychus urticae Koch in field-grown strawberries (see [4]).
Many scholars have been devoted to the analysis of impulsive differential equation models describing IPM strategies and some rich results have been obtained (see [6][7][8][9][10][11][12][13][14][15]17]).They assumed that the disease incidence rate should be distinguished; as far as disease transmission is concerned, nonlinear, bilinear, and standard incidence rates have often been used in establishing ecoepidemic models, which depends on different infective disease and environment.Georgescu and Zhang (see [10]) investigated a predator-pest model with incidence rate given by (), Pang and Chen (see [12]) discussed an  model with bilinear incidence rate , Wang et al. (see [13]) analyzed an  model with incidence rate given by (), and so forth.Main results of these theses have focused on conditions of pest eradication and permanence of the system.According to the authors' knowledge, at present stage, there are few studies of general incidence rate.So one of the goals of this paper is to generalize the incidence rate.
The functional response between pests and natural enemies plays an important role in assessing dynamical behavior of the system.People use natural enemy, as in some sense like a pesticide, to control pest via augmentation or releasing natural enemy once the quantity of pest has reached or exceeded the economic threshold (see [9,10,14,15]).Shi et al.
(see [14]) analyzed a predator-pest model with disease in the pest and functional response given by Holling-II type and the time-dependent impulsive strategy including release of infective pest individuals and those natural predators at different point in time; the threshold on pest eradication was obtained.However, little of paper has been devoted to analysis of models which combine release of infective pest individuals and those natural predators.The approach to biological control which we adopted is to release both infective pest individuals and natural predators periodically with constant amount at different point in time; what is more, the functional response is also a more general form.Motivated by the valuable contributions of Georgescu and Zhang [10], Wang et al. [13], and Shi et al. [14], in this paper general IPM model will be considered as follows: The model is based on the following assumptions: (H 1 ) The pest is divided into the susceptible and the infective, and the infective cannot produce offsprings as a result of the disease, but the infective still consume crop.The incidence rate of the infective is given by function (, ).The growth rate of the susceptible is assumed to function (, ).(H 3 ) The natural enemy only hunts the susceptible and the functional response is given by function ℎ(, );  > 0 is the conversion rate.
(H 4 ) Positive constants  and  are the death rates of the infective pest and the natural enemy, respectively.
In model (1), () and () denote the density of the susceptible pest and the infective pest (prey) population, respectively.() is the density of natural enemy (predator) population.For model (1), in this paper we will investigate global stability of the susceptible pest eradication periodic solution and the permanence of model (1).In Section 2, the positivity and boundedness of solutions are presented.In Section 3, by using the Floquet theory for impulsive differential equations, the theorem on the global asymptotic stability of the susceptible pest eradication periodic solution is established.In Section 4, by using the persistence theory of dynamical systems, the theorem on the permanence of model ( 1) is established.In Section 5, we will give the numerical simulations to illustrate the main results obtained in this paper.Finally, in last section a brief discussion and some possible future researches are proposed.
In the following, we introduce some necessary definitions and lemma on the persistence of dynamical systems, which will be used for the discussion of permanence of model (1).For more details, see [19,20].

Global Stability of Susceptible Pest Eradication Periodic Solution
From Lemma 3, we know that model (1) has a susceptible pest eradication periodic solution (0,  * (),  * ()).On the global asymptotic stability of this periodic solution, we have the following theorem.

Corollary 7. When the right-hand functions in model
then condition ( 13) is equivalent to the following form: where , , , , , and  are positive constants.
Remark 8.In (31), if  1 = 0, then  2 > /, which means that if only natural enemies are released periodically, then the release amount must be larger than / to ensure the eradication of the pest.If  2 = 0, then the release amount must satisfy the inequality to ensure the eradication of the pest.

Permanence of the Model
Theorem 9. Assuming that then model ( 1) is permanent.
Proof.Since the impulsive effects in model ( 1) are periodic, model (1) can be regarded as periodic model with period .Therefore, we can use the persistence theory of dynamical systems to discuss the permanence of model (1).Define Thus From Lemma 1, we claim that  and  0 are positively invariant with respect to model (1). 0 is a relatively closed set in .
Corollary 10.When functions (, ), (, ), and ℎ(, ) are given in (30), then condition (33) is equivalent to the following condition: Remark 11.In (50), if  1 = 0, then we have  2 < /, which means that if only natural enemies are released periodically and the amount is less than /, then the system is permanent and the pest will not be eradicated.If  2 = 0, then the release amount satisfies to ensure the system is permanent and the pest will not be eradicated.
Remark 12. Applying Theorem 1 given in [16], it is clear that when condition (33) holds, model ( 1) at least has one positive -periodic solution.
From the above example, we can guess that only inequality (33) holds; then model (1) has a unique positive T-periodic solution which is globally attractive.

Discussion
In this paper, a general ecoepidemic model with impulsive control strategy is proposed and its dynamical behavior is analyzed for the purpose of integrated pest management.Meanwhile, the model which the researchers obtained in [14] was generalized.By using Floquet theorem and theory of