Dynamics of a Stage Structured Pest Control Model in a Polluted Environment with Pulse Pollution Input

By using pollution model and impulsive delay differential equation, we formulate a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments and investigate the dynamics of such a system. We assume only that the natural enemies are affected by pollution, and we choose the method to kill the pest without harming natural enemies. Sufficient conditions for global attractivity of the natural enemy-extinction periodic solution and permanence of the system are obtained. Numerical simulations are presented to confirm our theoretical results.


Introduction
Nowadays, the problem of the world's environmental pollution is serious, which has a frustrating effect on the ecosystem damage in the direct or indirect ways.Pollution leads to the living environmental change and gene mutation.It results in not only birth defects and deformities but also population variability, which decreases the number of the population in the nature and even makes them extinct.In order to assess the risk of the populations exposed to a polluted environment, in recent years, mathematical models concerning this topic have been studied extensively including continuous pollution input and impulsive pollution input [1][2][3][4][5][6][7][8][9][10][11].
As we all know, the predator-prey system can be used to model the process of controlling the pests by spraying pesticides, as well as relying on their natural enemies.However, in a polluted environment, some natural enemies are affected by pollution seriously and pests almost are not affected.For example, frogs are the natural enemies of beetles, locusts, and mole cricket, but some chemical plants discard waste products into rivers for their convenience, which cause severe water contamination, seriously injures frog's reproductive system, and significantly decreases their fertility.Moreover, water pollution also causes large quantities of the fertilized eggs and tadpoles to die, resulting in the decrease of frogs.It is shown in a Sweden's new study that male tadpoles can eventually grow into female frogs only in the environment similar to the nature but full of pollutants with estrogen.However, some male frogs have ovaries but no fallopian tubes, and they finally turn into lifelong infertile frogs, which are called "Yin and Yang frog", and nearly one-third of the world's frog species may be extinct because of the environmental pollution.People must control the period and quantity of emission of pollution to prevent natural enemy from extinction.In addition, too much pesticide spraying will reduce pests significantly; meanwhile, it also causes serious environmental pollution.Therefore, when controlling pests, we had better choose the method to kill the pests without polluting the environment and harming natural enemies at regular intervals.
The predator-prey models with stage structure for the predator were introduced or investigated by Hastings and Wang [12][13][14].Since the immature predator takes  (which is called maturation time delay) units of time to mature, the death toll during the juvenile period should be considered, and time delays have important biological meanings in stage structured models.Recently, many models with time delay were extensively studied [15][16][17][18][19][20][21][22].
According to the above biological background, in this paper, we suggest an impulsive predator-prey pollution model with stage structured for predator by introducing a constant periodic pollutant input and proportional killing pest at different fixed moments to model the process of pest control and polluted environment.Recently, there has been quite a lot of literatures on the applications of impulsive differential equations on population [1,2,8,10,11,[20][21][22][23][24][25][26][27][28][29][30][31].To our knowledge, there have been no results on this topic in the literature.The questions that arise here are as follows: how do we control the emission of pollution to prevent the extinction of natural enemies?Under what condition can the system be permanent?How can we control pests effectively?
The organization of this paper is as follows.In the next section, we formulate our model and give several lemmas which are useful for our main results.In Section 3 and Section 4, the sufficient conditions for the global attractivity of the "natural enemy-extinction" periodic solution and permanence of the system are obtained.We give a brief discussion of our results in Section 5. Numerical simulations are presented to illustrate our theoretical results.

Model Formulation and Preliminaries
In this paper, we assume only that the natural enemies are affected by pollution and we choose the method to kill the pest without harming natural enemies.Then a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pests at different fixed moment is formulated as follows: where 0 ≤  ≤ The other parameters can be seen in [1].
Proof.Since  1 < 1, we have By Lemma 2, for sufficiently small enough  1 > 0, there exists a positive constant  1 such that ( Then we consider the following comparison system: According to Lemma 4, we know that is a unique globally asymptotically stable positive -periodic solution of system (17).By using comparison theorem of impulsive differential equation, there exist a positive integer  2 and a sufficiently small positive constant  2 such that for all  <  ≤ ( + 1),  >  2 , holds.From ( 15), (19), and the second equation of (2), we obtain that for  >  + , holds.

Discussion
In this paper, we discuss a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments.From Theorems 7, 9, and 10, we can observe that the extinction and permanence of the population are very much dependent on , , and .
To verify the theoretical results obtained in this paper, in the following we will give some numerical simulations and take  = 0.9;  = 0.8;  = 0.8;  = 0.4;  = 0.5;  = 0.8;  = 0.3;  = 0.1;  = 0.5;  2 = 0.2;  1 = 0.1;  = 0.1;  = 0.2; ℎ = 2;  = 0.2;  = 0.4;  = 1 (see Figure 1), and here we can compute  2 = 1.005252 > 1, and from Theorem 10 we know the system (2) is permanent.If we decrease the period of pulsing  = 0.3 ( 1 = 0.755095 < 1) or increase the pollution input amount to  = 0.8 ( 1 = 0.996079 < 1), and other parameters are the same with those in Figure 1, the natural enemy will be extinct (see Figures 2 and 3).If we increase the harvesting rate of pests to  = 0.5, and other parameters are the same with those in Figure 1, then  1 = 0.413158 < 1, and the natural enemies will also be extinct (see Figure 4).Our results indicate that if impulsive period  is short or  or  is too large, the natural enemy will go extinct, but we wish to protect natural enemy from extinction, so we should harvest the pests reasonably and control the period and quantity of emission of pollution into the environment efficiently.This offers us some reasonable suggestions for pest management.