Optimal Control Policies of Pests for Hybrid Dynamical Systems

and Applied Analysis 3 0 2 4 6 8 10 12 14 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 2 4 6 8 10 12 14 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 5 10 15 20 25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65


Introduction
A pest is an insect which is detrimental to humans or human concerns (as agriculture or livestock production).In its broadest sense, a pest is a competitor of humanity.Often insects are regarded as pests as they cause damage to agriculture by feeding on crops or parasitizing livestock, such as codling moth on apples or boll weevil on cotton.An animal could also be a pest when it causes damage to a wild ecosystem or carries germs within human habitats.Examples of these include those organisms which are vector-borne human diseases, such as rats and fleas which carry the plague disease, mosquitoes which are vector-borne malaria, and ticks which carry Lyme disease.The most serious pests (in the order of economic importance) are insects.Pesticides are chemicals and other agents (e.g., beneficial microorganisms) that are used to control or protect other organisms from insect pests.To control these insect pests, farmers rely strongly on intervention with chemical pesticides, which remain a significant component of the cost of production and ecological problems from pesticide resistance in key pests.In order to address the issue, researchers are increasingly embracing more components of the integrated pest management (IPM) [1][2][3][4][5][6] systems approach that is always applied in ecology.
With the rise of interdisciplinary research, the mathematical ecology has also emerged and developed rapidly.A variety of mathematical methods can be used in ecological science.There have been numerous publications [7][8][9][10][11][12][13][14][15] over the last ten years using ecological mathematical model to research IPM strategy (spraying pesticides and introducing additional natural enemy into a pest-natural enemy system).When we study the dynamic property between the pest and its natural enemy (predator-prey), one of the most important components of the predator-prey relationship is the socalled functional responses.In [7][8][9][10][11][12][13][14][15], the Holling functional responses and Beddington-DeAngelis functional response are introduced.The Holling type extends the range of values of  and  over which the feeding term is realistic.However, in some situations, the increase of the feeding rate is not proportional to the increase of the predator density, as a result of mutual interference between predators, which decreases the efficiency of predation [16].In addition, it is shown that most of the mathematical models on IPM include only one pest and one natural enemy.In [1], Finch and Collier's study concerning IPM strategies in field vegetable crops focuses on two key pests, the cabbage root fly (Delia radicum) and the carrot fly (Psila rosae), the two major root feeding pests.Thus, we consider a continuous three-level food web model with we apply the impulsive differential equations (IDE) [20][21][22] to integrate system (1).In [13], the authors constructed a predator-prey impulsive system to show the process of releasing natural enemies periodically and spraying pesticides twice at different fixed times in a period.The authors, obviously, avoid the side effects of pesticides on natural enemies existing in [7][8][9].But only two pulses in a period cannot effectively control pests.And most of the research is single species of pests while few papers have discussed multispecies of pests.However, most real pests communities are more complex than the community previously analysed by them.In the present paper, we make the following improvements: (a) in the modeling, we introduce a onenatural enemy and two-pest model, and the natural enemy shows optimal foraging between pest  1 and pest  2 , which is a well-known behavior of many predators [23]; (b) in the control strategy, we can control two pests more selectively by controlling pulse frequencies appropriately.

Model Formulation and Auxiliary Lemmas
Considering the previous factors, firstly, two models of different control strategies are discussed as follows.
Case 1.We suppose pesticides are sprayed several times in a release period, and the kill rates of pesticides to pests (   ,  = 1, 2) and natural enemies ( 3  ) are different in different impulsive moments in a release period.That is, we consider the following model:
Case 2. We suppose that the natural enemies are released several times in a spraying period.The release number of natural enemies (  ) and an effective kill rate to natural enemies (  ) are different in different impulsive moments in a spraying period.That is, we consider the following model: where  1 =  0 <  1 <  2 < ⋅ ⋅ ⋅ <   <  (+1) = ( + 1) 1 , ,  ∈  + ;  = 0, 1, 2, . . ., ;  (+1) −   =  1 /( + 1); 0 ≤   1 < 1 ( = 1, 2, 3) represents an effective kill rate to pest  1 , pest  2 and natural enemy  at time  =   , respectively.  represents reduced proportion of natural enemies owing to the delay effect of pesticides and eating the deleterious pests; represents natural enemy number of additional release at  =   ;  1 is a spraying period;   is the th releasing moment in the spraying th period.Other parameters are the same as those in system (1).
Furthermore, some essential notations, definitions and lemmas are given as follows.

Dynamical Analysis of Case 1 and Its Biological Implications
For Case 1, the basic properties of the following subsystem: play a key role in analyzing the pest control.
It is shown in the Appendix that there exists a globally stable periodic solution   () for the subsystem (6).Therefore, the complete expression for the pest-eradication periodic solution of system (6) over the ( − 1)th time interval ( − 1) ≤  ≤  is given by (0, 0,   ()).Furthermore, if the following threshold condition: is satisfied, then the pest-eradication periodic solution (0, 0,   ()) is globally asymptotically stable, where and ) . ( , and 1− 3  =  3 , where  = 1, 2, . . ., .What we want to address in the following is how control tactics including residual rates  1 ,  2 , and  3 release constant , timing of pesticide application , and timing of release period  affect the threshold condition   .Firstly, in the following section, we take Figure 2 as an example to control pests by the control strategy of Case 1.It implies that only the pest  2 needs be eradicated.We firstly use the traditional control method, spraying insecticides, in order to know well the impact of insecticides on pests and natural enemy species.In general, pesticides tend to be harmful to most natural enemies [24], which may be associated with the acute poisoning.It is significant to understand the acute poisoning of insecticides to natural enemies for the research of IPM strategy. We only apply insecticides but we do not release natural enemies as shown in Figure 3.The simulation results indicate that pesticide applications (number of pesticide applications  = 1 in a period shown in Figure 3(a)) do not lead to the extinction of pest  2 , and on the contrary, they can result in the recurrence of pest  1 , and with the increase of the number , the quantityof both pest  1 and pest  2 increases (see Figure 3(b)).Only when the dosages of pesticides are increased enough can the pests become extinct, but natural enemies also become extinct at the same time (see Figure 3(c)).This shows that the extinction of pests needs plenty of pesticides.Nevertheless, pesticide abuse can bring about environmental contamination, which can also result in human exposure through consumption of residues of pesticides in food and, possibly, drinking water.While developed countries have systems already in place to register pesticides and control their trade and usage, this is not always the case elsewhere, especially in China.Moreover, the pesticides have a serious impact on the natural enemies (Figures 3(a)-3(c)), and the repeated use of the same pesticides can result in one or more population pest outburst (Figures 3(a)-3(b)).The previous results show that pest control of multispecies is much more complicated than single pests [7][8][9].
By the previous analysis, the additional release of natural enemies is an indispensable part for pest control.Without loss of generality, we assume that natural enemies have food preference phenomenon with pest  1 ; that is,  1 >  2 , the intrinsic growth rate  1 ≤  2 , and the residual ratio of pests  1 ≤  2 (here the reason is explained in the following section) after spraying; thus, according to Theorem A.2,  1 <  2 and the condition of pest extinction only needs  2 < 1.
In Figure 4, we fix the other parameters of  2 and let the residual ratio  3 vary.The simulation results indicate that if the pesticide poisons the natural enemies with a relatively low residual ratio  3 (e.g.,  3 = 0.9), the threshold value  2 is a monotonically increasing function with respect to the number of pesticide applications  (Figure 4(a)).This further explains that if the pesticide has a severe impact on the natural enemies, repeated use of the same pesticides can result in pest resurgence.If the residual ratio  3 is slightly increased from 0.9 to 0.92, Figure 4(b) shows that the threshold value  2 is not monotonic with respect to the number of pesticide applications .So in this case we must carefully select the number of pesticide applications (one to three events in this case).If the pesticides do not kill the natural enemies so much, Figures 4(c) and 4(d) clarify that the threshold value  2 is a monotonically decreasing function with respect to the number of pesticide applications .In Figures 5(a)-5(d), similarly, we fix the other parameters of  2 and let the release period  vary.The simulation results indicate that the small change of release period  can lead to the change of the number of pesticide applications .All these simulations show that the releasing period, the number of times of spraying pesticides within this period, and the residual ratio of natural enemies are crucial to eradicate pest  1 and pest  2 .Figure 6 shows the relationship between the controllable parameters and the threshold condition  2 .All simulation results demonstrate that  2 seems to be quite sensitive to small changes in residual ratio  1 and  3 , release constant , and release period .By the simulation results, we obtain that the optimum time and frequency of pesticide application; the protection of natural enemies and pesticides choice are the key factors to pest outburst or eradication.The results may provide a theoretical basis for agricultural practitioners to guide them to spray pesticides and release natural enemies more efficiently.

Dynamical Analysis of Case 2 and Its Biological Implications
In this section, similar to the study method of system (2), we will give the dynamic property of system (3).
According to Floquet theory [16], if the following threshold condition: is satisfied, then the pest-eradication periodic solution (0, 0,   1 ()) of system (3) is globally asymptotically stable, where ) , For Case 2, there are  times releasing natural enemy during spraying period  1 .Denote   = ,   = , where  = 1, 2, . . ., .Since the release of natural enemies in this case is more frequent than spraying pesticides, the side effects of pesticides on the natural enemy population are largely reduced.Moreover, the threshold condition  2 can be strongly affected by the additional release of natural enemies.In Figure 7, we fix all the other parameters and choose a different releasing constant  and different release times .
The simulation results indicate that slight perturbation of the release constant  can rapidly reduce the threshold value  2 (Figure 7), while increasing the number of natural enemy releases as well.This shows that repeated releases of a small number of natural enemies in key time of the season can effectively control the pest outburst.In practice, an example of Liriomyza sativae and Trialeurodes vaporariorum occurs in heliogreenhouse.The parasitic rate of parasitic wasps which is their natural enemy can amount to over fifty percent without drugs.Thence, periodic releases of the parasitic wasps have been used to control the Liriomyza sativae and Trialeurodes vaporariorum in Anshan city in Liaoning Province, China, where greenhouse agriculture is developing rapidly.
Remark 4. In this paper, we suppose that the natural enemies can first select which are their favorite prey between pest  1 and pest  2 .It means that the favorite natural enemies may be a profitable pest to them.Thence, as seen in [19], the profitable pest is classified as palatable and the other as unpalatable.
In the following section, by numerical simulation, we will explain under what condition the natural enemies can prey on pest  1 or pest  2 .

Hybrid Impulsive Model with Economic Threshold
As the previous simulation indicates, pesticides may seriously influence the survival of natural enemies.They may impact natural enemies indirectly by killing or contaminating their hosts or prey.It is essential to avoid pesticide abuse when biological control is feasible, as shown in systems ( 2) and (3).Probably the best method for reducing the side effects of pesticides on natural enemies is to apply pesticides only when the sum of density of two pest populations reaches the economic threshold (ET), since a small number of insect pests may have compensation effect on crops [25].Thence, we formulate the model as follows: Figure 5: The effects of number of pesticide applications and release period  on the threshold level  2 with the parameter value  3 = 0.93 and other parameters the same as Figure 4.
where  1 ,  2 , and  3 are the same as in Section 5 and ET is the economic threshold. is the releasing period of natural enemies.
The previously mentioned facts show that the effect among the intrinsic growth rate of pests,  1 and  2 , the predation capacity of natural enemies (or functional response parameter),  1 and  2 , residual ratio  1 ,  2 , and  3 , releasing quantity , and other factors (such as ET) may determine dynamic behavior of pests and natural enemy species.How do these key parameters affect the control strategies?In particular, what we want to achieve is to study how the ET,  1 or  2 , and controllable parameters (such as ) affect the control strategies.
For a fixed ET, by simulation, we obtain the result that the successful control strategies are affected by the predation capacity of natural enemy to different types of pests.To show this, we vary the key parameter  1 while the other parameters are fixed as those in Figure 8.In Figure 8(a), for the predation capacity of natural enemy  1 = 0.74, the simulation result indicates that the sum of density of the two pests population never reaches the given ET, which implies that  1 ≥ 0.74 is free from spraying.If we set  1 = 0.737, Figure 8(b) indicates that the system is free from chemical control after spraying pesticides.If we set  1 = 0.736 or  1 = 0.732  or  1 = 0.73, Figures 8(c)-8(e) indicate that the system is free from chemical control after three, four, or five pesticide applications.If we further reduce the predation capacity of natural enemy and set  1 = 0.71, the pest outbreak frequency is sharply increased, as shown in Figure 8(f).As mentioned in Remark 4, in Figure 8, we will expound the interdependent relationship among the natural enemy, pest  1 , and pest  2 .
In the beginning, the density of pest  2 is larger, and it is regarded as palatable for the natural enemy,which implies that  2 ≥  1 , as shown in Figure 8(f), the pests can break out.However, with the release of natural enemies, an expanded population of natural enemies causes the reduction of pest 2, but when pest  2 falls below a certain critical value  * 2 [19], the natural enemy begins to eat not only pest  2 but also pest  1 , which largens  1 and makes pests gradually no longer break out as shown in Figures 8(e)-8(a).Meanwhile, it causes an immediate recovery of pest  2 , and when pest  2 raises above the certain critical value  * 2 , the natural enemy begins to eat palatable pest  2 again, which forms a cycle (it is also called switching between pest  1 and pest  2 by natural enemy [26]).The previous analysis illustrates that if the natural enemy and the two pests meet the above relationship, we only need to control pest  2 falling below the certain critical value without supervising pest  1 .
Finally, we will introduce the definition of pest outbreak duration (period) and analyse the relationship between pest outbreak period and the controllable parameters.We denote the time points at which the solution reaches ET as   ( = 1, 2, . ..).If mod(  , ) = 0, a chemical control is applied at   , and after that a biological control is also applied at the same time.If mod(  , ) ̸ = 0, only a chemical control programme is applied.Further, denote with  0 = 0 as pests-outbreak duration (or period), where  may be finite or infinite which depends on the solutions of the models.The relationship among , ET,  1 ,  2 ,  3 , or  and mean outbreak period of pests can be calculated from model (11) and formula ( 12) numerically (Figure 9).Mean pest outbreaks period is an average over several pest outbreak (here outbreaks indicate that the sum of densities of the pest  1 and pest  2 reaches the given ET).Model (11) predicts that the pests do not break out if the natural enemies are released more transitorily ( ≤ 2, Figure 9(d)) and the mean outbreak period is decreasing as the release period  or residual ratios of the pests  1 and  2 increase (Figures 9(a), 9(b), and 9(d)).Conversely, model (11) predicts that with the increase of ET or residual ratios of the natural enemy  3 , the mean outbreak period becomes longer (Figures 9(c) and 9(e)).In Figure 9(f), let the release period  and the other parameters fixed, and let the release constant  vary.This indicates that when the release period  is smaller (here  = 4), with the increase of release quantity, the pests will not break out (here  ≥ 1.5).And, more remarkably, the mean outbreak period can suddenly jump from a small value to a larger value at some critical points of  3 , , and , which implies that the protection of natural enemies, the selection of releasing time, and the quantity may be crucial in prolonging the pest outbreak period.Moreover, the different  2 or  3 or different values of the release constant  may have the same mean outbreak period (Figures 9(b), 9(c) and 9(f)).For system (11), the relationship between pest outbreak period and other parameters such as  1 ,  2 ,  1 , and  2 can be researched similarly.

Discussion
The agricultural pests management plays a decisive role on the survival of people all over the world especially that the impacts of extreme climate change are severer for pest control.For example, the armyworm which is the typical pest threatening corn growth in fall has been widely seen in North China Plain and Northeast China producing regions in August 2012.The leaves of corn stalks in portions of the above regions have been eaten up, cutting corn harvest Figure 9: The relationship between mean outbreak period and the parameters  1 ,  2 ,  3 , , ET, and  of model (11).The other parameters are the same as Figure 8.
prospect.The pests, common pests, became the most serious threat to the production of corn this summer in the country's major grain-producing regions.This is mainly due to frequent cyclonic activities since mid-July provided favorable conditions to the migration of the armyworms, and then heavy rainfall forced them to stay in the north and northeastern parts of the country.The pest outbreak occurred at the same time as severe droughts in the United States, where the driest conditions in more than half a century have battered corn and soybean crops, causing an upsurge in global grain prices [27,28].
Thence, it is particularly significant to explore an effective control strategy.In this paper, based on the IPM strategies, we give three different control strategies, which improve traditional IPM control strategies.For system (2), in a releasing natural enemies period, we spray pesticides several times.By the theoretical derivation to system (2), the critical value of pests eradication is figured out.When only using the traditional control method, insecticides, by the numerical simulation, we obtain that the pest  1 and pest  2 may break out with the increase of spray times, which is different from the control of a single pest.To better understand how the controllable parameters (here , ,  1 ,  2 , and  3 ) impact the pest control, by the numerical simulation, we give the relationship between the critical value of pests eradication and the controllable parameters.All these results express that the selection of spray times and the protection of the natural enemies are of vital importance for pests eradication.From system (3), we know the real embodiment of the significance of protecting natural enemies.By the analysis of systems ( 2) and (3), in summary, when insecticides are used excessively, the pests are killed and the natural enemies of the pests are wiped out.In the absence of natural enemies, the surviving population of insect pests multiply rapidly and reach epidemic proportions.Indiscriminate use of pesticides also leads to the development of resistance in pests.This occurs as a result of killing the susceptible genotypes and selecting the more resistant genotypes at every pesticide application event.After several years of using the same pesticide, there would come a time when that particular pesticide will have no effect on the pests because they have developed resistance to the pesticide.Considering the factors, we formulate model (11), which is to apply pesticides only when necessary.By model (11), we also obtain some important conclusions.
Most real communities are more complex than the community analysed here.Therefore, in the future, the factors on the pests of more species and natural enemies, resistance to the pesticide of pests and so on should be considered in the