Qualitative Analysis of a Pest Management Gompertz Model with Interval State Feedback Impulsive Control

An integrated pest management Gompertz model with interval impulsive control is put forward.Through pest density monitoring, an integrated control strategy is adopted; that is when pest density reaches an environmental damage level, pesticide is used as a control method; when pest density is lower than the damage level, predators as its natural enemy are released in case pest density is higher than the slightly harmful level and predator density is below its maintainable level. The analysis on the existence of order-1 or order-2 periodic orbit is carried out by the construction of Poincaré map of semicontinuous dynamical system.The stability and attractiveness of the periodic state are obtained by geometry approach, which ensures a certain robustness of control.The analytical results presented in the work are validated by numerical simulations for a specific model.


Introduction
Agricultural pests are harmful to the crops, and thus pest management plays an important role in agricultural sustainable development, which also becomes an interesting and significant topic in real life.The traditional and efficient method for pest control is to spray pesticide, which can quickly kill the pest.However, unrestrained use of persistent pesticide not only increases the incidence of pesticide-resistant pest varieties but also inflicts harmful effects on humans through the accumulation of hazardous chemicals in their food chain [1].An alternative way is the biological control by launching predators or enhancing predators' genes to improve the effectiveness of the pest control [2].This can be achieved by mass production and periodic release of natural enemies of the pest.However, the cultivation of the natural enemy in laboratories is uneasy, and the cost is very high in general.Integrated pest management (IPM) is the comprehensive utilization of agricultural, biological, chemical, and physical methods to control pests with the goal of controlling the number of the pests under an economic threshold (ET) [3][4][5].Compared to natural growth process, spraying pesticides (or releasing the natural enemies) can cause the density of pests (or the natural enemies) to change sharply in a short time, which results in the discontinuity of the system states.
In the farmland and forest, the state of pests is always monitored.The administrative authority makes decision (i.e., whether the control should be taken and what kind of control should be adopted) according to the monitored pest level [6].Based on the adopted control action, a series of integrated pest management models have been built in the literature, for example, the periodic release of predators [7][8][9]; the periodic release of pests infected by a disease [10]; the periodic release of predators and infected pests [11,12]; the periodic release of infected pests combined with periodic applications of pesticides [13]; the periodic release of predators and pests combined with periodic applications of pesticides [14][15][16]; and state-dependent release of predators combined with applications of pesticides [3][4][5][17][18][19][20][21][22].
As far as the state-dependent impulsive control is concerned, the earlier works in pest management belong to Tang et al. [3][4][5], where the biological and chemical controls are assumed to be taken at the pest economic threshold (ET).Notice that, in real applications, the biological control and chemical control sometimes are adopted at different pest levels; some researchers began to investigate the pest control models by assuming that releasing natural enemies and spraying pesticide are taken at different pest thresholds [23][24][25][26][27][28].From theoretical and practical points of view, the pest control model with twice impulsive controls looks more reasonable than the model with one impulsive control, and it can be seen as an extension of the one-impulsive-control models.
The ideas of involving biological and chemical controls at different prey densities is interesting and also has practical significance.Biological control with an appropriate yield of release of the predator is adopted in advance, to extend the time for pest density increasing to the damage level (i.e., reduce the operating frequency of adopting chemical control or pesticide).But there exists a problem in modelling the real system; that is, the biological control is adopted when the pest density reaches the first control level, but for a higher pest density between the biological and chemical control levels, there is no control strategy adopted.This is obviously unreasonable.Since the biological control and chemical control are activated at different pest levels, a more reasonable model should also consider the control action when the pest density lies between the two levels; that is to say, when the pest density increases to the chemical level, the chemical control has to be carried out, which causes a certain proportion to pest and predator to be killed.When the pest density increases to or exceeds the biological control activation level but is lower than the chemical control level, the biological control is sufficient, which is also necessary when the predator density is lower than its maintainable level, while in case of low pest density (i.e., below the biological control excitation level) it is not necessity to take any control action.Motivated by this control strategy, the current work presents and studies a preypredator system involving interval state impulsive control.
This paper is organized as follows.In Section 2, a pest control prey-predator model with interval impulsive control is put forward.In Section 3, the Poincaré map, successor function, and some basic definitions are given, followed by a detailed dynamics analysis in case of the chemical control strength.In Section 4, numerical simulations are carried out with a specific model to verify the theoretical results step by step.Finally, conclusions are presented in Section 5.

Dynamic Analysis of System (1)
As illustrated in Figure 1, system (1) has three equilibria (0, 0), (, 0), and ( * ,  * ), where  * = /( − ℎ) and  * =  ln(( − ℎ)/)/ if ( 1 ) :  >  ≜ /( − ℎ) and  > ℎ holds.The reference level   () is established by a statistics method in practice, which reflects a certain consistency of the system; that means the biological control taken at any level should take the same or similar effect in determining the system's dynamics.Thus, in this study a trajectory of the solution of system (1) starting from the point  0 on  = ℎ 1 is selected as a predator reference.To make practical sense, the study is restricted in the left region of ( * ,  * ); that is, 0 < ℎ 1 < ℎ 2 <  * .

Existence of Periodic Solution.
The discussions will be divided into three cases according to the magnitude between ℎ 1 and (1 − )ℎ 2 : the position of the trajectory T0  1 , the magnitude of  and , and the location of the point  +  .
(2) Stability of the Order-1 Periodic Solution.Suppose the impulsive point of  2 is  + 2 ; obviously, the trajectory of system (1) starting from any point will arrive at the segment which leads to a contradiction.Thus, the order-2 periodic solution is unique.
(2) The Stability of the Order-2 Periodic Solution.Assume that  + 1 and  + 3 are the impulsive points of  1 and  3 under the pulses Δ = (1 − ) and Δ = (1 − ), respectively.The trajectory  0  1 meets with  = (1−)ℎ 2 at the point  2 in the opposite direction;   0 is the impulsive point of  0 .Obviously, the trajectory of system (1) starting from any point will arrive at the segment

Conclusion
In this work, a pest control prey-predator model was analyzed, where the biological control and chemical control are        considered at different thresholds.At the early stage of the pest damage outbreaks, the biological control is adopted in case of the predator density in the environment lower than its maintainable level.Once the pest density reaches a critical level, the chemical control with a given strength is taken since higher pest density may cause a serious damage to environment.Different to the models in literature, the proposed model is more consistent with practice.
The theoretical analysis indicated that the yield of releases of the predator plays a key role in determining the existence of order-1 periodic orbit.The practical significance to studying the existence of order-1 or order-2 periodic solution lies in that it could provide a possibility to determine the frequency of using chemical pesticide and yield of releases of the predator, which makes the control a periodic one without realtime monitoring of the species while keeping the prey density below the damage level.The stability and attractiveness could ensure a certain robustness of control; that is, even though the species density is detected inaccurately or with a deviation, the system will be eventually stable at the periodic solution under the control action.
Thus, it is only necessary to consider the tendency of the trajectory starting from  2  + 2 .If   + For the case 0 < (1 − )ℎ 2 < ℎ 1 , if the order-2 periodic solution exists, it is unique, orbitally asymptotically stable, and globally attractive.
it is only necessary to consider the tendency of the trajectory starting from  + This means that {  +  } is a monotone increasing sequence with an upper bound   +  , and {  +  } is a monotone decreasing sequence with a lower bound   +  ; then, there exist   + and   + such that   +  →   + and {  +  } →   + when  → ∞.But ( + ) = ( + ) = 0 implies that  + =  + =  +