A Pest Management Model with Stage Structure and Impulsive State Feedback Control

A pest management model with stage structure and impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semicontinuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.


Introduction
Banana leaves diseases are divided into epiphyte and virus.Banana bunchy top disease (i.e., Prawn banana, Green banana, Banana) is one of virus diseases, caused by Banana bunchy top virus.Banana farmers call it an incurable disease.Banana aphids are the major propagation medium of banana virus diseases.The development of banana aphids includes three stages: egg, nymph, and adult (winged form).Eggs do not carry and spread virus.Nymphs transmit virus to healthy plants only through short-distance crawling since genitalia and wings of nymphs are not fully developed yet, and therefore infected nymphs have slight infective power.After 4 instars, nymphs grow into adults which have fully developed genitalia and wings and can oviposit and transmit virus to healthy plants through migrating after piercing and sucking the virus of diseased plants, so infected adults have strong infective power.To avoid the outbreak of banana aphids, we will use ovicides to kill eggs or use insecticides to kill nymphs and adults.
In pest management, we spray pesticides only when pest density increases to a certain level called ET (economic threshold, i.e., pest population density at which control measures should be adopted to prevent an increasing pest population from reaching the economic injury level).ET is the index of pest density.Crop output will not decrease much when pest density is lower than ET; thus, we need not adopt any control measure.Once pest density rises to ET, some measures must be carried out to prevent EIL (economic injure tolerate level) from happening.To control pests, such a measure for spraying pesticides is always adopted when pest density arrives at a given ET.
Considering that immature pests cause a minor damage to crops, in this paper, we will spray insecticides when the density of immature pests increases to ET, which is a more effective preventive measure than we do when the density of mature pests increases to ET. Usually, insecticides have specificity; in other words, insecticides (such as 2000 to 2500 times dilution of acetamiprid 3% EC, 15000 times dilution of imidacloprid 70% WG, 1000 times dilution of omethoate 40% EC, and 2500 to 3000 times dilution of sumicidin 20% EC) can only kill nymphs and adults but cannot kill eggs.Therefore, a pest management model with stage structure and impulsive state feedback control is constructed as follows: where (), () denote the proportions of immature pests (nymphs) and mature pests (adults) at time , respectively,  =  1  2 denotes the transformation rate from mature to immature pests, where  1 denotes the birth rate of mature pests,  2 denotes the transformation rate from eggs to immature pests,  denotes the transformation rate from immature to mature pests, ,  denote the death rate of immature and mature pests, respectively,  1 ,  2 , , , ,  are positive constants, 0 <  < 1, 0 <  < 1 are the ratios of killing immature and mature pests by spraying pesticides, respectively, and  * denotes ET.At present, for stage structure pest management model with impulsive effect, extinction and permanence have been proved by using Floquet theorem and comparison theorem [1][2][3][4].For impulsive state feedback control systems, the sufficient condition for the existence and the orbitally asymptotically stability of the order-1 periodic solutions have been obtained by differential equation geometry theory, the method of successor function, and analog of Poincarè criterion [5][6][7][8][9][10][11][12][13][14].However, for the pest management model with stage structure and impulsive state feedback control, almost no one investigates.In this paper, we try to obtain a new judgement method for the stability of the order-1 periodic solution by referencing the stability analysis of limit cycles for continuous systems.This is a superior method, by which the more perfect and simple conclusions than the others are obtained.
In the next section, we give some preliminaries.In Section 3, we get the sufficient condition for the existence of the order-1 periodic solution of system (1) by differential equation geometry theory and successor function.In Section 4, referencing the stability analysis of the limit cycles for continuous dynamic systems, we prove the order-1 periodic solution of system (1) is orbitally asymptotically stable under some conditions.In Section 5, we analyze numerically the theoretical results obtained.
Definition 3 (see [15]).Suppose  is the phase set of system (1),  is the impulse set of system (1), and both  and  are straight lines (see Figure 1).The intersection point of  and -axis is , the distance between point  ( ∈ ) and point  is noted by ,  1 denotes the intersection point of trajectory passing through point  and , phase point of  1 is  1 ( 1 ∈ ), and the distance between  1 and  is noted by  1 .One defines subsequent point of  as  1 , and the successor function of  is () =  1 − .
According to Lemma 5, we can get the following lemma.Lemma 6 (see [15]).Assume continuous dynamical system (, Ψ); if there exist two points ,  in the phase set such that successor function () > 0, () < 0, we can find a point  between  and  in the phase set satisfying () = 0.So there must exist an order-1 periodic solution passing through point .
Suppose  ∈  is the phase point of ; then   <   , where   ,   is the  coordinate of , , respectively.Choose   1 between  and ; the trajectory Γ 1 passing through  1 intersects the phase set  at  1 after impulse effect, and then  1 is the subsequent point of  1 .Since distinct trajectories do not intersect,  1 must be below ; we have  1 <   <  1 , where  1 ,  1 is the  coordinate of  1 ,  1 , respectively.Therefore, ( 1 ) =  1 −  1 < 0. Suppose the trajectory Γ 2 passing through  intersects the phase set  at  after impulse effect; then  is the subsequent point of , and there are two cases.Case 1.If  is above , then   >   , where   ,   is the  coordinate of , , respectively; we have () =   −  > 0. By Lemma 6, there exists a point  ∈  1 ⊂  such that () = 0. Therefore, there exists an order-1 periodic solution of system (1) passing through .The proof is completed.
Case 2. If  is below , then   <   ; we have () =   −   < 0 (see Figure 5).In addition, the trajectory Γ 3 passing through  intersects the phase set  at  after impulse The existence of order-1 periodic solution of system (1) in the trapezoid .
effect; then  is the subsequent point of , and  must be below  1 and above  because distinct trajectories do not intersect.Therefore, we have () =   −   =   > 0, where   ,   is the  coordinate of , , respectively.By Lemma 6, there exists a point  ∈  ⊂  such that () = 0.The proof is completed.Secondly, suppose  3 intersects ,  at , , respectively, and  1 intersects  at . Draw a straight line which is perpendicular to  and -axis; the foot points are , , respectively (see Figure 6).Let us analyze the existence of order-1 periodic solution of system (1) in the trapezoid .
Suppose ,  are the phase points of , , respectively.On the one hand, the trajectory Γ 4 passing through  intersects the phase set  at  1 after impulse effect; then  1 is the subsequent point of , and  1 must be below  because distinct trajectories do not intersect.We have () =  1 −   < 0, where  1 ,   is the  coordinate of  1 , , respectively.
On the other hand, choose a point  ∈  between  and .The trajectory Γ 5 passing through  intersects the phase set  at  1 after impulse effect, and then  1 is the subsequent point of , and  1 must be above  because distinct trajectories do not intersect.We have () =  1 −   > 0, where  1 ,   is the  coordinate of  1 , , respectively.
By Lemma 6, there exists a point  ∈  ⊂  such that () = 0. Therefore, there exists an order-1 periodic solution of system (1) passing through .The proof is completed.
Proposition 11 (königs).Assume  = () is a continuous transform from line segment  to itself;  = 0 is a fixed point under the transform.If the part near origin of curve  = () on the plane (, ) lies in the interior of the domain the fixed point  = 0 is stable (unstable).
Proof.We prove firstly that the fixed point  = 0 is stable.
Choose  > 0 sufficiently small such that for any point  in noncentral neighborhood  0 (0; ) of the fixed point  = 0, and we have For any point range From Figure 7, we have
In the same way, we prove the fixed point  = 0 is unstable.The proof is completed.
From Figure 8, assume the closed orbit consisting of the curve Â and line segment  is the order-1 periodic solution of system (20), denoted by Γ, where  ∈ ,  ∈ ,  is the phase set, and  is impulse set.Draw normal line  passing through  ∈ Γ and establish coordinate system (, ) on point .Choose any point  ∈  in small enough neighborhood of .The trajectory starting from  intersects vertically -axis at   and intersects impulse set  at .  denotes the phase point of , the trajectory passing through point  intersects vertically -axis at  +1 as  increases.
Assume rectangular coordinate of  is ((), ()); then for   , there is the relation between its rectangular coordinates (, ) and curvilinear coordinates (, ): where where  0 ,  0 denote the values of ,  at the point , respectively; we have From (20), it is easy that we have and hence Since there is a zero solution  = 0 for (24), when there exist continuous partial derivatives for functions , , there exists the continuous partial derivative of (, ) with respect to  also; (24) is written as In order to calculate we first get where  0 ,  0 ,  0 ,  0 denote partial derivatives of ,  when  = 0, respectively.Since  =  0 ,  =  0 when  = 0, it is easy to know  0   +  0   = 0.By ( 24) and ( 27), we have where () denotes the curvature of orthogonal trajectory at  for system (1).Therefore, the approximate equation of ( 25) is whose solution is Theorem 13.Assume ℎ is the length of curve Â which is a section of the order-1 periodic solution Γ of system (1).The order-1 periodic solution Γ is stable when Proof.Let us investigate trajectory    +1 (see Figure 8).In the coordinate system (, ), the ordinate of   is denoted by  0 and the ordinate of  is denoted by .From (30), we have when ∫ ℎ 0 () < 0, where ℎ is the length of curve Â.By Propositions 10 and 11, the order-1 periodic solution Γ is stable.
Let  = √ 2 0 +  2 0 ; the left of (31) can be rewritten as Consider the integral along the periodic solution Γ  of continuous system and we suppose the integral along the order-1 periodic solution Γ of semicontinuous system has the same result.
Lemma 15.If function (, ) is continuous and differentiable, the integral along the order-1 periodic solution of system (1) satisfies where period of the order-1 periodic solution is .
According to (34), we have the following theorem.
Theorem 16.If the integral along the order-1 periodic solution Γ of system (1) satisfies Γ is stable.
Theorem 17.The order-1 periodic solution of system ( 1) is stable. Proof.Since by Theorem 16, the order-1 periodic solution of system (1) is stable.The proof is completed.
To verify the theoretical results obtained in this paper, we choose  as the parameter and analyze numerically the following cases.
Case 1.Let  = 1.7,  = 1.8,  = 0.8,  = 0.9,  * = 50,  = 0,  = 0,  0 = 50,  0 = 10; we have  = 0.26 (see Figure 11).According to the above discussion, (0, 0) is an asymptotically stable node when  > 0. It implies that (), () tend to be extinct as  increases without any control measures.According to Theorem 7, if  < 0, that is,  −  < 0, there exists an order-1 periodic solution (see Figure 12).We can observe that there exists an order-1 periodic solution of system (1) which lies between the phase set and the impulse set (i.e., between 45 50).Figures 12 and 13 give the time series and phase portraits when  <  ( = 0.1,  = 0.3) and  >  ( = 0.8,  = 0.3), respectively, and show different positions of the periodic solution under different parameter values and different initial values.Furthermore, the phase portrait of Figure 12 indicates that the mature pests always keep increasing, but Figure 13 indicates that the mature pests firstly decrease and then begin to increase.Therefore, the control parameter  ( >  or  < ) can result in different change in density of mature pests and different efficiencies of killing mature pests by spraying pesticides which will give a conclusion theoretically to the researchers in killing mature pests.Researchers should give suitable control parameter  and appropriate initial values in order to obtain a steady and optimal control.In fact, immature pests (nymphs) are more easily to be killed by pesticides than mature pests (adults); thus, Figure 13 is more feasible than Figure 12.Time series portraits of Figures 12 and 13 show that the order-1 periodic solution of system (1) is stable, and it is consistent with Theorem 17.The numerical analysis illustrates that we can achieve the aim of controlling immature and mature pests by impulsively spraying pesticides when immature pests density increases to  * .
According to the obtained conclusions, we can predict the cycle time without repeated measurements, which can save a lot of labor and material resources.Obviously, the model with impulsive state feedback control is closer to the reality than the periodic impulsive model where there is no density dependence.

Figure 4 :
Figure 4: The existence of order-1 periodic solution of system (1) in the domain Δ when   >   .

Figure 5 :
Figure 5: The existence of order-1 periodic solution of system (1) in the domain Δ when   <   .