# Optimal Control of Agricultural Insects with a Stage-Structured Model

Authors: Baolin Kang; Mingfeng He; Bing Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A stage-structured pest control model with impulse effects by state feedback control is formulated, and a semicontinuous dynamic system and its successor functions are defined. The sufficient conditions of existence and attractiveness of order one periodic solution are obtained by the method of successor functions. The superiority of the state feedback control strategy in this paper is that we only need to monitor the sum of immature and mature pest populations. Moreover, our results show that our method used in this paper is more efficient and easier than the existing methods.


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## Body

## 1. Introduction

Insects are among the most diverse group of animals on the planet and include more than a million described species and represent more than half of all known living organisms. The number of extant species is estimated to be at between six and ten million and potentially represents over 90 percent of the differing metazoan life forms on Earth. They may be found in nearly all environments, although only a small number of species occur in the oceans, a habitat dominated by another arthropod group, the crustaceans. Some insects damage crops by feeding on sap, leaves, or fruits, a few bite humans and livestock, alive and dead, to feed on blood and some are capable of transmitting diseases to humans, pets and livestock. So, humans regard certain insects as pests, and attempt to control them using a host of techniques. Pest control is very important because out of control pests can wreak havoc. For example, many countries in the world suffer deeply from plagues of locusts each year.In pest management, we can control the pest population by many methods, such as spraying pesticides and releasing natural enemies. Among the methods, chemical pest control is still the predominant type of pest control today, although its long-term effects led to a renewed interest in traditional and biological pest controls towards the end of the 20th century. For most pests, such as aphid and silkworm, the survival rate has a relation with their age and development stage. In recent years, various stage-structure models have been proposed and studied for populations [1–12]. We suppose a stage-structure model as follows:
(1)dxdt=-ax+by,dydt=cx-dy,
where x(t) and y(t) represent the densities of immature and mature pest populations, respectively; a, b, c, and d are positive constants; a denotes the sum of the rate of an immature population turning into a mature population and the death rate of the immature pest population; b and d denote the birth rate of the immature pest population and the death rate of the mature pest population, respectively; c denotes the rate of an immature population turning into a mature population. The dynamics property of system (1) is very simple. The unique equilibrium point of system (1) is O(0,0). In the case of ad-bc>0, O(0,0) is stable, so no pest control measure is needed. But in the case of ad-bc<0, O(0,0) is a saddle with a separatrix Γ:k1x, where k1=((a-d)+(a-d)2+4bc)/2b. So the effective measures are necessary to control the pest population.Apparently, when the pesticides are sprayed, it is natural to assume that both the mature and immature pest populations diminish abruptly; that is, impulse occurs, and we can use an impulsive differential equation to model the process of spraying pesticides. The theory of impulsive differential systems has been developed by numerous mathematicians [13–15]. There are three kinds of typical impulsive differential equations [13]: (1) systems with impulses at fixed time; (2) systems with impulses at variable time; (3) autonomous systems with impulses. In recent years, most of researches on IDEs concern systems of types 1 [16–23]. However, the drawbacks of type (1) lie in neglecting growth law of the pest and the cost of management. We must control the amount of pesticide and use them in the crucial phase in order to maximize the benefits of pesticides. Therefore we should do the insect survey well, establish the reasonable control index, reduce the times of spraying, and improve the economic and ecological benefits. Today, some researchers also propose the way of the state feedback strategy by IDEs systems of types 3, such as in [21] and the economic threshold (ET) is mentioned and defined in [22]. However, the shortcoming of state feedback control strategy in [21], is the that pesticides might be applied only when the density of the mature pests population reaches ET, which can lead to the sum of pests beyond the ET because of the overlook of immature pests. Considering the previous factors, we formulate the model as follows:
(2)dxdt=-ax+by,dydt=cx-dy,x+y≠h,Δx=-αx,Δy=-βy,x+y=h,
where a, b, c, d, h>0. Δx=x(t+)-x(t) and Δy=y(t+)-y(t); α and β∈(0,1) represent the proportion of killed immature and mature pests by spraying pesticides, respectively; h represents the ET of the pest totality. System (2) has been studied by Jiang et al. [23], and they obtained the sufficient conditions of existence and stability of periodic solutions by means of the sequence convergence rule and the analogue of the Poincare criterion. In this paper, we improve the demonstration method of [23] via the successor function.The organization of this paper is as follows. In the next section, we give some important definitions and auxiliary lemmas. In Section3, the existence of one order periodic solution is obtained by using successor function. In Section 4, we discuss the attractiveness of order one periodic solution under its existence conditions by means of sequence convergence rules and qualitative analysis. Finally, by numerical simulation, we can confirm our theoretical conclusions and give a brief discussion.


## 2. Definitions and Lemmas

Definition 1.
A triple(X,Π,R+) is said to be a semidynamical system if X is a metric space, R+ is the set of all nonnegative real, and Π(P,t):X×R+→X is a continuous map such that:(i)
Π(P,0)=P for all P∈X;(ii)
Π(P,t) is continuous for t and s;(iii)
Π(Π(P,t),s)=Π(P,t+s) for all P∈X and t, s∈R+.
Sometimes a semidynamical system(X,Π,R+) is denoted by (X,Π).Definition 2.
Assume that(i)
(X,Π) is a semidynamical system;(ii)
M is a nonempty subset of X;(iii)
functionI:M→X is continuous and for any P∈M, and there exists a ε>0 such that for any 0<|t|<ε, Π(P,t)∉M.
Then,(X,Π,M,I) is called an impulsive semidynamical system.For anyP∈X, the function ΠP:R+→X defined as ΠP(t)=Π(P,t) is continuous, and we call ΠP(t) the trajectory passing through point P. The set C+(P)={Π(P,t)∣0≤t<+∞} is called positive semitrajectory of point P, for any subset M of X; we define M+(P)=C+(P)∩M-{P}. The set C-(P)={Π(P,t)∣-∞<t≤0} is called the negative semitrajectory of point P.LetΠ(P,t) be the trajectory of differential dynamical system dx/dt=f(x) passing through the point P. Now we give the trajectory Π~(P,t) of the following impulsive semidynamical system:
(3)dxdt=f(x),x∉M,x+=I(x),x∈M.Definition 3.
Let(X,Π,M,I) be an impulsive semidynamical system corresponding to the system (3), and, for any P∈X-M, the trajectory Π~(P,t):(X-M)×R+→X passing through point P is defined as follows:ifM+(p)=∅,Π~(P,t)=Π(P,t), for t∈R+;ifM+(p)≠∅, there exists a P1∈M+(P), and T0∈R+ such that Π(P,T0)=P1∈M and 0<t<T0, Π(P,t)∉M. Let I(P1)=I(Π(P,T0))=P1+∈N; then for 0≤t≤T0,
(4)Π~(P,t)={Π(P,t),0≤t<T0,P1+,t=T0,ifM+(P1)=∅, Π~(P1+,t)=Π(P1+,t-T0), t≥T0;ifM+(P1+)≠∅, there exists a P2∈M+(P1+), and T1∈R+ such that Π(P1+,T1)=P2∈M and T0<t<T0+T1, Π(P1+,t)∉M. Let I(P2)=I(Π(P1+),T1)=P2+∈N; then for T0≤t≤T0+T1,
(5)Π~(P,t)={Π(P1+,t-T0),T0≤t<T0+T1,P2+,t=T0+T1.
Repeat the above process; then there exists anN0 such that M+(PN0+)=∅ or for any n∈Z+={1,2,…,}, M+(Pn+)≠∅. So, we can obtain a finite or infinite impulsive point sequence {Pn} in M satisfying Pn=F(Pn-1+)=Π(Pn-1+,Tn-1); here Pn-1+=I(Pn)∈N,Tn-1∈R+. Then we have
(6)Π~(P,t)={Π(P,t),0≤t<T1,Π(P1+,t-T1),T1≤t<T1+T2,⋮Π(Pn-1+,t-∑k=1n-1Tk),∑k=1n-1Tk≤t<∑k=1nTk,Π(PN0+,t-∑k=1N0Tk),t≥∑k=1N0Tk
or
(7)Π~(P,t)={Π(P,t),0≤t<T1,Π(P1+,t-T1),T1≤t<T1+T2,⋮Π(Pn-1+,t-∑k=1n-1Tk),∑k=1n-1Tk≤t<∑k=1nTk.⋮
In system (2), M={(x,y)∈R+∣x+y=h,x≥0,y≥0}, N=I(M)={(x,y)∈R+∣x=(1-α)h, y=(1-β)h}. For the convenience, if P∈X-M,F(P) is defined as the first point of intersection of C+(P) and M; that is, there exists a t1∈R+ such that F(P)=Π(P,t1)∈M, and for 0<t<t1, Π(P,t)∉M; if B∈X-N,R(B) is defined as the first point of intersection of C-(P) and N; that is, there exists a t2∈R+ such that R(B)=Π(P,-t2)∈N, and for -t2<t<0, Π(B,t)∉N. Furthermore, the isocline dx/dt=0 is denoted by L1; the isocline dy/dt=0 is denoted by L2. The saddle separatrix k1x is denoted by Γ. Let B be the intersection of the saddle separatrix Γ, and let N and C be the intersection of coordinate axis x and set N. Next we will give the definition of the successor function of semidynamical system (2). First, we define a new number axis in set N. On straight line y-((1-β)x/(1-α))=(1-β)h that corresponds to set N, take the origin at point B and BC→ as positive direction and unit length to be consistent with coordinate axis x or y; then we obtain a number axis l. For any x∈l, l(x)∈N.Definition 4.
A trajectoryΠ~(P0,t) is called order one periodic solution with period T if there exists a point P0∈N and T>0 such that P=F(P0)=Π(P0,T)∈M and P+=I(P)=P0.Definition 5.
Supposeg:N→N to be a map. For any x∈N, there exists a t1>0 such that F(x)=Π(x,t1)=x1∈M, x1+=I(x1)∈N; then g(x)=l(x1+)-l(x) is called the successor function of point x, and the point x1+ is called the successor point of x.Remark 6.
Ifg(x0)=0, the trajectory Π(x0,t) with initial point x0 is an order one periodic solution of system (2).According to the continuity of compound function, we have the following.Lemma 7.
The successor functiong(x) is continuous.In system (2), if there exist two points x1∈N, x2∈N satisfying g(x1)g(x2)<0, by the zero point theorem of continuous function in the closed interval, we know that there must be a point x0∈N which is between x1 and x2 such that g(x0)=0, so we have the following lemmas.Lemma 8 (existence theorem of order one periodic solution).
If there exist two pointsx1∈N, x2∈N, which are both in the left B or right B, satisfying g(x1)g(x2)<0, it has an order one periodic solution in system (2).Lemma 9.
Letl1={(x,y)∣y=kx,x≥0,y≥0}, l2={(x,y)∣x+y=h,x≥0,y≥0}=M, l3={(x,y)∣y-((1-β)x/(1-α))=(1-β)h,x≥0,y≥0}=N. Denote F∈M as the intersect of l1 and l2 and G∈N as the intersect of l1 and l3; then,(i)
ifα>β,l(F+)<l(G);(ii)
ifα=β,l(F+)=l(G);(iii)
ifα<β,l(F+)>l(G).Proof.
We focus on proving case (i), by similar method we can prove case (ii) and (iii). By computing, we can easily obtainF(h/(k+1),kh/(k+1)), G((1-α)(1-β)h/(k(1-α)+(1-β)), k(1-α)(1-β)h/(k(1-α)+(1-β))), and F+(h(1-α)/(k+1), k(1-β)h/(k+1)). If α>β; then (1-α)(1-β)h/(k(1-α)+(1-β))>h(1-α)/(k+1) and k(1-α)(1-β)h/(k(1-α) + (1-β))<k(1-β)h/(k+1). So, l(F+)<l(G). This completes the proof.


## 3. Existence of Order One Periodic Solution of System (2)

In this section, we will investigate the existence of order one periodic solution of system (2) by using the successor function.We note that system (2) is an impulsive semidynamical system, where M={(x,y)∣x+y=h, (x,y)∈R2+} is the closed subset of R2+ and I:(x,y)∈M→((1-α)x and  (1-β)y)∈N is a continuous function. For any point E, let xe denote its abscissa, and let ye be its ordinate. If E(xe,ye)∈M, then pulse occurs at point E, and the impulsive function transfers the point E into E+. Without loss of generality, unless otherwise specified we assume that the initial point of the trajectory lies in N.Now we discuss the existence of order one periodic solution of system (2) from the following three cases.Case I (α>β (see Figure 1)).
In this case, according to Lemma9, l(A+)<l(B) and set F(A+)=A1(xA1,yA1)∈M; then impulse occurs at point A1. Suppose point A1 is subject to impulsive effects to point A1+((1-α)xA1,(1-β)yA1)∈N. In view of vector field and disjointness of any two trajectories, we know xA>xA1 and yA<yA1, so we have xA+>xA1+ and yA+<yA1+; then l(A1+)<l(A+), and the successor function of A+ is g(A+)=l(A1+)-l(A+)<0; there exists a trajectory Γ1 intersecting to M at point A3 which is subject to impulsive effects to point A3+∈N. Set F(A3+)=A2∈M; then impulse occurs at point A2. Suppose point A2 is subject to impulsive effects to point A2+((1-α)xA2,(1-β)yA2)∈N. We know xA2>xA3 and yA2<yA3, so we have xA2+>xA3+ and yA2+<yA3+; then l(A2+)>l(A3+), and the successor function of A3+ is g(A3+)=l(A2+)-l(A3+)>0.Figure 1By Lemma8, we know there exists an order one periodic solution of system (2), whose initial point is between A3+ and A+ in set N.Case II (α=β (see Figure 2)).
In this case, according to Lemma9, the point A+ coincides with B, and the successor function of B is g(B)=l(B)-l(A+)=0, so the segment BA is an order one periodic solution of system (2).Figure 2Case III (α<β (see Figure 3)).
In this case, according to Lemma9, l(A+)>l(B) and set F(A+)=A1′(xA1′,yA1′)∈M; then impulse occurs at point A1′. Suppose point A1′ is subject to impulsive effects to point A1′+((1-α)xA1′,(1-β)yA1′)∈N. We know xA<xA1′ and yA>yA1′, so we have xA+<xA1′+ and yA+>yA1′+; then l(A1′+)>l(A+), and the successor function of A+ is g(A+)=l(A1′+)-l(A+)>0; there exists a trajectory Γ1 intersecting to M at point A3′ which is subject to impulsive effects to point A3′+∈N. Set F(A3′+)=A2′∈M; then impulse occurs at point A2′. Suppose point A2′ is subject to impulsive effects to point A2′((1-α)xA2′,(1-β)yA2′)∈N. We know xA2′<xA3′ and yA2′>yA3′, so we have xA2′+<xA3′+ and yA2′+>yA3′+; then l(A2′+)<l(A3′+), and the successor function of A3′+ is g(A3′+)=l(A2′+)-l(A3′+)<0.Figure 3By Lemma8 we know there exists an order one periodic solution of system (2), whose initial point is between A+ and A3′+ in set N.Thus we have the following theorem.Theorem 10.
Ifad-bc<0, there exists an order one periodic solution in system (2).


## 4. Attractiveness of Order One Periodic Solution of System (2)

In this section, we discuss the attractiveness of system (2).Theorem 11.
Assume that conditionsα>β and ad-bc<0 hold; then,(a)
there exist order one periodic solutions of system (2) in the region AA3A3+A+.(b)
If the periodic solution is unique, it is attractive, where the attractive region is in the regionAA3A3+A+ and the coordinate A(h/(k1+1),k1h/(k1+1)), A3(dh/(c+d),ch/(c+d)),A3+(dh(1-α)/(c+d),ch(1-β)/(c+d)), A+(h(1-α)/(k+1),kh(1-β)/(k+1)) (see Figure 1).Proof.
(a) By the derivation of Theorem10 and the conditions α>β, we know g(A+)=l(A1+)-l(A+)<0, g(A3+)=l(A2+)-l(A3+)>0, so there exists an order one periodic solution at least of system (2) whose initial point is between A3+ and A+ in set N. According to the continuity of successor function and the zero point theorem of continuous function, we can easily prove there exists odd-numbered order one periodic solution of system (2).
(b) If the periodic solution is unique, assume trajectoryP+P^ and segment PP+ are the periodic solution of system (2) with initial point P+∈N (see Figure 4). Next, we will prove the periodic solution is attractive.
On the one hand, setF(C)=A1; then owning to the disjointness of any two trajectories, we have xA1<xP and yA1>yP; thus, xC<xA1+<xP+ and yC>yA1+>yP+. So, we obtain l(A1+)>l(C) and g(C)=l(A1+)-l(C)>0; set F(A1+)=A2; we know xA1<xA2<xP and yA1>yA2>yP; then we have xA1+<xA2+<xP+ and yA1+>yA2+>yP+. So l(A2+)>l(A1+) and g(A1+)=l(A2+)-l(A1+)>0. This process is continuing; then we get a sequence {Ak+}k=1,2,… of set N satisfying xA1+<xA2+<⋯<xAk+<⋯<xP+ and yA1+>yA2+>⋯>yAk+>⋯>yP+, so l(Ak+1+)>l(Ak+), and g(Ak+)=l(Ak+1+)-l(Ak+)>0. Series {yAk+}k=1,2,… decreases monotonously and has lower bound, and series {xAk+}k=1,2,… increases monotonously and has upper bound, so limk→+∞xAk+ and limk→+∞yAk+ exist. Next we will prove limk→+∞xAk+=xP+ and limk→+∞yAk+=yP+. Set A*+=limk→+∞Ak+; that is, we will prove A*+=P+. Otherwise, A*+≠P+, set F(A*+)=A~; then we have xA~<xP,xA~+<xP+ and yA~>yP,yA~+>yP+. Since g(A*+)≥0 and A*+≠P+, we have g(A*+)=l(A~+)-l(A*+)>0; thus, we have xA*+<xA~+<xP and yA*+>yA~+>yP holds. Set F(A~+)=A~~; we know xA~<xA~~<xP,yA~>yA~~>yP, and xA~+<xA~~+<xP+, yA~+>yA~~+>yP+; then we have g(A~+)=l(A~~+)-l(A~+)>0, this is contradicted to the fact that A*+ is a limit of sequence {Ak+}k=1,2,…. So we obtain A*+=P+ and limk→+∞xAk+=xP+ and limk→+∞yAk+=yP+.
On the other hand, setF(D)=B1; then owning to the disjointness of any two trajectories, we have xB1>xP and yB1<yP; thus, xD>xB1+>xP+ and yD<yB1+<yP+. So, g(D)=l(B1+)-l(D)>0; set F(B1+)=B2. Because xD>xB1+>xP+ and yD<yB1+<yP+, we know xB1>xB2>xP and yB1<yB2<yP; then we have xB1+>xB2+>xP+ and yB1+<yB2+<yP+. So l(B2+)<l(B1+) and g(B1+)=l(B2+)-l(B1+)<0; this process is continuing; then we get a sequence {Bk+}k=1,2,… of set N satisfying xB1+>xB2+>⋯>xBk+>⋯>xP+ and yB1+<yB2+<⋯<yBk+<⋯<yP+ so l(Bk+1+)<l(Bk+) and g(Bk+)=l(Bk+1+)-l(Bk+)<0. Series {xBk+}k=1,2,… decreases monotonously and has lower bound and series {yBk+}k=1,2,… increases monotonously and has upper bound, so limk→+∞xBk+ and limk→+∞yBk+ exist. Next we will prove limk→+∞xBk+=xP+ and limk→+∞yBk+=yP+. Set B*+=limk→+∞Bk+; that is, we will prove B*+=P+. Otherwise, B*+≠P+, set F(B*+)=B~; then we have xB~>xP,xB~+>xP+, and yB~<yP,yB~+<yP+. Since g(B*+)≥0 and B*+≤P+; we have g(B*+)=l(B~+)-l(B*+)<0; thus, we have xB*+>xB~+>xP, and yB*+<yB~+<yP holds. Set F(B~+)=B~~, we know xB~>xB~~>xP, yB~<yB~~<yP, and xB~+>xB~~+>xP+, yB~+<yB~~+<yP+; then we have g(B~+)=l(B~~+)-l(B~+)<0, this is contradicted to the fact that B*+ is a limit of sequence {Bk+}k=1,2,…. So we obtain B*+=P+ and limk→+∞xBk+=xP+ and limk→+∞yBk+=yP+. Since the trajectory initiating any point of Ω0={(x,y)∣x≥0, y≥0, x(0)+y(0)<h} is certain to intersect with set N; next we only need to prove that the trajectory initiating any point of N will ultimately tend to the unique order one periodic solution P+PP+^.
Any pointU∈N must be in some interval [l(B(k+1)+),l(Bk+))k=2,3,…, [l(B1+),l(C)),(l(C),l(A1+)],(l(Ak+),l(Ak+1+)]k=2,3,…. Without loss of generality, we assume U∈[l(Bk+1+),l(Bk+)). The trajectory initiating point U moves between trajectory Bk+Bk+1^ and Bk+1+Bk+2^ and intersects with M at some point between Bk+2 and Bk+1; under the impulsive effects it jumps to the point of N which is between [l(Bk+1+),l(Bk+2+)); then trajectory Π~(U,t) continues to move between trajectory Bk+1+Bk+2^ and Bk+2+Bk+3^. This process can be continued unlimitedly. Since limk→+∞xBk+=xP+ and limk→+∞yBk+=yP+, the intersection sequence of trajectory Π~(U,t) and set N will ultimately tend to point P+. Thus the trajectory initiating any point of N will ultimately tend to the unique order one periodic solution P+PP+^.
From the above analysis, we know the trajectory initiating any point ofN will ultimately tend to order one periodic solution P+PP+^. Therefore, order one periodic solution P+PP+^ is attractive in the region Ω0={(x,y)∣x≥0,y≥0,x(0)+y(0)<h}. This completes the proof. Like the analysis of Theorem 11, we have the following theorem.Figure 4Theorem 12.
Assume that conditionsα>β and ad-bc<0 hold; then,(a)
there exists an order one periodic solutions of system (2) in the region A3′AA+A3′+.(b)
If the periodic solution is unique, it is attractive, where the attractive region is in the regionA3′AA+A3′+ and the coordinate A3′(ah/(a+b),bh/(a+b)), A(h/(k1+1),k1h/(k1+1)),A+(h(1-α)/(k+1),kh(1-β)/(k+1))A3′+(ah(1-α)/(a+b),bh(1-β)/(a+b)) (see Figure 3).Theorem 13.
ifα=β and ad-bc<0 hold, in view of vector field and Lemma 9, the segment BA is a unique order one periodic solution of system (2) which is attractive in the region Ω0={(x,y)∣x≥0,y≥0,x(0)+y(0)<h}.Proof.
In view of vector field and Lemma9, we can easily prove the theorem; thus, the proof is omitted.


## 5. Conclusion

To control pest, a stage-structure model with state-dependent impulsive control was investigated. We defined the successor functions of semicontinuous dynamic system and gave a new method to judge the existence of order one periodic solution of such system. By means of sequence convergence rules and qualitative analysis, we give sufficient conditions of the existence of order one periodic solution, as well as global attractiveness of order one periodic solution. To verify the theoretical results obtained in this paper, we will give some numerical simulations. Leta=0.6, b=0.8, c=0.5, d=0.2, h=1. By comparing Figures 1, 2, and 3 with Figures 5(a), 5(b), and 5(c), it is obvious that our new method to judge the existence of order one periodic solution is precise. From Figures 6(a) and 6(b), using our spraying strategy, pest population can be controled under the economic threshold, whose method reduces insecticide application and protects environment. It is the first time to put forward the method. Our results have provided some fundamental theoretical conclusions that could be of applied importance.Phase portrait of system (2) to see the effect of the proportion of killed immature and mature α and β with a=0.6, b=0.8, c=0.5, d=0.2, and  h=1. (a) α>β. (b) α=β. (c) α<β.
(a)
(b)
(c)Permanence of system (2) to see the effect of the the proportion of killed immature and mature α and β. Other parameters are the same with those in Figure 5. (a) Time-series of the population x(t). (b) Time-series of the population y(t).
(a)
(b)Remark 14.
The attractiveness of order one periodic solution of system (2) can be shown by Figure 7. We take some different initial values, for example, (0.25,0.55), (0.25,0.15) and (0.4,0.15), respectively. By numerical simulations, we find that the trajectories initiating different values finally tend to a unique order one periodic solution of system (2). This indicates that the order one periodic solution of system (2) is globally attractive and we guess that it is unique order one periodic solution of system (2).Figure 7
The global attractiveness of system (2) with parameters a=0.6, b=0.8, c=0.5, d=0.2, h=0.2, α=0.5, β=0.3. The initial values for trajectory L1, L2 and L3 are (0.25,0.55), (0.25,0.15) and (0.4,0.15), respectively.



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