# Stability Analysis and Control Optimization of a Prey-Predator Model with Linear Feedback Control

Authors: Yaning Li; Yan Li; Yu Liu; Huidong Cheng
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The application of pest management involves two thresholds when the chemical control and biological control are adopted, respectively. Our purpose is to provide an appropriate balance between the chemical control and biological control. Therefore, a Smith predator-prey system for integrated pest management is established in this paper. In this model, the intensity of implementation of biological control and chemical control depends linearly on the selected control level (threshold). Firstly, the existence and uniqueness of the order-one periodic solution (i.e., OOPS) are proved by means of the subsequent function method to confirm the feasibility of the biological and chemical control strategy of pest management. Secondly, the stability of system is proved by the limit method of the successor points’ sequences and the analogue of the Poincaré criterion. Moreover, an optimization strategy is formulated to reduce the total cost and obtain the best level of pest control. Finally, the numerical simulation of a specific model is performed.


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## Body

## 1. Introduction

In the practical production, effective control of pests is a very important issue of the world, which catches attention of scholars for pest management method [1–7]. Integrated pest management (IPM), also known as integrated pest control (IPC), is an effective approach that integrates biological, chemical tactics, and physical methods for pests control [8–11]. Due to population dynamics and its related environment, IPM utilize effective methods and techniques comprehensively to reduce the level of economic harm caused by pests. The aim of IPM is to control the density of the insects under the economic threshold by integrated usage of less harmful pesticides and biological control methods for maximizing the protection of the ecosystem.In mathematics, impulsive differential equations (IDES) is such a powerful tool to describe these phenomena that rapid changes in biological populations are caused by the variety of the pests control by artificial intervention [12–22]. In recent years, the theoretical studies on IDES have produced a lot of good research results [23–34]. Based on the theoretical research, some scholars have introduced impulsive differential equations in Lotka-Volterra system such as the regular release of predators [35–37]; the periodic release of infected pests [38–40]; the periodic release of predators together with regular spray of pesticides [41–43]; the periodic release of predators and infected pests together with regular spray of pesticides [39, 44]. In the practical application, the two control measures can be adopted at two different levels of pest density concerning this case. Nie et al. [45], Tian et al. [46], Zhao et al. [47], and Zhang et al. [48] studied the following predator-prey system and assumed that different control measures were adopted at different thresholds,(1)u′t=auln⁡Ku-buv,v′t=vt-d+λbu,u≠h1,h2,oru=h1,v>v∗,Δut=0,Δvt=γ,u=h1,v=v∗,Δut=-δut,Δvt=-ϱvt+τ,u=h2,where the intrinsic growth rate of prey is denoted by a, the environment carrying capacity is denoted by K, the predation rate by natural enemies is denoted by b, and the transformation rate and the death rate of predator are denoted by λ and d, respectively. The η is a positive parameter, and the effect of pesticide to predator and prey species is denoted by ϱ and δ, respectively. The releasing quantity of natural enemy v(t) are denoted by γ and τ, respectively.It is of great practical significance to adopt biological and chemical control strategies based on the different pest thresholds. But an important issue in this process should be pointed out, in which the biological control is carried out when the density of pest denoted byu(t) reaches the threshold h1, and when the density u(t) reaches the threshold h2, the integrated control strategy is adopted. But no strategy adopted for the density of pest denoted by u(t)=h, where h1<h<h2, which is obviously unreasonable. In addition, from an economic and practical point of view, the control taken at threshold h1 seems to be early and the amount of releasing predators will also be huge, while the control taken at threshold h2 seems to be late and the intensity of chemical control will also be high. Considering the above problems, we should choose a pest control method between h1 and h2.An outline of this paper is as follows. In next section, a pest management Smith model is formulated. Then the existence, uniqueness, and the asymptotically orbit stability of order-one periodic solution (OOPS) of system (7) are proved in Section 3. In Section 4, an optimization problem is formulated and obtained the minimized total cost in pest control. The theoretical results are verified by numerical simulations in Section 5. Finally, a conclusion is drawn.


## 2. Model Formulation

In biological mathematics, Logistic model [10](2)dxdt=rx1-xKis a classical mathematical model, where the predator and prey densities at time t are denoted by y(t) and x(t). r denotes the intrinsic rate of growth and K denotes the maximum environment carrying capacity, while system (2) is based on the assumption that the relative growth rate dx/xdt of the population size is linear function 1-x/K. In 1963, F.E.Smith found that the data about the population of Daphnia did not conform to the linear function [49]. Thus, Smith assumed that the relative growth rate of population density at time t is proportional to the amount of remaining food; i.e.,(3)1xdxdt=r1-HtT,where H(t) is the rate of food demand of the population at time t; T is the rate of demand for food in a population saturated state. Smith assumed that the food required to keep the population is c1x(t) and the food required for the population to reproduce is c2dx/dt. That is to say, (4)Hx=c1xt+c2dxdt.Then (5)dxdt=rx1-T-c1xT+rc2x.Considering the demand for food of population reproduction, the Smith model uses the hyperbolic function (T-c1x)/(T+rc2x) instead of the linear function in the Logistic model. Thus, the Smith model is a further improvement of Logistic model. With the absence of predators, the per capita growth rate lgrow of the pest is assumed to be the Smith growth [49] model.(6)lgrow=rxK-xK-r/cx.By the control strategy, the following predator-prey Smith system is investigated in this paper:(7)dxdt=mxt-rxt2K+dxt-qxtyt,dydt=μxtyt-lyt,x<h,Δxt=-αxxt,Δyt=-βxyt+σx,x=h,y≤y¯h,where the releasing amount of the predator is denoted by σ(x) and σ(h1)=σmax, σ(h2)=σmin, where 0≤σmin<σmax. The strength of chemical control to the prey is α(x) and that to the predator is β(x), where the parameters σ(x),α(x),β(x) are continuous functions and satisfies α(h2)=αmax, β(h2)=βmax. A pest control level h is between h1 and h2. y¯h denotes the level of the predator at a lower density. By calculation we obtain y¯h=(λKq-l)/q(Kqλ-dl), where rK≜m,r/c≜d,λq≜μ<l are constants. When the density of predator is below y¯h, the chemical control is taken. Clearly, the control strategy of system (7) changes into the biological control strategy of system (1) when parameters α(x), β(x), and σ(x), x of system (7), are chosen 0, 0, γ, h1, respectively. When parameters α(x), β(x), and σ(x), x of system (7), are chosen δ, ϱ, τh2, respectively, the control strategy of system (7) turns into the integrated control strategy of system (1). Therefore, system (7) is the further promotion of system (1).In our paper,σ(x), α(x), and β(x) are assumed to have the following linear form [10](8)σx=σmax-σmax-σminx-h1h2-h1,αx=αmaxx-h1h2-h1βx=βmaxx-h1h2-h1


## 3. Dynamical Analysis of System (7)

In this section, we dynamically analyze system (7) to study the existence, uniqueness and orbital asymptotical stability of the OOPS. For convenience, OOPS is used to represent the order-one periodic solution.


### 3.1. Qualitative Analysis of System (7)

We first study the following continuous system of system (7); i.e.,(9)dxtdt=mxt-rxt2K+dxt-qxtyt,dytdt=μxtyt-lyt.Let(10)mxt-rxt2K+dxt-qxtyt=0,μxtyt-lyt=0.Then we get three equilibria O(0,0), E0(m/r,0), and E∗(x∗,y∗), where (11)x∗=lμ,y∗=μm-lrqμK+dl.Let (12)I:lμ>Δ-rKrd,where Δ=r2K2+mKrd. Thus, we get the following theorem.Theorem 1.
The positive equilibrium pointE∗(x∗,y∗) is locally asymptotically stable, if (I) holds.Proof.
At the pointE∗(x∗,y∗), the Jacobian matrix is (13)JE∗=mK-rx∗dx∗+2KK+dx∗2-qy∗-qx∗μy∗0,then (14)detJE∗=qμx∗y∗>0,tr⁡JE∗=mK-rx∗dx∗+2KK+dx∗2-qy∗.
When (I) holds, thentr(J(E∗))<0. Thus the point E∗ is locally asymptotically stable.Theorem 2.
Ifμx≤l holds, then the point E∗ is globally asymptotically stable.Proof.
LetB=1/x, then we have (15)D=∂PB∂x+∂QB∂x=-rK+dx-dm-rxK+dx2+μ-lx=-rK-dmK+dx2+μ-lxwhen μx≤l, then D<0.
By the method in [48], the point E∗(x∗,y∗) is globally asymptotically stable (see Figure 1.)Figure 1
The phase diagram of system (7) with m=1, r=0.3, K=2, d=0.5, q=0.6, μ=0.5, and l=0.4.


### 3.2. Existence and Uniqueness of the Periodic Orbit of System (7)

For convenience, letH(x,y)=H0 denote the first integral of system (7), where the implicit function H(x,y)=H(x0,y0) is divided into upper and lower branches by isoclinic line dx/dt=0 denoted by yH+(x,P0) and yH-(x,P0), where the starting point is P0. The impulsive set of system (7) is denoted by ∑M={(x,y)∣x=h,0≤y≤yh¯} and the phase set is denoted by ∑N={(x,y)∣x=(1-α(h))h,y≥0}. Let L1={(x,y)∣x=l/μ,y≥0} and L2={(x,y)∣0≤x,y=(m-rx)/q(K+dx)} as the isoclinic line dy/dt=0 and dx/dt=0, respectively. For any point A(xA,yA), where xA and yA are denoted as the abscissa and ordinate of point A, respectively. By the location of the threshold h and positive equilibrium point E∗, we get the following theorem.Theorem 3.
If0<h1<h<min⁡{x∗,h2} holds, then a uniqueness OOPS exists in system (7).Proof.
Case I (0<h1<h<min⁡x∗,h2). In view of Theorems 1 and 2, for any point Q0∈Ω where Ω={(x,y)∣0≤x≤(1-α(h))h,y∈R} the trajectory yH-(x,Q0) has an intersection point with phase set ∑N. Thus, we discuss the trajectory tendency of the initial point on the phase set ∑N.
Assuming the intersection point of phase set∑N and isoclinic line dx/dt=0 is point A(xA,yA), where xA=(1-α(h))h and yA=(m-rh1-αh)/q[K+dh(1-α)]. The trajectory yH-((1-α)h,A) intersects with pulse set ΣM at point A-(xA-,yA-), then the impulsive function can transfer the point A-(xA-,yA-) into the point A+(xA+,yA+). Thus, we have (16)yA+=1-βhyA-+σh=1-βhyH-h,A+σh;define (17)σ¯h≜yA+-1-βhyH-h,A.By the magnitudes between σ(h) and σ¯(h), one has
(i)σ(h)≤σ¯.
Ifσ(h)=σ¯, the subsequent function of point A(xA,yA) is g(A)=0, then the trajectory AA-A+^ is an OOPS.
Ifσ(h)<σ¯, the point A+(xA+,yA+) under the point A(xA,yA), thus the subsequent function of point A is (18)gA=yA+-yA<0.The phase set ∑N intersects with x-axis at point B(xB,0), where xB=h(1-α(h)). By the orbit tendency, yH-(x,B) intersects with the impulsive set ∑M at the point B-(xB-,yB-) which jumps to the point B+(xB+,yB+). Obviously, the point B(h(1-α(h)),0) is under the point B+(xB+,yB+). The subsequent function of the point B(h(1-α(h)),0) is (19)gB=yB+-yB>0.
According to the continuity of subsequent function, there must be a pointC between point A and B, which makes (20)gC=0.(See Figures 2(a) and 2(b).)
(ii)σ(h)>σ¯
Ifσ(h)>σ¯, then the point A+(xA+,yA+) must be above the point A(xA,yA). Thus the subsequent function g(A)>0. The orbit yH-(x,A+) will intersect with impulsive set ∑M at point A1-(xA1-,yA1-), then hits phase set ∑N at point A1+(xA1+,yA1+). Clearly, the point A1-(xA1-,yA1-) is under the point A-(xA-,yA-). Thus the point A1+(xA1+,yA1+) must be under the point A+(xA+,yA+). The subsequent function of point A+(xA+,yA+) is (21)gA+=yA1+-yA+<0.So there must be a point D∈∑N, such that g(D)=0 (see Figure 2(c)).
Now, the uniqueness of OOPS of system (7) is to be discussed.
Assuming thatP1,P2∈QA^, then orbit P1P1-P1+^ and P2P2-P2+^ are OOPS, where yP1<yP2.(22)yP1+=1-βhhyP1-+σh,yP2+=1-βhhyP2-+σh,
Assume(23)δP1P2x=yH-x,P2-yH-x,P1,x∈1-βhh,h,then (24)δP1P2′x=yH-′x,P2-yH-′x,P1=μxy-lymx-rx2/K+dx-qxy=μx-lK+dxxyP2m-rx-qyP2K+dx-yP1m-rx-qyP1K+dx=μx-lK+dxxωξyP2-yP1,where(25)ωy=ym-rx-qyK+dx,and(26)ω′y=yr+dqym-rx-qyK+dx2>0.
SoδP1P2(x)<0, x∈[(1-α)h,h]; that is to say, δP1P2(x) is a decreasing function. For x∈[(1-α)h,h], then δP1P2(h)<δP1P2((1-α)h).
Thus(27)σh=yP2+-1-βyP2-=yP2-1-βyP2-=yP1+δP1P21-αh-1-βyP1-+δP1P2h=yP1-1-βyP1-+δP1P21-αh-1-βδP1P2h>yP1-1-βyP1-=σh.which is a contradiction.
Whenσ(h)>σ¯, then there exists a point P∈AA+¯ such that yP+=yP. Thus, for any P′∈BA+¯, the subsequent function of point P′ is (28)gP′=yP′+-yP′=1-βyP′-+σh-yP′=1-βyP′--yP′+yP-1-βyP-=yP-yP′+1-βyP′--yP-.
According to the proof above, we haveyP-yP′>yP-yP′-. Thus, if yP′<yP, then g(P′)<0.
IfyP′>yP, then g(P′)>0. Thus the uniqueness of OOPS of system (7) in case of σ(h)>σ¯ is proved. The proof is completed.Figure 2
The existence of the OOPS of system (7). (a) Discussion in Case (i) if σ(h)=σ-. (b) Discussion in Case (i) if σ(h)<σ-. (c) Discussion in Case (ii).
(a) (b) (c)Theorem 4.
Ifmax⁡x∗,h1<h<h2 holds, then we have two cases. If h>h0 holds, then system (7) has no OOPS. If h<h0 holds, then system (7) has a unique OOPS, where h0=max⁡{x∣0<yH-(x,B)≤h2}.Proof.
Case II ( max ⁡ x ∗ , h 1 < h ≤ h 2). Two cases are discussed according the magnitude of h and h0.
(i) Ifh>h0, the trajectory yH-(x,B) tending to point E∗(x∗,y∗) is based on the global asymptotical stability of E∗(x∗,y∗). For any P(x,y)∈Ω, the trajectory yH-(x,P) tends to E∗(x∗,y∗). (See Figure 3.)
(ii) Ifh<h0, the trajectory yH-(x,B) must intersect with the impulsive set ∑M at the point B-(xB-,yB-), which jumps to the point B+(xB+,yB+). The subsequent function of point B(xB,yB) is g(B)=yB+-yB>0. Similar to the proof of Case I, when σ≤σ¯, an OOPS exists in system (7). If σ>σ¯, we also can prove that an OOPS exists in system (7) by same method of Case I. (See Figure 4.)Figure 3
The existence of the OOPS of system (2) if h>h0 in Case (II).Figure 4
The existence of the OOPS of system (7) if h<h0 in Case (II).
(a) (b) (c)


## 3.1. Qualitative Analysis of System (7)

We first study the following continuous system of system (7); i.e.,(9)dxtdt=mxt-rxt2K+dxt-qxtyt,dytdt=μxtyt-lyt.Let(10)mxt-rxt2K+dxt-qxtyt=0,μxtyt-lyt=0.Then we get three equilibria O(0,0), E0(m/r,0), and E∗(x∗,y∗), where (11)x∗=lμ,y∗=μm-lrqμK+dl.Let (12)I:lμ>Δ-rKrd,where Δ=r2K2+mKrd. Thus, we get the following theorem.Theorem 1.
The positive equilibrium pointE∗(x∗,y∗) is locally asymptotically stable, if (I) holds.Proof.
At the pointE∗(x∗,y∗), the Jacobian matrix is (13)JE∗=mK-rx∗dx∗+2KK+dx∗2-qy∗-qx∗μy∗0,then (14)detJE∗=qμx∗y∗>0,tr⁡JE∗=mK-rx∗dx∗+2KK+dx∗2-qy∗.
When (I) holds, thentr(J(E∗))<0. Thus the point E∗ is locally asymptotically stable.Theorem 2.
Ifμx≤l holds, then the point E∗ is globally asymptotically stable.Proof.
LetB=1/x, then we have (15)D=∂PB∂x+∂QB∂x=-rK+dx-dm-rxK+dx2+μ-lx=-rK-dmK+dx2+μ-lxwhen μx≤l, then D<0.
By the method in [48], the point E∗(x∗,y∗) is globally asymptotically stable (see Figure 1.)Figure 1
The phase diagram of system (7) with m=1, r=0.3, K=2, d=0.5, q=0.6, μ=0.5, and l=0.4.


## 3.2. Existence and Uniqueness of the Periodic Orbit of System (7)

For convenience, letH(x,y)=H0 denote the first integral of system (7), where the implicit function H(x,y)=H(x0,y0) is divided into upper and lower branches by isoclinic line dx/dt=0 denoted by yH+(x,P0) and yH-(x,P0), where the starting point is P0. The impulsive set of system (7) is denoted by ∑M={(x,y)∣x=h,0≤y≤yh¯} and the phase set is denoted by ∑N={(x,y)∣x=(1-α(h))h,y≥0}. Let L1={(x,y)∣x=l/μ,y≥0} and L2={(x,y)∣0≤x,y=(m-rx)/q(K+dx)} as the isoclinic line dy/dt=0 and dx/dt=0, respectively. For any point A(xA,yA), where xA and yA are denoted as the abscissa and ordinate of point A, respectively. By the location of the threshold h and positive equilibrium point E∗, we get the following theorem.Theorem 3.
If0<h1<h<min⁡{x∗,h2} holds, then a uniqueness OOPS exists in system (7).Proof.
Case I (0<h1<h<min⁡x∗,h2). In view of Theorems 1 and 2, for any point Q0∈Ω where Ω={(x,y)∣0≤x≤(1-α(h))h,y∈R} the trajectory yH-(x,Q0) has an intersection point with phase set ∑N. Thus, we discuss the trajectory tendency of the initial point on the phase set ∑N.
Assuming the intersection point of phase set∑N and isoclinic line dx/dt=0 is point A(xA,yA), where xA=(1-α(h))h and yA=(m-rh1-αh)/q[K+dh(1-α)]. The trajectory yH-((1-α)h,A) intersects with pulse set ΣM at point A-(xA-,yA-), then the impulsive function can transfer the point A-(xA-,yA-) into the point A+(xA+,yA+). Thus, we have (16)yA+=1-βhyA-+σh=1-βhyH-h,A+σh;define (17)σ¯h≜yA+-1-βhyH-h,A.By the magnitudes between σ(h) and σ¯(h), one has
(i)σ(h)≤σ¯.
Ifσ(h)=σ¯, the subsequent function of point A(xA,yA) is g(A)=0, then the trajectory AA-A+^ is an OOPS.
Ifσ(h)<σ¯, the point A+(xA+,yA+) under the point A(xA,yA), thus the subsequent function of point A is (18)gA=yA+-yA<0.The phase set ∑N intersects with x-axis at point B(xB,0), where xB=h(1-α(h)). By the orbit tendency, yH-(x,B) intersects with the impulsive set ∑M at the point B-(xB-,yB-) which jumps to the point B+(xB+,yB+). Obviously, the point B(h(1-α(h)),0) is under the point B+(xB+,yB+). The subsequent function of the point B(h(1-α(h)),0) is (19)gB=yB+-yB>0.
According to the continuity of subsequent function, there must be a pointC between point A and B, which makes (20)gC=0.(See Figures 2(a) and 2(b).)
(ii)σ(h)>σ¯
Ifσ(h)>σ¯, then the point A+(xA+,yA+) must be above the point A(xA,yA). Thus the subsequent function g(A)>0. The orbit yH-(x,A+) will intersect with impulsive set ∑M at point A1-(xA1-,yA1-), then hits phase set ∑N at point A1+(xA1+,yA1+). Clearly, the point A1-(xA1-,yA1-) is under the point A-(xA-,yA-). Thus the point A1+(xA1+,yA1+) must be under the point A+(xA+,yA+). The subsequent function of point A+(xA+,yA+) is (21)gA+=yA1+-yA+<0.So there must be a point D∈∑N, such that g(D)=0 (see Figure 2(c)).
Now, the uniqueness of OOPS of system (7) is to be discussed.
Assuming thatP1,P2∈QA^, then orbit P1P1-P1+^ and P2P2-P2+^ are OOPS, where yP1<yP2.(22)yP1+=1-βhhyP1-+σh,yP2+=1-βhhyP2-+σh,
Assume(23)δP1P2x=yH-x,P2-yH-x,P1,x∈1-βhh,h,then (24)δP1P2′x=yH-′x,P2-yH-′x,P1=μxy-lymx-rx2/K+dx-qxy=μx-lK+dxxyP2m-rx-qyP2K+dx-yP1m-rx-qyP1K+dx=μx-lK+dxxωξyP2-yP1,where(25)ωy=ym-rx-qyK+dx,and(26)ω′y=yr+dqym-rx-qyK+dx2>0.
SoδP1P2(x)<0, x∈[(1-α)h,h]; that is to say, δP1P2(x) is a decreasing function. For x∈[(1-α)h,h], then δP1P2(h)<δP1P2((1-α)h).
Thus(27)σh=yP2+-1-βyP2-=yP2-1-βyP2-=yP1+δP1P21-αh-1-βyP1-+δP1P2h=yP1-1-βyP1-+δP1P21-αh-1-βδP1P2h>yP1-1-βyP1-=σh.which is a contradiction.
Whenσ(h)>σ¯, then there exists a point P∈AA+¯ such that yP+=yP. Thus, for any P′∈BA+¯, the subsequent function of point P′ is (28)gP′=yP′+-yP′=1-βyP′-+σh-yP′=1-βyP′--yP′+yP-1-βyP-=yP-yP′+1-βyP′--yP-.
According to the proof above, we haveyP-yP′>yP-yP′-. Thus, if yP′<yP, then g(P′)<0.
IfyP′>yP, then g(P′)>0. Thus the uniqueness of OOPS of system (7) in case of σ(h)>σ¯ is proved. The proof is completed.Figure 2
The existence of the OOPS of system (7). (a) Discussion in Case (i) if σ(h)=σ-. (b) Discussion in Case (i) if σ(h)<σ-. (c) Discussion in Case (ii).
(a) (b) (c)Theorem 4.
Ifmax⁡x∗,h1<h<h2 holds, then we have two cases. If h>h0 holds, then system (7) has no OOPS. If h<h0 holds, then system (7) has a unique OOPS, where h0=max⁡{x∣0<yH-(x,B)≤h2}.Proof.
Case II ( max ⁡ x ∗ , h 1 < h ≤ h 2). Two cases are discussed according the magnitude of h and h0.
(i) Ifh>h0, the trajectory yH-(x,B) tending to point E∗(x∗,y∗) is based on the global asymptotical stability of E∗(x∗,y∗). For any P(x,y)∈Ω, the trajectory yH-(x,P) tends to E∗(x∗,y∗). (See Figure 3.)
(ii) Ifh<h0, the trajectory yH-(x,B) must intersect with the impulsive set ∑M at the point B-(xB-,yB-), which jumps to the point B+(xB+,yB+). The subsequent function of point B(xB,yB) is g(B)=yB+-yB>0. Similar to the proof of Case I, when σ≤σ¯, an OOPS exists in system (7). If σ>σ¯, we also can prove that an OOPS exists in system (7) by same method of Case I. (See Figure 4.)Figure 3
The existence of the OOPS of system (2) if h>h0 in Case (II).Figure 4
The existence of the OOPS of system (7) if h<h0 in Case (II).
(a) (b) (c)


## 4. The Orbital Asymptotical Stability of OOPS of System (7)

According to the discussion above, a unique OOPS exists in system (7), denoted by PP-P+^. Then we get the following theorem.Theorem 5.
Ifσ≤σ¯, then the OOPS of system (7) is orbitally asymptotically stable and globally attractive to the point E∗.Proof.
We choose arbitrary pointA0 on the phase set ∑N. If A0∈N/AP¯, then after several pulse effects the trajectory will jump to the segment AP¯. Thus we assume that A0∈AP¯; the trajectory yH-(x,A0) will hit the impulsive set ∑M at point A1-, which jumps to the point A1+. The trajectory yH-(x,A1+) will intersect with impulsive set ∑M at point A2- and then jumps to the point A2+. Repeat the process above; we get a point sequences {Ak+}, where k=1,2,3,⋯ such that (29)yA>yA0>yA1+>⋯>yAk+>⋯≥yP.
The sequenceAk+k=0,1,2,⋯ is a monotonic decreasing sequence with lower bound yP. According to the monotonic bounded theorem, there must exist a limit yP′ such that limk→∞yAk+=yP′, which means that (30)gP′=glimk→∞⁡yAk+=limk→∞⁡gyAk+=limk→∞⁡yAk+1+-yAk+=0.Since g(A)=0, if and only if A=P, then P′=P. That is to say limk→∞yAk+=yP.
Similarly, we can use the above method to get an increasing point sequencesBk+k=0,1,2,⋯ such that (31)yB<yB0<yB1+<⋯<yBk+<⋯≤yP.There must exist a limit yP′ such that limk→∞yBk+=yP′, which means that (32)gP′=glimk→∞⁡yBk+=limk→∞⁡gyBk+=limk→∞⁡yBk+1+-yBk+=0.Since g(B)=0, if and only if B=P, then P′=P. That is to say, limk→∞yBk+=yP. By the arbitrariness of the point A0 and B0, one has (33)limk→∞⁡yAk+=limk→∞⁡yBk+=yP.
Thus the OOPS of system (7) is orbitally asymptotically stable and globally attractive (see Figure 5).Figure 5
The orbitally asymptotically stability of the OOPS of system (7).Theorem 6.
Ifσ>σ¯ and γ1[m-rη0-qγ0(K+dη0)](1-β)(K+dη1)/γ0(m-rη1-qγ1)(K+dη0)<1, then the OOPS of system (7) is orbitally asymptotically stable.Proof.
Letx=η(t),y=γ(t) is a T-periodic orbit of system (7) and η0=η(0),γ0=γ(0),η1=η(T),γ1=γ(T),η1+=η(T+),γ1+=γ(T+), then (34)η1+=η0=1-αh,γ1+=γ0=1-βγ1+σ.Let (35)Px,y=mx-rx2K+dx-qxy,Qx,y=μxy-ly,ζx,y=-αx,ξx,y=-βy+σ.Then (36)∂ζ∂x=-α,∂ξ∂x=-β,∂ζ∂y=0,∂ξ∂y=-β,∂φ∂x=1,∂φ∂y=0Δ1=P+∂ξ/∂y∂φ/∂x-∂ξ/∂x∂φ/∂y+∂φ/∂x+Q+∂ζ/∂x∂φ/∂y-∂ζ/∂y∂φ/∂x+∂φ/∂yP∂φ/∂x+Q∂φ/∂y,=Pη1+,γ1+-β-0+1+Qη1+,γ1+-α×0-r×0+0Pη1,γ1×1+Qη1,γ1×0=η0m-rη0/K+dη0-qγ01-βη1m-rη1/K+dη1-qγ1.and (37)
∫
0
T
∂
P
∂
x
+
∂
Q
∂
y
d
t
=
∫
0
T
m
-
2
r
x
K
+
d
x
-
d
m
x
-
r
x
2
K
+
d
x
2
-
q
y
+
μ
x
-
l
d
t
=
∫
0
T
m
-
r
x
K
+
d
x
-
r
x
K
+
d
x
-
d
m
x
-
r
x
2
K
+
d
x
2
-
q
y
+
μ
x
-
l
d
t
=
∫
0
T
x
˙
x
t
+
y
˙
y
t
-
∫
0
T
r
x
K
+
d
x
+
d
m
x
-
r
x
2
K
+
d
x
2
d
t
=
ln
⁡
x
T
y
T
x
0
y
0
-
∫
0
T
r
x
K
+
d
x
+
d
m
x
-
r
x
2
K
+
d
x
2
d
t
=
ln
⁡
x
T
y
T
x
0
y
0
-
∫
0
T
d
m
x
+
r
K
x
K
+
d
x
2
d
t
.
μ
2
=
Δ
1
∫
0
T
∂
P
∂
x
+
∂
Q
∂
y
d
t
=
m
η
0
-
r
η
0
2
-
q
η
0
γ
0
K
+
d
η
0
1
-
β
K
+
d
η
0
K
+
d
η
1
m
η
1
-
r
η
1
2
-
q
η
1
γ
1
η
1
γ
1
η
0
γ
0
exp
⁡
∫
0
T
-
d
m
x
-
r
K
x
K
+
d
x
2
=
γ
1
m
-
r
η
0
-
q
γ
0
K
+
d
η
0
1
-
β
K
+
d
η
1
γ
0
m
-
r
η
1
-
q
γ
1
K
+
d
η
0
exp
⁡
∫
0
T
-
d
m
x
-
r
K
x
K
+
d
x
2
.
Thus, whenσ>σ¯ and γ1[m-rη0-qγ0(K+dη0)](1-β)(K+dη1)/γ0(m-rη1-qγ1)(K+dη0)<1, μ2<1. Therefore, the OOPS is orbitally asymptotically stable.


## 5. Numerical Simulations and Optimization of Pest Control Level

### 5.1. Numerical Simulations

In this section, the feasibility of our conclusions is verified by an example. Letm=1, r=0.3, K=2, d=0.5, q=0.6, μ=0.5, and l=0.4. By calculation, the equilibrium point E∗ of system (7) is E∗(0.8,0.528). Parameter values are taken into system (7), then(38)dxdt=xt-0.3xt22+0.5xt-0.6xtyt,dydt=0.5xtyt-0.4yt,x<h,Δxt=-αxxt,Δyt=-βxyt+σx,x=h,y≤y¯hLeth=0.7 satisfy the condition 0<h1<h<min⁡{x∗,h2} and the initial value be (0.4,0.4). Let σmax=1.5, δmin=0.1, αmax=1.75, βmax=2, h1=0, and h2=2. A directed calculation yields that α0.7=0.15, β0.7=0.5, and σ0.7¯=0.2975. Let σ=0.26, σ=σ0.7¯=0.2975, and σ=0.3. Figures 6(a), 6(b), and 6(c) show that a unique and asymptotically stable OOPS exists in system (7).Figure 6
Numerical simulations in case0<h1<h<min⁡{x∗,h2}. (a) Phase portrait of x(t) and y(t) on h=0.7. (b) Time series of x(t). (c) Time series of y(t).
(a) (b) (c)Leth=0.9 satisfy the condition 0<max⁡{x∗,h1}<h<h0<h2 and the initial value be (0.5,1). A directed calculation yields that α0.9=0.5, β0.9=0.3, and σ0.9¯=0.269. Let σ=0.24, σ=σ0.9¯=0.269, and σ=0.3. Figures 7(a), 7(b), and 7(c) show that system (7) has a unique and asymptotically stable OOPS.Figure 7
Numerical simulations in case0<max⁡{x∗,h1}<h<h0<h2. (a) Phase portrait of x(t) and y(t) on h=0.9. (b) Time series of x(t). (c) Time series of y(t).
(a) (b) (c)For the case of0<max⁡{x∗,h1}<h0<h≤h2, for example, h=1.5 and the orbit of system (7) starts from (0.5,1), we get α1.5=0.5, β0.9=0.3 and σ1.5=0.24 by calculation. Figures 8(a), 8(b), and 8(c) show that system (7) has no OOPS.Figure 8
Numerical simulations in case<max⁡{x∗,h1}<h0<h≤h2. (a) Phase portrait of x(t) and y(t) on h=1.5. (b) Time series of x(t). (c) Time series of y(t).
(a) (b) (c)


### 5.2. Determination and Optimization of Pest Control Level

The goal to investigate the existence of OOPS of system (7) lies in that it can obtain the possibility of determining the frequency of releasing predators and spraying pesticides, which makes the density of pest below the damage level. Although the density of prey is inaccurate or biased, the system will eventually undergo periodic changes under the effective control. The following problems are considered to determine the optimal frequency for releasing predators and chemical controls.Assuming that unit cost of releasing predator is denoted byι1 and the unit cost of spraying pesticides is denoted by ι2, which include the price of chemical agent and the price of the damage to environment. Our goal is to reduce the unit cost in this process. In one period, the total cost is denoted by Fι, which is a function about α(h) (i.e., chemical control strength) and σ(h) (i.e., yield of releases of predator). Then Fι(h)=ι1σ(h)+ι2α(h). So the optimization model is formulated as(39)maxFιhThs.t.h1≤h≤h2The optimization problem is solved to yield the optimal pest levelh∗, which the optimal release rate of predator is σ∗=σh∗, the optimal strength of chemical control is α∗=αh∗, and the optimal impulse period of chemical control is T∗=T(σ∗,α∗). However, the optimum pest control level h∗ is dependent on the ratio of ω≜ι2/ι1. The impulse period T varies with the threshold h, as shown in Figure 9(a). And Figure 9(b) shows the variation of cost per unit time F/T and the period T with the pest control level h, where ι1=1000, ι2=1000, i.e., ω=1. The optimal pest level is h∗=0.9, the optimal strength of chemical control is αh∗=0.788, and the optimal release rate of predator is σh∗=0.87. It is important to note that the optimum economic threshold h is dependent on ω, as is illustrated in Figure 10.Figure 9
The variety in the periodT and the profit per unit time F/T on the threshold h. (a) The variety in the period T on the threshold h. (b) The profits per unit time F/T on the threshold h.
(a) (b)Figure 10
The change in the cost per unit timeF/T on the control level h for ι2/ι1=2.5,1,5,10.


## 5.1. Numerical Simulations

In this section, the feasibility of our conclusions is verified by an example. Letm=1, r=0.3, K=2, d=0.5, q=0.6, μ=0.5, and l=0.4. By calculation, the equilibrium point E∗ of system (7) is E∗(0.8,0.528). Parameter values are taken into system (7), then(38)dxdt=xt-0.3xt22+0.5xt-0.6xtyt,dydt=0.5xtyt-0.4yt,x<h,Δxt=-αxxt,Δyt=-βxyt+σx,x=h,y≤y¯hLeth=0.7 satisfy the condition 0<h1<h<min⁡{x∗,h2} and the initial value be (0.4,0.4). Let σmax=1.5, δmin=0.1, αmax=1.75, βmax=2, h1=0, and h2=2. A directed calculation yields that α0.7=0.15, β0.7=0.5, and σ0.7¯=0.2975. Let σ=0.26, σ=σ0.7¯=0.2975, and σ=0.3. Figures 6(a), 6(b), and 6(c) show that a unique and asymptotically stable OOPS exists in system (7).Figure 6
Numerical simulations in case0<h1<h<min⁡{x∗,h2}. (a) Phase portrait of x(t) and y(t) on h=0.7. (b) Time series of x(t). (c) Time series of y(t).
(a) (b) (c)Leth=0.9 satisfy the condition 0<max⁡{x∗,h1}<h<h0<h2 and the initial value be (0.5,1). A directed calculation yields that α0.9=0.5, β0.9=0.3, and σ0.9¯=0.269. Let σ=0.24, σ=σ0.9¯=0.269, and σ=0.3. Figures 7(a), 7(b), and 7(c) show that system (7) has a unique and asymptotically stable OOPS.Figure 7
Numerical simulations in case0<max⁡{x∗,h1}<h<h0<h2. (a) Phase portrait of x(t) and y(t) on h=0.9. (b) Time series of x(t). (c) Time series of y(t).
(a) (b) (c)For the case of0<max⁡{x∗,h1}<h0<h≤h2, for example, h=1.5 and the orbit of system (7) starts from (0.5,1), we get α1.5=0.5, β0.9=0.3 and σ1.5=0.24 by calculation. Figures 8(a), 8(b), and 8(c) show that system (7) has no OOPS.Figure 8
Numerical simulations in case<max⁡{x∗,h1}<h0<h≤h2. (a) Phase portrait of x(t) and y(t) on h=1.5. (b) Time series of x(t). (c) Time series of y(t).
(a) (b) (c)


## 5.2. Determination and Optimization of Pest Control Level

The goal to investigate the existence of OOPS of system (7) lies in that it can obtain the possibility of determining the frequency of releasing predators and spraying pesticides, which makes the density of pest below the damage level. Although the density of prey is inaccurate or biased, the system will eventually undergo periodic changes under the effective control. The following problems are considered to determine the optimal frequency for releasing predators and chemical controls.Assuming that unit cost of releasing predator is denoted byι1 and the unit cost of spraying pesticides is denoted by ι2, which include the price of chemical agent and the price of the damage to environment. Our goal is to reduce the unit cost in this process. In one period, the total cost is denoted by Fι, which is a function about α(h) (i.e., chemical control strength) and σ(h) (i.e., yield of releases of predator). Then Fι(h)=ι1σ(h)+ι2α(h). So the optimization model is formulated as(39)maxFιhThs.t.h1≤h≤h2The optimization problem is solved to yield the optimal pest levelh∗, which the optimal release rate of predator is σ∗=σh∗, the optimal strength of chemical control is α∗=αh∗, and the optimal impulse period of chemical control is T∗=T(σ∗,α∗). However, the optimum pest control level h∗ is dependent on the ratio of ω≜ι2/ι1. The impulse period T varies with the threshold h, as shown in Figure 9(a). And Figure 9(b) shows the variation of cost per unit time F/T and the period T with the pest control level h, where ι1=1000, ι2=1000, i.e., ω=1. The optimal pest level is h∗=0.9, the optimal strength of chemical control is αh∗=0.788, and the optimal release rate of predator is σh∗=0.87. It is important to note that the optimum economic threshold h is dependent on ω, as is illustrated in Figure 10.Figure 9
The variety in the periodT and the profit per unit time F/T on the threshold h. (a) The variety in the period T on the threshold h. (b) The profits per unit time F/T on the threshold h.
(a) (b)Figure 10
The change in the cost per unit timeF/T on the control level h for ι2/ι1=2.5,1,5,10.


## 6. Conclusion

A Smith prey-predator system with linear feedback control for integrated pest management is investigated in this paper. Integrated control strategy is more practical which can maximize the protection of the ecological environment and reduce the cost of pest management. First, the method of subsequent function and differential equation geometry theory are used to prove the existence, uniqueness, and stability of the OOPS of system (7). Second, a specific example is given to verify the conclusion of the impulsive strategy. Last, an optimized problem is formulated and the minimized total cost in pest control is obtained. However, the optimized results have some deviations which need to be further improved.



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*Source: 4945728-2018-12-05.xml*