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Research article

An efficient modified HS conjugate gradient algorithm in machine learning

  • Received: 19 September 2024 Revised: 01 November 2024 Accepted: 08 November 2024 Published: 15 November 2024
  • The Hestenes-Stiefe (HS) conjugate gradient method is very effective in resolving larger-scale sophisticated smoothing optimization tasks due to its low computational requirements and high computational efficiency. Additionally, the algorithm has been employed in practical applications to address image restoration and machine learning issues. In this paper, the authors proposed an improved Hestenes-Stiefe conjugate gradient algorithm having characteristics like: ⅰ) The algorithm depicts the decreasing features and trust region properties free of conditionalities. ⅱ) The algorithm satisfies global convergence. ⅲ) The algorithm can be applied to tackle the image restoration problem, monotone nonlinear equations, and machine learning problems. Numerical results revealed that the proffered technique is a competitive method.

    Citation: Gonglin Yuan, Minjie Huang. An efficient modified HS conjugate gradient algorithm in machine learning[J]. Electronic Research Archive, 2024, 32(11): 6175-6199. doi: 10.3934/era.2024287

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  • The Hestenes-Stiefe (HS) conjugate gradient method is very effective in resolving larger-scale sophisticated smoothing optimization tasks due to its low computational requirements and high computational efficiency. Additionally, the algorithm has been employed in practical applications to address image restoration and machine learning issues. In this paper, the authors proposed an improved Hestenes-Stiefe conjugate gradient algorithm having characteristics like: ⅰ) The algorithm depicts the decreasing features and trust region properties free of conditionalities. ⅱ) The algorithm satisfies global convergence. ⅲ) The algorithm can be applied to tackle the image restoration problem, monotone nonlinear equations, and machine learning problems. Numerical results revealed that the proffered technique is a competitive method.



    One of the most significant trends in global agricultural development is the ecological management of pests. From the perspective of ecosystem integrity, reducing and controlling pests through biological and ecological control are of great significance for the construction of ecological civilization. Biological and ecological control can reduce management cost, maintain ecological stability, and avoid environmental pollution and damage to biodiversity. As a large agricultural country, China places a premium on green prevention and control within its agricultural sector and proposed the National Strategic Plan for Quality Agriculture (2018–2022), which proposes to implement green prevention and control actions instead of chemical control and achieve a coverage rate of more than 50% for green prevention and control of major crop pests. The Crop Pests Regulations on the Prevention and Control of Crop Pests prioritizes the endorsement and support of green prevention and control technologies such as ecological management, fosters the widespread application of information technology and biotechnology, and propels the advancement of intelligent, specialized, and green prevention and control efforts [1]. Therefore, the simulation of pest dynamic behavior and the research of control strategies are helpful for more scientific and reasonable pest management.

    In a natural ecosystem, the predator-prey relationship is one of the most important relationships, and has become a main topic in ecological research and widely studied by scholars in recent years. Depending on the problem under consideration and the biological background, related research can be divided into two forms: ordinary differential [2,3,4,5] and partial differential [6,7,8,9,10,11]. The earliest work on the mathematical modeling of predation relationships dates back to the twentieth century, named as the Lotka-Volterra model [12,13]. Subsequently, scholars have extended the Lotka-Volterra model in different directions such as introducing different types of growth functions [14,15,16] and different forms of functional response [17,18,19,20]. The Gompertz model [14] is one of the most frequently used sigmoid models fitted to growth data and other. Scholars have fitted the Gompertz model to everything from plant growth, bird growth, fish growth, and growth of other animals [21,22,23]. Compared with the logistic model, it is more suitable for pest or disease curve fitting with S-shaped curve asymmetry, and fast development at first and slow development later. In addition, the Holling-Ⅱ functional response function is the most commonly employed one, in which the searching rate is considered as a constant. Nevertheless, in the real world, the density of the prey and the predator's searching environment can affect the predator's searching speed. Consequently, Hassell et al. [24] proposed a saturated searching rate. Guo et al. [25] introduced a fishery model with the Smith growth rate and the Holling-Ⅱ functional response with a variable searching rate. In this work, a pest-natural enemy model with the Gomportz growth rate and a variable searching rate is investigated.

    To prevent the spread of pests, effective control action should be implemented before the pests cause a certain amount of damage to the environment and crops. One way is to slow down the spreading speed of pests by setting a warning threshold, and when the density of pests exceeds this threshold level, an integrated control measure is imposed on the system. This kind of control system can be modeled by a Filippov system, which has been recognized by scholars and widely used in the study of concrete models with one threshold [26,27,28,29,30,31], a ratio-dependent threshold [32], or two thresholds [33]. In this study, we will also focus on Filippov predation models with dual thresholds. In addition, considering the instantaneous behavior of the control, an integrated pest-management strategy with threshold control is adopted, which is an instantaneous intervention imposed on the system and always taken as a practical approach for pest management. In recent years, there has been a lot of research and application of impulsive differential equations (IDEs) in population dynamics to model the instantaneous intervention activities. There are mainly six types of models involved in the research: periodic [34,35,36,37], prey-dependent [38,39,40,41,42,43], predator-dependent [44], ratio-dependent [45], nonlinear prey-dependent [46], and combined prey-predator dependent [47,48,49,50,51]. In the context of integrated pest management, setting a threshold for pest population density to control its spread is crucial. Therefore, in this study, we introduce a pest economic threshold: when the pest population density exceeds this threshold, we will intervene manually, which includes not only spraying pesticides but also releasing natural enemies.

    The article is organized in the following way: In Section 2, an integrated pest-management model with a variable searching rate based on double-threshold control is proposed. In Section 3, a dynamical analysis of the continuous system is performed, including the positivity and boundedness of the solutions, the existence and local stability of equilibrium points, and the dynamic behavior of the Filippov pest-management model with double thresholds. In Section 4, the complex dynamic behavior of the system induced by the economic threshold feedback control is focused on. In Section 5, numerical simulations are carried out to illustrate the main results of the above two sections step by step and to illustrate the practical implications. Finally, a summary of the research work is presented, and future research directions are discussed.

    A pest-natural enemy Gomportz model with a variable searching rate and Holling-Ⅱ functional response is considered:

    {dxdt=rx(lnKlnx)b(x)xy1+hx,dydt=b(x)exy1+hxdy, (2.1)

    where x (y) represents the pest's (natural enemies) density, respectively; r represents the pest's intrinsic growth rate; K represents the pest's environmental carrying capacity; b(x)=bx/(x+g) represents the variable searching rate [24,25] with maximum searching rate b and saturated constant g; e represents the conversion efficiency; and d represents the predator's natural mortality. All parameters are positive, and b, e and d are less than one. In addition, it requires that ebdh>0, i.e., the natural enemy species can survive when pests are abundant.

    To prevent the rapid spread of pests, two control methods are adapted: one is the continuous control with two thresholds, that is, when the pest density is below the pest warning threshold xET, no control measures need to be taken, when the pest and natural enemy densities satisfy x>xET and y<yET, the control action by spraying pesticides and releasing a part (q1) of the natural enemies is taken, which causes the death of pests (p1) and natural enemies (q2), when their densities satisfy x>xET,y>yET, only spraying pesticides is adapted. Based on the above control strategy, the impulsive control system can be formulated as follows:

    {dxdt=rx(lnKlnx)bx2y(1+hx)(x+g)δ1(x,y)x,dydt=ebx2y(1+hx)(x+g)dy+δ2(x,y)y, (2.2)

    where

    (δ1(x,y),δ2(x,y))={(0,0),x<xET,(p1,q1q2),x>xET,y<yET,(p1,q2),x>xET,y>yET. (2.3)

    Another one is an intermittent control with an economic threshold, that is, when the pest's density is below an economic threshold, no control action is implemented. Once the pest's density reaches the economic threshold, the control action by spraying pesticides and releasing a nonlinear volume τ1+ly of natural enemies is taken, which causes the death of pests (p1) and natural enemies (q2), where τ and l>0 are the formal parameters of the maximum volume of predators, respectively. Based on this control strategy, we can formulate the impulsive control system as follows:

    {dxdt=rx(lnKlnx)bx2y(1+hx)(x+g)dydt=ebx2y(1+hx)(x+g)dy}x<xET,x(t+)=(1p1)x(t)y(t+)=(1q2)y(t)+τ1+ly(t)}x=xET. (2.4)

    The aim of this study focuses on analyzing the effects of different control measures on the dynamics of Models (2.2) and (2.4), respectively.

    Consider a piecewise-continuous system

    (dxdtdydt)={F1(x,y) if (x,y)S1,F2(x,y) if (x,y)S2, (2.5)

    where

    S1={(x,y)R+:H(x,y)>0},S2={(x,y)R+:H(x,y)<0}

    and discontinuous demarcation is

    Σ={(x,y)R+:H(x,y)=0}.

    Let FiH=H,Fi, where , is the standard scalar product. Then FmiH=(Fm1iH),Fi. Thus the discontinuous demarcation Σ can be distinguished into three regions: 1) sliding region: Σs={(x,y)Σ:F1H<0andF2H>0}; 2) crossing region: Σc={(x,y)Σ:F1HF2H>0}; 3) escaping region: Σe={(x,y)Σ:F1H>0andF2H<0}.

    The dynamics of system (2.5) along Σs is determined by

    (dxdtdydt)=Fs(x,y)(x,y)Σs

    where Fs=λF1+(1λ)F2 with λ=F2HF2HF1H(0.1).

    Definition 1 ([24]). For system (2.5), E is a real equilibrium if i{1,2} so that Fi(E)=0, ESi; E is a virtual equilibrium if i,j{1,2},ij, so that Fi(E)=0, ESj; and E is a pseudo-equilibrium if Fs(E)=λF1(E)+(1λ)F2(E)=0,H(E)=0, and λ=F2HF2HF1H(0,1).

    For the given planar model

    {dxdt=χ1(x,y),dydt=χ2(x,y)ω(x,y)0,Δx=I1(x,y),Δy=I2(x,y)ω(x,y)=0, (2.6)

    we have:

    Definition 2 (Order-k periodic solution [50,51]). The solution ˜z(t)=(˜x(t),˜y(t)) is called periodic if there exists n(1) satisfying ˜zn=˜z0. Furthermore, ˜z is an order-k T-periodic solution with kmin{j|1jn,˜zj=˜z0}.

    Lemma 1 (Stability criterion [50,51]). The order-k T-periodic solution z(t)=(ξ(t),η(t))T is orbitally asymptotically stable if |μq|<1, where

    μk=kj=1Δjexp(T0[χ1x+χ2y](ξ(t),η(t))dt),

    with

    Δj=χ+1(I2yωxI2xωy+ωx)+χ+2(I1xωyI1yωx+ωy)χ1ωx+χ2ωy,

    χ+1=χ1(ξ(θ+j),η(θ+j)), χ+2=χ2(ξ(θ+j),η(θ+j)), and χ1, χ2, I1x, I1y, I2x, I2y, ωx, ωy are calculated at (ξ(θj),η(θj)).

    For convenience, denote

    f1(x,y)r(lnKlnx)bxy(1+hx)(x+g),f2(x)ebx2(1+hx)(x+g)d,χ1(x,y)=xf1(x,y),χ2(x,y)=yf2(x).

    Since

    x(t)=x(0)exp(t0f1(x,y)ds)0,y(t)=y(0)exp(t0f2(x)ds)0,

    then all solutions (x(t),y(t)) of Model (2.1) with x(0)>0 and y(0)>0 are positive in the region D={(x(t),y(t))|0<xK,y0}.

    Theorem 1. For Model (2.1), the solutions are ultimately bounded and uniform in the region D1.

    Proof. Define ι(x(t),y(t))x(t)+y(t). Then

    dιdt=dxdt+dydt=rx(lnKlnx)(1e)bx2y(1+hx)(x+g)dy.

    Take 0<θmin{r,d}, and there is

    dιdt+θιrx(lnKlnx)+θxσ(x).

    Obviously, σ(x)=r(lnKrlnx1)θ. If 0<x<Keθr1, then σ(x)>0. If x>Keθr1, then σ(x)<0. Then σ(x) has a maximum σ. Thus ddt(ισθ)θ(ισθ), and then

    0ι(x(t),y(t))(1eθt)σθ+ι(x(0),y(0))eθt.

    For t, there is 0ι(x(t),y(t))σθ. Therefore, the solutions of Model (2.1) are uniformly bounded in the region

    D1={(x,y)D:x(t)+y(t)σθ}D.

    For Model (2.1), the boundary equilibrium EK(K,0) always exists. Define

    ˉb(d;p1)=d(Kep1r+g)(1+hKep1r)/(eK2e2p1r),Δ(d)=d2(1+gh)2+4dg(ebdh),U(x)=(x+g)(1+hx)+(ghx2)(lnKlnx).

    Theorem 2. For Model (2.1), if b<ˉb(d;0), then EB(K,0) is locally asymptotically stable. If b>ˉb(d;0), there exists a coexistence equilibrium, denoted as E1=(x1,y1), which is locally asymptotically stable if U(x1)>0, where

    x1=d(1+gh)+Δ(d)2(ebdh),y1=r(lnKlnx1)(x1+g)(1+hx1)bx1.

    Proof. For Model (2.1), we have

    J=(r(lnKlnx)rbxy(x+hgx+2g)[(x+g)(1+hx)]2bx2(x+g)(1+hx)ebxy(x+hgx+2g)[(x+g)(1+hx)]2ebx2(x+g)(1+hx)d).

    1) For EK(K,0), we have

    J|(K,0)=(rbK2(K+g)(1+hK)0ebK2(K+g)(1+hK)d).

    Then λ1=r<0 and λ2=ebK2(K+g)(1+hx)d. Therefore, EB(K,0) is locally asymptotically stable if b<b(d;0).

    2) Since

    f1x=rxby(ghx2)[(x+g)(1+hx)]2,f1y=bx(x+g)(1+hx),f2x=ebx(x+hgx+2g)[(x+g)(1+hx)]2,

    then for E1, we have

    λ1λ2=x1y1f1yf2x>0,λ1+λ2=x1f1x.

    If U(x1)>0 holds, then λ1λ2>0,λ1+λ2<0, i.e., E1 is locally asymptotically stable.

    Let

    F1(x,y)=(rx(lnKlnx)bx2y(1+hx)(x+g),ebx2y(1+hx)(x+g)dy)T,F2(x,y)=(rx(lnKlnx)bx2y(1+hx)(x+g)p1x,ebx2y(1+hx)(x+g)dy+(q1q2)y)T,F3(x,y)=(rx(lnKlnx)bx2y(1+hx)(x+g)p1x,ebx2y(1+hx)(x+g)dyq2y)T.

    Then systems (2.2) and (2.3) can be described as

    (dxdtdydt)=Fi(x,y),(x,y)Gi,i=1,2,3, (3.1)

    where

    G1={(x,y)R2+:x<xET},G2={(x,y)R2+:x>xET,y<yET},G3={(x,y)R2+:x>xET,y>yET}.

    The switching boundaries are, respectively,

    Σ1={(x,y)R2+:x=xET,y<yET},Σ2={(x,y)R2+:x=xET,y>yET},Σ3={(x,y)R2+:x>xET,y=yET}.

    Let n1=(1,0) and n2=(0,1) be the normal vector for Σ1 and Σ3. If ΣijΣi such that the trajectory of Fi(x,y) approaches or moves away from Σi (i{1,2,3}) on both sides, then a sliding domain exists, and the dynamics on Σi can be determined by means of the Filippov convex method.

    The dynamic behavior of the model in G1 can be referred to Section 3.2. The model in G2 is described as follows:

    {dxdt=rx(lnKlnx)bx2y(1+hx)(x+g)p1x,dydt=ebx2y(1+hx)(x+g)dy+(q1q2)y. (3.2)

    Theorem 3. Model (3.2) always has an equilibrium E¯B(Kep1r,0). If q1<q2+d and b<b(dq1+q2;p1), then E¯B(Kep1r,0) is locally asymptotically stable. If q1<q2+d and b>ˉb(d+q2q1,p1), Model (3.2) has a coexistence equilibrium, denoted as E2=(x2,y2), which is locally asymptotically stable if U(x2)>0, where

    x2=(dq1+q2)(1+gh)+Δ(dq1+q2)2[eb+h(q1q2d)],y2=[r(lnKlnx2)p1](x2+g)(1+hx2)bx2.

    Similarly, the model in G3 is described as follows:

    {dxdt=rx(lnKlnx)bx2y(1+hx)(x+g)p1x,dydt=ebx2y(1+hx)(x+g)dyq2y. (3.3)

    Theorem 4. Model (3.3) always has an equilibrium E¯B(Kep1r,0). If b<b(d+q2;p1), then E¯B is locally asymptotically stable. If b>b(d+q2;p1), Model (3.3) has a coexistence equilibrium, denoted as E3=(x3,y3), which is locally asymptotically stable if U(x3)>0, where

    x3=(d+q2)(1+gh)+Δ(dq1+q2)2[ebh(d+q2)],y3=[r(lnKlnx3)p1](x3+g)(1+hx3)bx3.

    It is assumed that

    (H1) p1<r;

    (H2) d(1+gh)+Δ(d)2(ebdh)<K;

    (H3) q1q2d<0,(dq1+q2)(1+gh)+Δ(dq1+q2)2[eb+h(q1q2d)]<Kep1r;

    (H4) (d+q2)(1+gh)+Δ(d+q2)2[ebh(d+q2)]<Kep1r.

    For Model (2.2), we have x1<x2<x3 when q1<q2 and x2<x1<x3 when q2<q1<q2+d.

    Define

    yET1=[r(lnKlnxET)p1](xET+g)(1+hxET)bxET,yET2=r(lnKlnxET)(xET+g)(1+hxET)bxET,

    where yET2>0 and yET1<yET2.

    First, we will discuss the sliding mode domain on Σ1 and the corresponding dynamics. Since

    <F1,n1>|(x,y)Σ1=xET[r(lnKlnxET)bxETy(1+hxET)(xET+g)],<F2,n1>|(x,y)Σ1=xET[r(lnKlnxET)bxETy(1+hxET)(xET+g)p1], (3.4)

    then the sliding mode domain on Σ1 does not exist if yET<yET1. When yET>yET1, we have

    Σ11={(x,y)Σ1|max{0,yET1}<y<min{yET2,yET}}. (3.5)

    Next, the Filippov convex method is used, i.e.,

    dXdt=λF1+(1λ)F2,(x,y)Σ11, (3.6)

    where

    λ=<F2,n1><F2,n1><F1,n1>,

    and the sliding mode dynamics of Eq (3.1) along Σ11 is determined by the following system:

    {dxdt=0,dydt=[ebxET2(1+hxET)(xET+g)d]y+q1q2p1[r(lnKlnxET)bxETy(1+hxET)(xET+g)]y. (3.7)

    Let ς1=ebxET2+(1+hxET)(xET+g)[r(q1q2)p1(lnKlnxET)d]. Then a positive equilibrium Ea1(xET,ya1) exists, where ya1=p1ς1bxET(q1q2)>0. Therefore

    ya1yET2=p1bxET(q1q2)[ebxET2d(1+hxET)(xET+g)].

    If x1<xET, then ya1>yET2, i.e., Ea1 is not located in Σ11, and then Ea1 is not a pseudo-equilibrium. If x1>xET, then ya1<yET2.

    Similarly, we have

    ya1yET1=p1bxET(q1q2)[ebxET2+(q1q2d)(1+hxET)(xET+g)].

    If x2>xET, then ya1<yET1, i.e., Ea1 is not located in Σ11, and then Ea1 is not a pseudo-equilibrium. If x2<xET, then ya1>yET1. Therefore, yET1<ya1<yET2. When ya1<yET, Ea1 is the pseudo-equilibrium.

    Second, we will discuss the sliding mode domain on Σ2 and the dynamic characteristics on the sliding mode. Since

    <F1,n1>|(x,y)Σ2=xET[r(lnKlnxET)bxETy(1+hxET)(xET+g)],<F3,n1>|(x,y)Σ2=xET[r(lnKlnxET)bxETy(1+hxET)(xET+g)p1], (3.8)

    then the sliding mode domain on Σ2 does not exist if yET>yET2; When yET<yET2, we have

    Σ22={(x,y)Σ2|max{yET1,yET}<y<yET2}. (3.9)

    Therefore, when yET>yET2, there is no sliding mode domain on Σ2. When yET<yET2, the sliding mode domain of the system (3.1) on Σ2 can be expressed as Eq (3.9).

    According to the Filippov convex method, we have

    dXdt=λF1+(1λ)F3,(x,y)Σ22, (3.10)

    where

    λ=<F3,n1><F3,n1><F1,n1>,

    and the sliding mode dynamics of equation (3.1) along Σ22 is determined by the following system:

    {dxdt=0,dydt=[ebxET2(1+hxET)(xET+g)d]yq2p1[r(lnKlnxET)bxETy(1+hxET)(xET+g)]y. (3.11)

    Let ς2=(1+hxET)(xET+g)[rq2p1(lnKlnxET)+d]ebxET2. Then the system (3.11) has a positive equilibrium Ea2(xET,ya2), where ya2=p1ς2q2bxET>0. Obviously,

    ya2yET2=p1q2bxET[ebxET2d(1+hxET)(xET+g)].

    If x1>xET, then ya2>yET2, i.e., Ea2 is not located in Σ22. If x1<xET, then ya2<yET2. Similarly, we have

    ya2yET1=p1q2bxET[ebxET2(d+q2)(1+hxET)(xET+g)].

    If x3<xET, then ya2<yET1, i.e, Ea2 is not located in Σ22. If x3>xET, then ya2>yET1. Therefore yET1<ya2<yET2. If yET1<yET<ya2 or yET<yET1, then Ea2 is located in Σ22 and is a pseudo-equilibrium.

    Finally, we will discuss the sliding mode domain on Σ3 and the dynamic characteristics of the sliding mode. We have

    <F2,n2>|(x,y)Σ3=yET[ebx2(1+hx)(x+g)d+q1q2],<F3,n2>|(x,y)Σ3=yET[ebx2(1+hx)(x+g)dq2]. (3.12)

    According to Eq (3.12), <F2,n2>|(x,y)Σ3><F3,n2>|(x,y)Σ3. If x3<xET, then the system (3.1) does not have a sliding mode domain on Σ3. If x2<xET<x3, it is found through Eq (3.12) that the system (3.1) can be expressed in the sliding mode domain on Σ3 as

    Σ33={(x,y)Σ3|xET<x<x3}. (3.13)

    According to the Filippov convex method, we have

    dXdt=λF2+(1λ)F3,(x,y)Σ33, (3.14)

    where

    λ=<F3,n2><F3,n2><F2,n2>.

    The sliding mode dynamics of Eq (3.1) along Σ11 is determined by the following system:

    {dxdt=rx(lnKlnx)bx2yET(1+hx)(x+g)p1x,dydt=0. (3.15)

    Then

    [r(lnKlnx)p1](1+hx)(x+g)bxyET=0. (3.16)

    If the root x=xb>0 of Eq (3.16) satisfies Eq (3.13), then Eb(xb,yET) of the system (3.15) is a pseudo-equilibrium, and if it does not satisfy Eq (3.13), then Eb is not a pseudo-equilibrium.

    For Model (2.4), let

    y=ˆy(x)r(ln(K)ln(x))(1+hx)(x+g)bx.

    The curve y=ˆy(x) intersects with x=xET and x=(1p1)xET at P(xET,yP) (yP=ˆy(xET)) and R0((1p1)xET,yR0). The trajectory passing through P is denoted by γ1, and it goes backward and intersects y=ˆy(x) at H(xH,yH)(yH=ˆy(xH)). If xH<(1p1)xET, then denote Q1((1p1)xET,yQ1),Q2((1p1)xET,yQ2) as the intersection points between γ1 and x=(1p1)xET with yQ1<yQ2. The trajectory passing through R0 is denoted by γ2. If γ2{x=xET}, then denote R1(xET,yR1) as the intersection point between γ2 and x=xET. The curve γ2 defines a function y=y(x,yR0) on the interval [(1p1)xET,xET] with

    dydx=ebx2y(1+hx)(x+g)dyrx(lnKlnx)bx2y(1+hx)(x+g)φ(x,y),y((1p1)xET,yR0)=yR0,

    which takes the form

    y=y(x,yR0)=yR0+x(1p1)xETφ(u,y(u,yR0))du.

    For Model (2.4), we have M={(x,y)x=xET,y>0}. The trajectory of the system (2.4) with x0<xET can reach M1={(x,y)|x=xET,0yyP}M, which is called the effective impulse set, denoted by Meff. The corresponding effective phase set is denoted by Neff. Moreover, define M2={(x,y)x=xET,0yyR1}M1.

    Since Δy=q2y+τ1+ly, then define

    ρ(y)(1q2)y+τ1+ly.

    Obviously, the function ρ(y) reaches a minimum at y=y, where yτl(1q2)(1q2)l(1q2). Denote R(xET,y)M, and its phase point is R+((1p1)xET,ρ(y)).

    Define

    x1ETmax{xET|y(xET,R0)},x2ETmax{xET|y(xET,Q1)yQ2/2}.

    Denote

    τ11q2l,τ2(1q2)(1+lyP)2l,τ3(1q2)(1+lyR1)2l.

    The exact domains of M and N can be determined by sign(ρ(y)) and sign(y), which will be discussed in the following two situations:

    Case Ⅰ: x1ET<xETx2ET.

    For this situation, Meff=M1. To determine Neff, we are required to judge the magnitude between y and yP. Denote Λ=[0,yQ1][yQ2,+).

    ⅰ) ττ1, then y0. For y[0,yP], ρ0 holds, and then τρ(y)ρ(yP) for y[0,yP]. Denote Λ11=[τ,ρ(yP)], Λ1=ΛΛ11, and Neff=N1={(x+,y+)|x+=(1p1)xET,y+Λ1}.

    ⅱ) τ1<τ<τ2, then 0<y<yP. For y[0,y], ρ0 holds, and then ρ(y)ρ(y)τ for y[0,y]. Denote Λ21=[ρ(y),τ], Λ21=ΛΛ21, and N21={(x+,y+)|x+=(1p1)xET,y+Λ21}. Similarly, for y(y,yP], ρ>0 holds, i.e., ρ(y)<ρ(y)ρ(yP). Denote Λ22=(ρ(y),ρ(yP)], Λ22=ΛΛ22, and N22={(x+,y+)|x+=(1p1)xET,y+Λ22}. Thus, we have Neff=N2=N21N22.

    ⅲ) ττ2, then yyP. For y[0,yP], ρ0 holds, i.e., ρ(yP)ρ(y)τ. Denote Λ33=[ρ(yP),τ] and Λ3=ΛΛ33. Then Neff=N3={(x+,y+)|x+=(1p1)xET,y+Λ3}.

    Case Ⅱ: xETx1ET.

    For this situation, Meff=M2. Similar to the discussion in case Ⅰ, we have

    ⅰ) ττ1. Then Neff=N4={(x+,y+)|x+=(1p1)xET,y+Λ4}, where Λ4=[τ,ρ(yR1)].

    ⅱ) τ1<τ<τ3. For y[0,y], we have N51={(x+,y+)|x+=(1p1)xET,y+Λ21}. Similarly, for y(y,yR1], denote Λ52=(ρ(y),ρ(yR1)], and then N52={(x+,y+)|x+=(1p1)xET,y+Λ52}. Therefore, Neff=N5=N51N52.

    ⅲ) ττ3. Then Neff=N6={(x+,y+)|x+=(1p1)xET,y+Λ6} with Λ6=[ρ(yR1),τ].

    Denote Gi(xET,yi)M, G+i((1p1)xET,y+i)N, i=0,1,2,..., where G+i=I(Gi). Since G+i and Gi+1 lie on the same trajectory γG+i, then we have yi+1=ϖ(y+i) and y+i+1=ψ(y+i), where

    ψ(y+i)(1q2)ϖ(y+i)+τ1+lϖ(y+i).

    If ˆyN such that ψ(ˆy)=ˆy, then Model (2.4) admits an order-1 periodic trajectory. Next, we will investigate the monotonicity of ψ(y) with τ>0 for situations Ⅰ and Ⅱ.

    Case Ⅰ: x1ET<xETx2ET.

    ⅰ) ττ1. ρ(y) monotonically increases on [0,yP], and then the map ψ(y) monotonically increases on [0,yQ1] and monotonically decreases on [yQ2,+).

    ⅱ) τ1<τ<τ2. we have ρ(y)<0 for y[0,y] and ρ(y)>0 for y(y,yP]. Denote yS1=min{y:ψ(y)=yR+}, yS2=max{y:ψ(y)=yR+}. Then ψ(y) monotonically increases on [yS1,yQ1], [yS2,+) and monotonically decreases on the interval [0,yS1], [yQ2,yS2], respectively.

    ⅲ) ττ2. ρ(y) monotonically decreases on [0,yP], and then the map ψ(y) monotonically decreases on [0,yQ1] and monotonically increases on [yQ2,+).

    Case Ⅱ: xETx1ET.

    ⅰ) ττ1. Then ψ(y)>0 for y[0,yR0] and ψ(y)<0 for y[yR0,+).

    ⅱ) τ1<τ<τ3. Denote yV1=min{y:ψ(y)=yR+} and yV2=max{y:ψ(y)=yR+}. Then ψ(y) monotonically decreases on [0,yV1] and [yR0,yV2], and monotonically increases on [yV1,yR0] and [yV2,+).

    ⅲ) ττ3. The map ψ(y) monotonically decreases on [0,yR0] and monotonically increases on [yR0,+).

    For Model (2.4) with τ=0, if y00, then y0 holds. Thus Model (2.4) is degenerated to

    {dxdt=rx(lnKlnx),x<xET,Δx=p1x(t),x=xET. (4.1)

    Let x=˜x(t) be the solution of equation

    dxdt=rx(lnKlnx)

    with initial value ˜x(0)=x0(1p1)xET. Define

    T01r(lnlnK(1p1)xETlnlnKxET).

    We have ˜x(T0)=xET and ˜x(T+0)=x0. Thus, z(t)=(˜x(t),0) ((k1)T0<tkT0, kN+) is a natural enemy extinction periodic trajectory.

    Theorem 5. The natural enemy extinction period trajectory z(t)=(˜x(t),0) ((k1)T0<tkT0, kN+) is orbitally asymptotically stable if q2>ˆq, where

    ˆq1τl(lnKlnxET)exp(T00(r(lnKln˜x)r+eb˜x2(˜x+g)(1+h˜x)d)dt)(1p1)(lnKln(1p1)xET).

    Proof. For Model (4.1), we have

    I1(x,y)=p1x,I2(x,y)=q2y+τ1+ly,ω(x,y)=xxET.

    Then

    χ1x=r(lnKlnx)rbxy(x+hgx+2g)[(x+g)(1+hx)]2,χ2y=ebx2(x+g)(1+hx)d,I1x=p1,I1y=0,I2x=0,I2y=q2lτ(1+ly)2,ωx=1,ωy=0.

    Through calculation, we have

    ˜κ=(1q2lτ)(1p1)(lnKln(1p1)xET)lnKlnxET

    and

    T0+(χ1x+χ2y)(˜x,0)dt=T00(r(lnKln˜x)r+eb˜x2(˜x+g)(1+h˜x)d)dt.

    Thus,

    ˆρ=˜κexp(T00(r(lnKln˜x)r+eb˜x2(˜x+g)(1+h˜x)d)dt).

    Therefore, if q2>ˆq, we have ˆρ<1, and by Lemma 1, z(t)=(˜x(t),0) ((k1)T0<tkT0, kN+) is orbitally asymptotically stable.

    Denote that the points P, R1 are mapped to the points P+((1p1)xET,(1q2)yP+τ1+lyP) and R+1((1p1)xET,(1q2)yR1+τ1+lyR1), respectively, after a single impulse. Denote W((1p1)xET,τ).

    Case Ⅰ: x1ET<xETx2ET.

    Define

    q21yQ22yP,l1+4yP(1q2)/(yQ1(1q2)yP)12yP,
    ˜τ1(1+lyP)(yQ1(1q2)yP),˜τ2(1+lyP)(yQ2(1q2)yP),
    ˆτ1=(1q2)[1+lyQ2/(1q2)]24l,ˆτ2=(1q2)[1+lyQ1/(1q2)]24l,τ4(1q2)1+lyPl.

    Obviously, τ4<τ2 and for q2q2, we have l>max{l,0}.

    1) For τ=˜τ2, we have ψ(yQ2)=yP+=yQ2.

    2) For τ>˜τ2, we have ψ(yQ2)=yP+>yQ2. Then

    ● 2-a) for ττ2, Wis the highest after the pulse, while P+ is the lowest after the pulse. Then ψ(τ)<τ, ψ(yP+)>yP+, and thus y(yP+,τ) such that ψ(y)=y.

    ● 2-b) for ˜τ2<τ<τ2, if τˆτ2, then ρ(y)yQ2. Since the point R+ is the lowest point after the pulse, then ψ(yR+)>yR+. If τ>τ4, we have τ>yP+. Then W is the highest point after the impulse, i.e., ψ(τ)<τ. If ττ4, we have τyP+. Then P+ is the highest point after the impulse, i.e., ψ(yP+)<yP+. Combine the above two aspects and it can be concluded that y(yR+,max{τ,yP+}) such that ψ(y)=y. While for τ<ˆτ2, we have ρ(y)<yQ2. In such a case, ψ is not defined on [ρ(y),yQ2) and it is uncertain whether a fixed point of ψ exists or not.

    3) When ˜τ1<τ<˜τ2, then yQ1<yP+<yQ2. If τˆτ1, then ρ(y)yQ1, i.e., ψ does not have a fixed point. While for τ<ˆτ1, we have ρ(y)<yQ1. In such a case, it is uncertain whether a fixed point of ψ exists or not.

    4) When 0<τ<˜τ1, and then ψ(yQ1)=yP+<yQ1. If τ>τ1, the point R+ is the lowest point after the pulse, then ψ(yR+)>yR+, i.e., ψ(y) has a fixed point on (yR+,yQ1). If ττ1, the point W is the lowest point after the pulse, and then ψ(τ)>τ, i.e., ψ(y) has a fixed point on (τ,yQ1).

    Case Ⅱ: xETx1ET.

    Define ˜τ(1+lyR1)(yR0(1q2)yR1).

    1) For τ=˜τ, we have ψ(yR0)=yR+1=yR0.

    2) For τ>˜τ, we have ψ(yR0)=yR+1>yR0. Then

    ● 2-a) for ττ3, since R+1 and W are the lowest and highest points after the pulse, then we have ψ(yR+1)>yR+1, ψ(τ)<τ, and thus y(yR+1,τ) such that ψ(y)=y;

    ● 2-b) for ˜τ<τ<τ3 and if τ>yR+1, then ψ(τ)<τ; if τyR+1, then ψ(yR+1)>yR+1. On the other hand, take the point A in a small neighborhood near the point R0, i.e., A(R0,δ). A is above R0. By the continuity of the impulse function and the Poincaré map, we have ψ(yA)>yA. Therefore, the map ψ(y) has a fixed point on (yA,max{yR+1,τ}).

    3) When τ<˜τ, then ψ(yR0)=yR+1<yR0. If τ>τ1, we have ψ(yR+)>yR+. If ττ1, we have ψ(τ)>τ. Combine the above two aspects and it can be concluded that y(max{yR+,τ},yR0) such that ψ(y)=y.

    To sum up, we have:

    Theorem 6. For the situation of xETx1ET, Model (2.4) admits an order-1 periodic trajectory. While for the situation of x1ET<xETx2ET, Model (2.4) admits an order-1 periodic trajectory if τ<˜τ1 or τmax{˜τ2,ˆτ2}.

    Let ˜z(t)=(ξ(t),η(t)) ((k1)T<tkT, kN+) be the T-periodic trajectory of the system (2.4) with initial values A0((1p1)xET,yA0). The trajectory intersects M at A0(ξ(T),η(T)), where ξ(T)=xET,η(T)=y0, and then it is pulsed to N at A+0(ξ(T+),η(T+)). Thus, ξ(T+)=(1p1)xET,η1(T+)=(1q2)y0+τ1+ly0=yA0.

    Theorem 7. The T-periodic trajectory ˜z(t)=(ξ(t),η(t)) ((k1)T<tkT, kN+) with initial values ((1p1)xET,(1q2)y0+τ1+ly0) is orbitally asymptotically stable if

    T0(r(lnKlnx)rbxy(x+hgx+2g)[(x+g)(1+hx)]2+ebx2(x+g)(1+hx)d)(ξ(t),η(t))dt<ln(ˆκ),

    where

    ˆκ=r(lnKlnxET)bxETy0(xET+g)(1+hxET)(1p1)(1q2lτ(1+ly0)2)[r(lnKln(1p1)xET)b(1p1)xET((1q2)y0+τ1+ly0)((1p1)xET+g)(1+h(1p1)xET)].

    Proof. The proof can be referred to that in Theorem 5 and is, therefore omitted.

    For the purpose of simulation, it is assumed that r=1.5, K=120, b=0.595, h=0.92, e=0.8, d=0.5 and g=0.8.

    When b=0.595, the interior equilibrium E1=(54.7,103.1) is locally asymptotically stable, as presented in Figure 1. When b increases to 0.61, a limit cycle occurs, as presented in Figure 2. The effect of the maximum search rate on pests and natural enemies in the coexistence steady state is presented in Figure 3, and it is obvious that x1 decreases with increasing b, while y1 increases and then decreases with increasing b. Therefore, increasing the search rate for pests helps to reduce the number of pests.

    Figure 1.  Tendencies of Model (2.1) in case of b=0.595.
    Figure 2.  Tendencies of Model (2.1) in case of b=0.61.
    Figure 3.  Relationship between the number of pests (natural enemies) and the maximum search rate at the steady state.

    When p1=0.25, q1=0.5 and q2=0.007, the positive equilibrium of the G1 region is E1=(54.707,103.1337), the positive equilibrium of the G2 region is E2=(0.1229,141.532), and the positive equilibrium of the G3 region is E3=(92.5247,20.443). When xET=100 and yET=110, there is x2<x3<xET, x1<xET, yET1=3.6997 and yET2=43.0879. The sliding mode domain of Model (3.1) on Σ1 can be represented as Σ11={(x,y)Σ1|3.6997<y<43.0879}, and there is no pseudo-equilibrium on Σ1, as illustrated in Figure 4(a). When xET=80 and yET=110, there is x2<xET<x3, x1<xET, yET1=45.3593 and yET2=77.0172. The sliding mode domain of Model (3.1) on Σ1 can be represented as Σ11={(x,y)Σ1|45.3593<y<77.0172}, and there is no pseudo-equilibrium on Σ1, as presented in Figure 4(b).

    Figure 4.  The dynamic behavior of Model (2.2) for yET=110 and different pest control levels: (a) xET=100; (b) xET=80.

    When q2=0.11, for xET=62.667, the periodic trajectory of Model (2.4) is presented by changing the killing rate p1 of the prey, the amount of predator released τ, and the value of the parameter l. When τ=0, p1=0.5 and l=0.02, the natural enemy extinction periodic trajectory is orbitally asymptotically stable (Figure 5(a)). To prevent the extinction of natural enemies, we are required to release natural enemies in an appropriate amount. When τ=5, the natural enemy extinction periodic trajectory loses its stability and an order-1 periodic trajectory occurs (Figure 5(b)).

    Figure 5.  Illustration of the trajectory of Model (2.4) for l=0.02 and different τ: (a) τ=0, (b) τ=5.
    Figure 6.  Presentation of the order-1 periodic trajectory of Model (2.4).

    Next, the accurate domains of M and N for different cases are presented as well as the order-1 periodic trajectory (Figure 6). The accurate domains of M and N are marked in red and blue solid lines, respectively. When p1=0.5, H lies on the left side of the phase set. The schematic diagram of the exact domain of the phase set and pulse set, and the order-1 periodic trajectories for different cases are presented in subfigures Figure 6(a)(c). When p1=0.6, H lies on the right side of the phase set. The schematic diagram of the accurate domain of M and N and the order-1 periodic trajectories for different cases are presented in subfigures Figure 6(d)(f).

    When p1=0.5, l=0.02, τ=10, b=0.595 and xET=50, E is locally asymptotically stable, and Model (2.4) admits an order-1 periodic trajectory for xET<x1ET, as presented in Figure 7.

    Figure 7.  Presentation of the order-1 periodic trajectory of Model (2.4) for xET<x1ET.

    Finally, order-n periodic solutions are presented for different τ and l. When b=0.595, E is locally asymptotically stable. For control parameters l=0.03, τ=120, xET=62.667 or l=0.002, τ=40, xET=62.667, Model (2.2) admits an order-k periodic trajectory, as presented in subfigures 8(a) and (b). When b=0.61, Model (2.1) admits a limit cycle. For xET=50, Model (2.2) admits an order-k periodic trajectory, as presented in subfigures 8(c)–8(f).

    Pests are important factors that harm agricultural production. In order to effectively control the spread of pests, a pest-natural enemy model with a variable search rate and threshold dependent feedback control was proposed. The dynamic properties such as the existence, positivity, and boundedness of solutions for continuous systems were discussed, and the results show that pests and natural enemies will not increase indefinitely due to system constraints (Theorem 1). In addition, it is shown that the natural enemy's searching rate b plays an important role in determining the dynamics of the system, i.e., when b is smaller than the level ˉb(d;0), the predators in the system will go to extinction and when b is greater than ˉb(d;0), there exists a steady state E at which the natural enemies and the pests in the system keep a balance. Moreover, the steady state is locally asymptotically stable as long as U(x)>0 (Theorems 2–4, Figure 1). When U(x)<0, the stability is lost and a limit cycle surrounding E is obtained (Figure 2). The relationship between the number of pests (natural enemies) and the maximum search rate at the steady state was presented in Figure 3.

    To prevent the spread of pests, two different types of control strategies were adopted. The first is a non-smooth control and the model is described by a Filippov system with two warning thresholds. By analyzing the sliding dynamics, we discussed the existence of pseudo-equilibrium Ep (Figure 4). The pseudo-equilibrium Ep is a new state of the control system at which the pests and the natural enemies keep a balance and the pest populations can be controlled at appropriate levels, which in turn indicates the effectiveness of the control. The second is an intermittent control with an economic threshold. When the pests reach the economic threshold, manual intervention is carried out by spraying pesticides and releasing a certain amount of natural enemies. For the control model, the accurate domain of the phase set was presented and the Poincaré map was constructed, through which the conditions for the existence of the order-1 periodic trajectories were presented (Theorems 5 and 6 and Figures 57). The order-1 periodic solution provides a possibility for periodic pest control, thus avoiding the need and difficulty of implementing pest population monitoring. The stability of the order-1 periodic trajectory was also verified (Theorems 5 and 7). This ensures the robustness of the control, and even if there is a condition monitoring error, it can still converge to the periodic solution of the system, thus providing a guarantee for the periodic control. We also presented the order-k periodic solutions in numerical simulations (Figure 8), which further explain the complexity of the control system and the necessity of maintaining the stability of the system. The results illustrate the complex dynamics of the proposed models, which can serve as a valuable reference for the advancement of sustainable agricultural practices and the control of pests.

    Figure 8.  Presentation of the order-k periodic trajectory of Model (2.4) for different situations.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The work was supported by the National Natural Science Foundation of China (No. 11401068).

    The authors declare that there is no known competing financial interests to influence the work in this paper.



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