
Via symbolic computation we deduce 97 new type series for powers of π related to Ramanujan-type series. Here are three typical examples:
∞∑k=0P(k)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−640320)3k=18×5574033√100055π
with
P(k)=637379600041024803108k2+657229991696087780968k+19850391655004126179,
∞∑k=1(3k+1)16k(2k+1)2k3(2kk)3=π2−82,
and
∞∑n=03n+1(−100)nn∑k=0(nk)2Tk(1,25)Tn−k(1,25)=258π,
where the generalized central trinomial coefficient Tk(b,c) denotes the coefficient of xk in the expansion of (x2+bx+c)k. We also formulate a general characterization of rational Ramanujan-type series for 1/π via congruences, and pose 117 new conjectural series for powers of π via looking for corresponding congruences. For example, we conjecture that
∞∑k=039480k+7321(−29700)kTk(14,1)Tk(11,−11)2=6795√5π.
Eighteen of the new series in this paper involve some imaginary quadratic fields with class number 8.
Citation: Zhi-Wei Sun. New series for powers of π and related congruences[J]. Electronic Research Archive, 2020, 28(3): 1273-1342. doi: 10.3934/era.2020070
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Via symbolic computation we deduce 97 new type series for powers of π related to Ramanujan-type series. Here are three typical examples:
∞∑k=0P(k)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−640320)3k=18×5574033√100055π
with
P(k)=637379600041024803108k2+657229991696087780968k+19850391655004126179,
∞∑k=1(3k+1)16k(2k+1)2k3(2kk)3=π2−82,
and
∞∑n=03n+1(−100)nn∑k=0(nk)2Tk(1,25)Tn−k(1,25)=258π,
where the generalized central trinomial coefficient Tk(b,c) denotes the coefficient of xk in the expansion of (x2+bx+c)k. We also formulate a general characterization of rational Ramanujan-type series for 1/π via congruences, and pose 117 new conjectural series for powers of π via looking for corresponding congruences. For example, we conjecture that
∞∑k=039480k+7321(−29700)kTk(14,1)Tk(11,−11)2=6795√5π.
Eighteen of the new series in this paper involve some imaginary quadratic fields with class number 8.
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(lnK−lnx)−b(x)xy1+hx,dydt=b(x)exy1+hx−dy, | (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 eb−dh>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(lnK−lnx)−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,q1−q2),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(lnK−lnx)−bx2y(1+hx)(x+g)dydt=ebx2y(1+hx)(x+g)−dy}x<xET,x(t+)=(1−p1)x(t)y(t+)=(1−q2)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=⟨∇(Fm−1iH),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:F1H⋅F2H>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+(1−λ)F2 with λ=F2HF2H−F1H∈(0.1).
Definition 1 ([24]). For system (2.5), E∗ is a real equilibrium if ∃i∈{1,2} so that Fi(E∗)=0, E∗∈Si; E∗ is a virtual equilibrium if ∃i,j∈{1,2},i≠j, so that Fi(E∗)=0, E∗∈Sj; and E∗ is a pseudo-equilibrium if Fs(E∗)=λF1(E∗)+(1−λ)F2(E∗)=0,H(E∗)=0, and λ=F2HF2H−F1H∈(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 k≜min{j|1≤j≤n,˜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=k∏j=1Δjexp(∫T0[∂χ1∂x+∂χ2∂y](ξ(t),η(t))dt), |
with
Δj=χ+1(∂I2∂y∂ω∂x−∂I2∂x∂ω∂y+∂ω∂x)+χ+2(∂I1∂x∂ω∂y−∂I1∂y∂ω∂x+∂ω∂y)χ1∂ω∂x+χ2∂ω∂y, |
χ+1=χ1(ξ(θ+j),η(θ+j)), χ+2=χ2(ξ(θ+j),η(θ+j)), and χ1, χ2, ∂I1∂x, ∂I1∂y, ∂I2∂x, ∂I2∂y, ∂ω∂x, ∂ω∂y are calculated at (ξ(θj),η(θj)).
For convenience, denote
f1(x,y)≜r(lnK−lnx)−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<x≤K,y≥0}.
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(lnK−lnx)−(1−e)bx2y(1+hx)(x+g)−dy. |
Take 0<θ≤min{r,d}, and there is
dιdt+θι≤rx(lnK−lnx)+θx≜σ(x). |
Obviously, σ′(x)=r(lnK−rlnx−1)−θ. If 0<x<Keθr−1, then σ′(x)>0. If x>Keθr−1, then σ′(x)<0. Then σ(x) has a maximum σ∗. Thus ddt(ι−σ∗θ)≤−θ(ι−σ∗θ), and then
0≤ι(x(t),y(t))≤(1−e−θt)σ∗θ+ι(x(0),y(0))e−θt. |
For t→∞, there is 0≤ι(x(t),y(t))≤σ∗θ. Therefore, the solutions of Model (2.1) are uniformly bounded in the region
D1={(x,y)∈D:x(t)+y(t)≤σ∗θ}⊂D. |
For Model (2.1), the boundary equilibrium EK(K,0) always exists. Define
ˉb(d;p1)=d(Ke−p1r+g)(1+hKe−p1r)/(eK2e−2p1r),Δ(d)=d2(1+gh)2+4dg(eb−dh),U(x)=(x+g)(1+hx)+(g−hx2)(lnK−lnx). |
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 E∗1=(x∗1,y∗1), which is locally asymptotically stable if U(x∗1)>0, where
x∗1=d(1+gh)+√Δ(d)2(eb−dh),y∗1=r(lnK−lnx∗1)(x∗1+g)(1+hx∗1)bx∗1. |
Proof. For Model (2.1), we have
J=(r(lnK−lnx)−r−bxy(x+hgx+2g)[(x+g)(1+hx)]2−bx2(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)=(−r−bK2(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=−rx−by(g−hx2)[(x+g)(1+hx)]2,f1y=−bx(x+g)(1+hx),f2x=ebx(x+hgx+2g)[(x+g)(1+hx)]2, |
then for E∗1, we have
λ1λ2=−x∗1y∗1f1yf2x>0,λ1+λ2=x∗1f1x. |
If U(x∗1)>0 holds, then λ1λ2>0,λ1+λ2<0, i.e., E∗1 is locally asymptotically stable.
Let
F1(x,y)=(rx(lnK−lnx)−bx2y(1+hx)(x+g),ebx2y(1+hx)(x+g)−dy)T,F2(x,y)=(rx(lnK−lnx)−bx2y(1+hx)(x+g)−p1x,ebx2y(1+hx)(x+g)−dy+(q1−q2)y)T,F3(x,y)=(rx(lnK−lnx)−bx2y(1+hx)(x+g)−p1x,ebx2y(1+hx)(x+g)−dy−q2y)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 ∃Σij⊂Σi 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(lnK−lnx)−bx2y(1+hx)(x+g)−p1x,dydt=ebx2y(1+hx)(x+g)−dy+(q1−q2)y. | (3.2) |
Theorem 3. Model (3.2) always has an equilibrium E¯B(Ke−p1r,0). If q1<q2+d and b<b(d−q1+q2;p1), then E¯B(Ke−p1r,0) is locally asymptotically stable. If q1<q2+d and b>ˉb(d+q2−q1,p1), Model (3.2) has a coexistence equilibrium, denoted as E∗2=(x∗2,y∗2), which is locally asymptotically stable if U(x∗2)>0, where
x∗2=(d−q1+q2)(1+gh)+√Δ(d−q1+q2)2[eb+h(q1−q2−d)],y∗2=[r(lnK−lnx∗2)−p1](x∗2+g)(1+hx∗2)bx∗2. |
Similarly, the model in G3 is described as follows:
{dxdt=rx(lnK−lnx)−bx2y(1+hx)(x+g)−p1x,dydt=ebx2y(1+hx)(x+g)−dy−q2y. | (3.3) |
Theorem 4. Model (3.3) always has an equilibrium E¯B(Ke−p1r,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 E∗3=(x∗3,y∗3), which is locally asymptotically stable if U(x∗3)>0, where
x∗3=(d+q2)(1+gh)+√Δ(d−q1+q2)2[eb−h(d+q2)],y∗3=[r(lnK−lnx∗3)−p1](x∗3+g)(1+hx∗3)bx∗3. |
It is assumed that
(H1) p1<r;
(H2) d(1+gh)+√Δ(d)2(eb−dh)<K;
(H3) q1−q2−d<0,(d−q1+q2)(1+gh)+√Δ(d−q1+q2)2[eb+h(q1−q2−d)]<Ke−p1r;
(H4) (d+q2)(1+gh)+√Δ(d+q2)2[eb−h(d+q2)]<Ke−p1r.
For Model (2.2), we have x∗1<x∗2<x∗3 when q1<q2 and x∗2<x∗1<x∗3 when q2<q1<q2+d.
Define
yET1=[r(lnK−lnxET)−p1](xET+g)(1+hxET)bxET,yET2=r(lnK−lnxET)(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(lnK−lnxET)−bxETy(1+hxET)(xET+g)],<F2,n1>|(x,y)∈Σ1=xET[r(lnK−lnxET)−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+(1−λ)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+q1−q2p1[r(lnK−lnxET)−bxETy(1+hxET)(xET+g)]y. | (3.7) |
Let ς1=ebxET2+(1+hxET)(xET+g)[r(q1−q2)p1(lnK−lnxET)−d]. Then a positive equilibrium Ea1(xET,ya1) exists, where ya1=p1ς1bxET(q1−q2)>0. Therefore
ya1−yET2=p1bxET(q1−q2)[ebxET2−d(1+hxET)(xET+g)]. |
If x∗1<xET, then ya1>yET2, i.e., Ea1 is not located in Σ11, and then Ea1 is not a pseudo-equilibrium. If x∗1>xET, then ya1<yET2.
Similarly, we have
ya1−yET1=p1bxET(q1−q2)[ebxET2+(q1−q2−d)(1+hxET)(xET+g)]. |
If x∗2>xET, then ya1<yET1, i.e., Ea1 is not located in Σ11, and then Ea1 is not a pseudo-equilibrium. If x∗2<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(lnK−lnxET)−bxETy(1+hxET)(xET+g)],<F3,n1>|(x,y)∈Σ2=xET[r(lnK−lnxET)−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+(1−λ)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]y−q2p1[r(lnK−lnxET)−bxETy(1+hxET)(xET+g)]y. | (3.11) |
Let ς2=(1+hxET)(xET+g)[rq2p1(lnK−lnxET)+d]−ebxET2. Then the system (3.11) has a positive equilibrium Ea2(xET,ya2), where ya2=p1ς2q2bxET>0. Obviously,
ya2−yET2=−p1q2bxET[ebxET2−d(1+hxET)(xET+g)]. |
If x∗1>xET, then ya2>yET2, i.e., Ea2 is not located in Σ22. If x∗1<xET, then ya2<yET2. Similarly, we have
ya2−yET1=−p1q2bxET[ebxET2−(d+q2)(1+hxET)(xET+g)]. |
If x∗3<xET, then ya2<yET1, i.e, Ea2 is not located in Σ22. If x∗3>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+q1−q2],<F3,n2>|(x,y)∈Σ3=yET[ebx2(1+hx)(x+g)−d−q2]. | (3.12) |
According to Eq (3.12), <F2,n2>|(x,y)∈Σ3><F3,n2>|(x,y)∈Σ3. If x∗3<xET, then the system (3.1) does not have a sliding mode domain on Σ3. If x∗2<xET<x∗3, 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<x∗3}. | (3.13) |
According to the Filippov convex method, we have
dXdt=λF2+(1−λ)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(lnK−lnx)−bx2yET(1+hx)(x+g)−p1x,dydt=0. | (3.15) |
Then
[r(lnK−lnx)−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=(1−p1)xET at P(xET,yP) (yP=ˆy(xET)) and R0((1−p1)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<(1−p1)xET, then denote Q1((1−p1)xET,yQ1),Q2((1−p1)xET,yQ2) as the intersection points between γ1 and x=(1−p1)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 [(1−p1)xET,xET] with
dydx=ebx2y(1+hx)(x+g)−dyrx(lnK−lnx)−bx2y(1+hx)(x+g)≜φ(x,y),y((1−p1)xET,yR0)=yR0, |
which takes the form
y=y(x,yR0)=yR0+∫x(1−p1)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,0≤y≤yP}⊂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,0≤y≤yR1}⊂M1.
Since Δy=−q2y+τ1+ly, then define
ρ(y)≜(1−q2)y+τ1+ly. |
Obviously, the function ρ(y) reaches a minimum at y=⌢y, where ⌢y≜√τl(1−q2)−(1−q2)l(1−q2). Denote R(xET,⌢y)∈M, and its phase point is R+((1−p1)xET,ρ(⌢y)).
Define
x1ET≜max{xET|y(xET,R0)},x2ET≜max{xET|y(xET,Q1)≥yQ2/2}. |
Denote
τ1≜1−q2l,τ2≜(1−q2)(1+lyP)2l,τ3≜(1−q2)(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<xET≤x2ET.
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) τ≥τ1, then ⌢y≤0. For ∀y∈[0,yP], ρ′≥0 holds, and then τ≤ρ(y)≤ρ(yP) for y∈[0,yP]. Denote Λ11=[τ,ρ(yP)], Λ1=Λ⋂Λ11, and Neff=N1={(x+,y+)|x+=(1−p1)xET,y+∈Λ1}.
ⅱ) τ1<τ<τ2, then 0<⌢y<yP. For ∀y∈[0,⌢y], ρ′≤0 holds, and then ρ(⌢y)≤ρ(y)≤τ for y∈[0,⌢y]. Denote Λ21=[ρ(⌢y),τ], Λ∗21=Λ⋂Λ21, and N21={(x+,y+)|x+=(1−p1)xET,y+∈Λ∗21}. Similarly, for ∀y∈(⌢y,yP], ρ′>0 holds, i.e., ρ(⌢y)<ρ(y)≤ρ(yP). Denote Λ22=(ρ(⌢y),ρ(yP)], Λ∗22=Λ⋂Λ22, and N22={(x+,y+)|x+=(1−p1)xET,y+∈Λ∗22}. Thus, we have Neff=N2=N21⋃N22.
ⅲ) τ≤τ2, then ⌢y≥yP. For ∀y∈[0,yP], ρ′≤0 holds, i.e., ρ(yP)≤ρ(y)≤τ. Denote Λ33=[ρ(yP),τ] and Λ3=Λ⋂Λ33. Then Neff=N3={(x+,y+)|x+=(1−p1)xET,y+∈Λ3}.
Case Ⅱ: xET≥x1ET.
For this situation, Meff=M2. Similar to the discussion in case Ⅰ, we have
ⅰ) τ≥τ1. Then Neff=N4={(x+,y+)|x+=(1−p1)xET,y+∈Λ4}, where Λ4=[τ,ρ(yR1)].
ⅱ) τ1<τ<τ3. For ∀y∈[0,⌢y], we have N51={(x+,y+)|x+=(1−p1)xET,y+∈Λ21}. Similarly, for ∀y∈(⌢y,yR1], denote Λ52=(ρ(⌢y),ρ(yR1)], and then N52={(x+,y+)|x+=(1−p1)xET,y+∈Λ52}. Therefore, Neff=N5=N51⋃N52.
ⅲ) τ≤τ3. Then Neff=N6={(x+,y+)|x+=(1−p1)xET,y+∈Λ6} with Λ6=[ρ(yR1),τ].
Denote Gi(xET,yi)∈M, G+i((1−p1)xET,y+i)∈N, i=0,1,2,..., where . Since and lie on the same trajectory , then we have and , where
If such that , then Model (2.4) admits an order-1 periodic trajectory. Next, we will investigate the monotonicity of with for situations Ⅰ and Ⅱ.
Case Ⅰ: .
ⅰ) . monotonically increases on , and then the map monotonically increases on and monotonically decreases on .
ⅱ) . we have for and for . Denote , . Then monotonically increases on , and monotonically decreases on the interval , , respectively.
ⅲ) . monotonically decreases on , and then the map monotonically decreases on and monotonically increases on .
Case Ⅱ: .
ⅰ) . Then for and for .
ⅱ) . Denote and . Then monotonically decreases on and , and monotonically increases on and .
ⅲ) . The map monotonically decreases on and monotonically increases on .
For Model (2.4) with , if , then holds. Thus Model (2.4) is degenerated to
(4.1) |
Let be the solution of equation
with initial value . Define
We have and . Thus, (, ) is a natural enemy extinction periodic trajectory.
Theorem 5. The natural enemy extinction period trajectory (, ) is orbitally asymptotically stable if , where
Proof. For Model (4.1), we have
Then
Through calculation, we have
and
Thus,
Therefore, if , we have , and by Lemma 1, (, ) is orbitally asymptotically stable.
Denote that the points , are mapped to the points and , respectively, after a single impulse. Denote .
Case Ⅰ: .
Define
Obviously, and for , we have .
1) For , we have .
2) For , we have . Then
● 2-a) for , is the highest after the pulse, while is the lowest after the pulse. Then , , and thus such that .
● 2-b) for , if , then . Since the point is the lowest point after the pulse, then . If , we have . Then is the highest point after the impulse, i.e., . If , we have . Then is the highest point after the impulse, i.e., . Combine the above two aspects and it can be concluded that such that . While for , we have . In such a case, is not defined on and it is uncertain whether a fixed point of exists or not.
3) When , then . If , then , i.e., does not have a fixed point. While for , we have . In such a case, it is uncertain whether a fixed point of exists or not.
4) When , and then . If , the point is the lowest point after the pulse, then , i.e., has a fixed point on . If , the point is the lowest point after the pulse, and then , i.e., has a fixed point on .
Case Ⅱ: .
Define .
1) For , we have .
2) For , we have . Then
● 2-a) for , since and are the lowest and highest points after the pulse, then we have , , and thus such that ;
● 2-b) for and if , then ; if , then . On the other hand, take the point in a small neighborhood near the point , i.e., . is above . By the continuity of the impulse function and the Poincaré map, we have . Therefore, the map has a fixed point on .
3) When , then . If , we have . If , we have . Combine the above two aspects and it can be concluded that such that .
To sum up, we have:
Theorem 6. For the situation of , Model (2.4) admits an order-1 periodic trajectory. While for the situation of , Model (2.4) admits an order-1 periodic trajectory if or .
Let (, ) be the -periodic trajectory of the system (2.4) with initial values . The trajectory intersects at , where , and then it is pulsed to at . Thus,
Theorem 7. The -periodic trajectory (, ) with initial values is orbitally asymptotically stable if
where
Proof. The proof can be referred to that in Theorem 5 and is, therefore omitted.
For the purpose of simulation, it is assumed that , , , , , and .
When , the interior equilibrium is locally asymptotically stable, as presented in Figure 1. When increases to , 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 decreases with increasing , while increases and then decreases with increasing . Therefore, increasing the search rate for pests helps to reduce the number of pests.
When , and , the positive equilibrium of the region is , the positive equilibrium of the region is , and the positive equilibrium of the region is . When and , there is , , and . The sliding mode domain of Model (3.1) on can be represented as , and there is no pseudo-equilibrium on , as illustrated in Figure 4(a). When and , there is , , and . The sliding mode domain of Model (3.1) on can be represented as , and there is no pseudo-equilibrium on , as presented in Figure 4(b).
When , for , the periodic trajectory of Model (2.4) is presented by changing the killing rate of the prey, the amount of predator released , and the value of the parameter . When , and , 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 , the natural enemy extinction periodic trajectory loses its stability and an order-1 periodic trajectory occurs (Figure 5(b)).
Next, the accurate domains of and for different cases are presented as well as the order-1 periodic trajectory (Figure 6). The accurate domains of and are marked in red and blue solid lines, respectively. When , 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 , lies on the right side of the phase set. The schematic diagram of the accurate domain of and and the order-1 periodic trajectories for different cases are presented in subfigures Figure 6(d)–(f).
When , , , and , is locally asymptotically stable, and Model (2.4) admits an order-1 periodic trajectory for , as presented in Figure 7.
Finally, order- periodic solutions are presented for different and . When , is locally asymptotically stable. For control parameters , , or , , , Model (2.2) admits an order- periodic trajectory, as presented in subfigures 8(a) and (b). When , Model (2.1) admits a limit cycle. For , Model (2.2) admits an order- 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 plays an important role in determining the dynamics of the system, i.e., when is smaller than the level , the predators in the system will go to extinction and when is greater than , there exists a steady state 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 (Theorems 2–4, Figure 1). When , the stability is lost and a limit cycle surrounding 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 (Figure 4). The pseudo-equilibrium 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 5–7). 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- 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.
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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