# Mathematical Modelling of Lesser Date Moth Using Sex Pheromone Traps and Natural Enemies

Authors: Moustafa El-Shahed; Asmaa M. Al-Dubiban
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a mathematical model for lesser date moth is proposed and analyzed. The interaction between the date palm tree, lesser date moth, and natural enemy has been investigated. The impact of sex pheromone traps on lesser date moth is demonstrated. Some sufficient conditions are obtained to ensure the local and global stability of equilibrium points. The occurrence of local bifurcation near the equilibrium points is performed using Sotomayor’s theorem. Theoretical results are illustrated using numerical simulations.


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## Body

## 1. Introduction

Lesser date moth is one of the most dangerous insect pests affecting date palm around the world. It was discovered in the Gulf region in the mid-eighties in the United Arab Emirates in 1986 and in Saudi Arabia in 1987. It is known that one-third of the production of date crops is lost as a result of the agricultural pests that infect the date palm, especially the lesser date moth [1–5].Control of the lesser date moth with insecticides is very difficult because it causes tunnels inside the stem of the palm in all directions and depths without any early signs of infection. Several methods have been adopted, including preventive and curative measures to control lesser date moth in date palm plantations around the world, including chemical and nonchemical control, as well as the use of pheromone traps as an effective tool for monitoring and attracting complete insects of lesser date moth. The sex pheromone trap is one of the most important methods used to combat lesser date moth. This method depends on collecting complete insects, killing them to prevent them, from completing their life cycle, and increasing their numbers [5, 6]. One of the most promising strategies for controlling lesser date moth is the use of a mating disruption using the sex pheromone traps [7–9]. Larvae predators are among the most important natural enemies that are used in many areas to combat some agricultural pests. Understanding the role of natural enemies in controlling lesser date moth is crucial to the difficulty in using other methods to control it. The lesser date moth is attacked by its natural enemy such as Goniozus swirskiana [10, 11]. The main objective of this paper is to propose and analyse a mathematical model of lesser date moth with sex pheromone traps taking into consideration the effect of the natural enemy on lesser date moth. The focus will be on the effect of sex pheromone traps parameter on lesser date moth. The paper is arranged as follows: in Section 2, the mathematical model is described and the boundedness of the solutions of lesser date moth system is verified. The local and global stability of the lesser date moth system is analyzed in Section 3. The local bifurcation conditions are derived in Section 4, and we find the existence of Hopf bifurcation around the coexistence equilibrium point. Finally, some numerical simulations are presented to verify the obtained theoretical results.


## 2. Mathematical Model

In this section, we construct a mathematical model describing the dynamic of lesser date moth (LDM) population based on its biological and ecological facts.(i)
LetPt represent the population density of date palm tree. We assume that the date palm grows with intrinsic growth rate r and carrying capacity k in the absence of lesser date moth.(ii)
We split the life cycle of LDM into two main stages: the larval stage and the adult stage. The larval stage is the most dangerous, as it feeds on the living tissue inside the trunk of the palm tree, which leads to the death of the date palm tree. LetLt represent the population density of the lesser date moth larvae. The adult stage is split into two compartments: the lesser date moth female Ft and lesser date moth male Mt.(iii)
LetWt represent the population density of natural enemy at time t.(iv)
The lesser date moth larvae emerge to female or male at the rate ofα. Among lesser date moth larvae which emerge, we suppose that a proportion ϵ becomes female, while complimentary part 1−ɛ becomes male.(v)
We consider the lesser date moth response on date palm as Holling type-II response functionβ1P/a+P, which denotes the rate of consumption of date palm per LDM. β1 is the predation rate of a date palm tree by lesser date moth. The parameter a is the half-saturation point, i.e., the value of P at which the predation rate is half the maximum value. e1 is the conversion rate of date palm biomass to LDM biomass, and we assume 0<e1<1 since the whole biomass of the date palm is not transformed to the biomass of the lesser date moth. It is also assumed that their natural enemy attacks the lesser date moth larvae, i.e., Goniozus swirskiana population, at a rate of β2, following Holling type-II functional response and b is the value of L at which the predation rate is half the maximum value. e2 is the conversion rate of lesser date moth biomass to natural enemy biomass and 0<e2<1.(vi)
In order to maintain a low level of lesser date moth, we consider controlling the use of female pheromone traps to disrupt male mating behavior. Firstly, mating between LDM males and LDM females is disturbed to reduce the chances of fertilization, which in turn reduces the number of offsprings. This is done using traps that are releasing a female pheromone lure to which males are attracted. This leads to a decrease in the number of males available for mating near females and reduces the chance of fertilization. Secondly, the LDM males, attracted with traps, are killed because traps contain pesticides that kill what is caught.(vii)
To take into account the effect of pheromone traps, we consider the approach proposed by Barclay and Van den Driessche [12], Barclay and Hendrichs [13], and Anguelov et al. [14]. That is, the strength of the trap is represented by the amount of pheromones released by an equivalent number of wild females. Following [14–17], we assume that the effect of the pheromone trap corresponds to the attraction of additional η females. In such a context, the total number of LDM females attracting LDM males is F+η. In particular, this means that males have a probability of η/F+η to be attracted to pheromone traps. The parameter γ represents the mortality rate of males attracted to pheromones traps.The model of lesser date moth with sex pheromone trap is described by the following system:(1)dPdt=rP1−Pk−β1PLa+P−μ1P,dLdt=e1β1PLa+P−αL−β2LWb+L−μ2L,dFdt=εαL−μ3F,dMdt=1−εαL−γηMF+η−μ4M,dWdt=e2β2LWb+L−μ5W,with initial values P0=P0≥0,L0=L0≥0,F0=F0≥0,M0=M0≥0,W0=W0≥0. The variables and parameters of system (1) are presented in Table 1.Table 1
Biological description of parameters used in system (1).
ParametersDescriptionrThe date palm tree intrinsic growth ratekThe carrying capacityαTransfer rate from larvae to adult lesser date mothβ1Predation intensity between date palm and larvaeβ2Predation intensity between larvae and natural enemye1Conversion coefficient for predation term betweenP and Le2Conversion coefficient for predation term betweenL and Wμ1Mortality rate of date palm treeμ2Mortality rate of lesser date moth larvaeμ3Mortality rate of lesser date moth femaleμ4Mortality rate of lesser date moth maleμ5Mortality rate of natural enemya,bHalf-saturation constantsηMaximal death rate by sex pheromone trapγThe capture rate for male by sex pheromone trap


### 2.1. Nonnegativity and Boundedness

The boundedness of the solutions of model (1) is given in the following.Proposition 1.
The population density of date palm treeP is always bounded from above by P1=k/rr−μ1.Proof.
According to the first equation of the lesser date moth system (1), one has dP/dt≤rP1−P/k, as a result dP/dt≤−r/kP2−k/rr−μ1P, then we have(2)dPdt+r−μ1P≤−rkP2−2krr−μ1P≤−krr−μ12.
Applying the comparison theorem, one obtainsP≤P1.Theorem 1.
All the solutions of lesser date moth model (1) starting in ℝ+5 are uniformly bounded.Proof.
Following [18–21], let Pt,Lt,Ft,Mt,Wt to be any solution of the system (1) with nonnegative initial conditions. Let Ht=Pt+Lt+Ft+Mt+Wt, then(3)dHdt≤rP1−Pk−μ1P−μ2L−μ3F−μ4M−μ5W−γηMF+η≤rP1−Pk−νH,≤−rkP2−kP−νH,≤−rkP−k22+rk4−νH,where ν=minμ1,μ2,μ3,μ4,μ5. Thus, dH/dt+νM≤rk/4. Applying the comparison theorem, one obtains 0≤Ht≤rk/4ν. Hence, all the solutions of lesser date moth system (1) that start in ℝ+5 are uniformly bounded in the region(4)Ω=P,L,F,M,W∈ℝ+5:H≤rk4ν+ξ,for anyξ>0.Using the next generation method [22], one can obtain the basic offspring number(5)R0=e1β1P1α+μ2a+P1.


## 2.1. Nonnegativity and Boundedness

The boundedness of the solutions of model (1) is given in the following.Proposition 1.
The population density of date palm treeP is always bounded from above by P1=k/rr−μ1.Proof.
According to the first equation of the lesser date moth system (1), one has dP/dt≤rP1−P/k, as a result dP/dt≤−r/kP2−k/rr−μ1P, then we have(2)dPdt+r−μ1P≤−rkP2−2krr−μ1P≤−krr−μ12.
Applying the comparison theorem, one obtainsP≤P1.Theorem 1.
All the solutions of lesser date moth model (1) starting in ℝ+5 are uniformly bounded.Proof.
Following [18–21], let Pt,Lt,Ft,Mt,Wt to be any solution of the system (1) with nonnegative initial conditions. Let Ht=Pt+Lt+Ft+Mt+Wt, then(3)dHdt≤rP1−Pk−μ1P−μ2L−μ3F−μ4M−μ5W−γηMF+η≤rP1−Pk−νH,≤−rkP2−kP−νH,≤−rkP−k22+rk4−νH,where ν=minμ1,μ2,μ3,μ4,μ5. Thus, dH/dt+νM≤rk/4. Applying the comparison theorem, one obtains 0≤Ht≤rk/4ν. Hence, all the solutions of lesser date moth system (1) that start in ℝ+5 are uniformly bounded in the region(4)Ω=P,L,F,M,W∈ℝ+5:H≤rk4ν+ξ,for anyξ>0.Using the next generation method [22], one can obtain the basic offspring number(5)R0=e1β1P1α+μ2a+P1.


## 3. Equilibria and Stability

The lesser date moth model (1) has the following three equilibrium points:(1)
E0=0,0,0,0,0, which always exists.(2)
The free lesser date moth equilibrium pointE1=P1,0,0,0,0. E1 exists if r>μ1.(3)
The free natural enemy equilibrium pointE2=P2,L2,F2,M2,0, where(6)P2=aα+μ2e1β1−α+μ2,L2=a+P2β1r−μ1−P2k,F2=ɛαL2μ3,M2=1−ɛαL2η+F2γη+μ4η+F2.The free natural enemy equilibrium point exists positively ifR0>1.(4)
The coexistence equilibrium pointE3=P3,L3,F3,M3,W3, where(7)L3=bμ5e2β2−μ5,F3=ɛbαμ5μ3e2β2−μ5,M3=1−ɛαF3+ηηγ+μ4F3+η,P3=−ar−kμ1+kr+ra+k−kμ12−4krβ1L32r,W3=b+L3e1β1P3−α+μ2a+P3β2a+P3.The coexistence equilibrium pointE3 exists if e2β2>μ5 and e1β1P3>α+μ2a+P3.The locally asymptotically stable equilibrium points of lesser date moth system (1) are now investigated. Following [23, 24], the Jacobian matrix is given as follows:(8)J=r−2rPk−μ1−aβ1La+P2−β1Pa+P000e1aβ1La+P2e1β1Pa+P−α−μ2−bβ2Wb+L200−β2Lb+L0ɛα−μ30001−ɛαγηMF+η2−γηF+η−μ400e2bβ2Wb+L200e2β2Lb+L−μ5.The stability of lesser date moth extinction equilibrium pointE0=0,0,0,0,0 is investigated as follows.Theorem 2.
Ifr<μ1, then E0 is locally asymptotically stable.Proof.
The Jacobian matrix of LDM system (1) at E0 is(9)JE0=r−μ100000−α+μ20000ɛα−μ30001−ɛα0−γ+μ400000−μ5.
The eigenvalues ofJE0 are λ1=r−μ1, λ2=−α+μ2, λ3=−μ3, λ4=−γ+μ4, and λ5=μ5. When r<μ1, all the eigenvalues of the Jacobian matrix JE0 near the equilibrium point E0 have negative real parts. Thus, due to the Routh–Hurwitz criterion, E0 is locally asymptotically stable if r<μ1.Theorem 3.
Ifr<μ1, then E0 is globally asymptotically stable.Proof.
Considering the following positive definite Lyapunov functionΨ1t=Pt+Lt+Ft+Mt+Wt, then the time derivative of Ψ1 along the solution of lesser date moth system (1), one obtains(10)dΨ1dt≤rP1−Pk−μ1P−μ2L−μ3F−μ4M−μ5W−γηMF+η≤r−μ1P.
Choosingr<μ1, one obtains dP/dt≤0. Thus, if r<μ1, then E0 is globally asymptotically stable.The stability of free lesser date moth equilibrium pointE1=P1,0,0,0,0 is investigated as follows.Theorem 4.
IfR0<1, then free lesser date moth equilibrium point E1 is locally asymptotically stable.Proof.
The Jacobian matrix of model (1) at E1 is(11)JE1=r−2rP1k−μ1−β1P1a+P10000e1βP1a+P1−α+μ20000ɛα−μ30001−ɛα0−γ−μ400000−μ5.
The eigenvalues ofJE1 are λ1=−μ5, λ2=−μ3, λ3=−γ+μ4, and λ4=−r+μ1. It can be observed that λ5<0, when e1βP1/a+P1<α+μ2 which is equivalent to R0<1. So, E1 is locally asymptotically stable if R0<1.Theorem 5.
Ife1β1P1<aμ2, then equilibrium point E1 is globally asymptotically stable.Proof.
The following positive definite Lyapunov function is considered:(12)Ψ2t=e1P−P1−P1lnPP1+12L2+Lt+Ft+Mt+1e2Wt.
By calculating the time derivative ofΨ2 along the solution of system (1), one obtains(13)dΨ2dt≤e1P−P1r1−Pk−μ1−β1La+P+L2e1β1Pa+P−α−μ2−β2Wb+L+e1β1PLa+P−μ2L≤−e1rkP−P12+e1β1P1a−μ2L+L2e1β1P1a+P1−α−μ2−β2WL2b+L≤−e1rkP−P12+e1β1P1a−μ2L+α+μ2R0−1L2.
Choosinge1β1P1<aμ2, one obtains dΨ2/dt≤0, and hence, the free lesser date moth equilibrium point E1 is globally asymptotically stable.
The stability of the free natural enemy equilibrium pointE2=P2,L2,F2,M2,0 is investigated as follows.Theorem 6.
Ifβ1L2/a+P22<r/k and e2β2L2/b+L2<μ5, then all the eigenvalues of the Jacobian matrix JE2 near the equilibrium point E2 have negative real parts. Thus, due to the Routh–Hurwitz criterion, the equilibrium point E2 is locally asymptotically stable.Proof.
The Jacobian matrix of LDM model (1) at E2 is(14)JE2=β1L2P2a+P22−rP2k−β1P2a+P2000ae1β1L2a+P22000−β2L2b+L20ɛα−μ30001−ɛαγηM2η+F22−γηη+F2−μ400000e2β2L2b+L2−μ5.
The first three eigenvalues ofJE2 are λ1=−μ3, λ2=−γη/F2+η+μ4, and λ3=e2β2L2/b+L2−μ5. The other roots are determined by(15)λ2+P2β1L2a+P22−rkλ+ae1β12P2L2a+P22=0.
Whenβ1L2/a+P22<r/k and e2β2L2/b+L2<μ5, then all the eigenvalues of the Jacobian matrix JE2 near the equilibrium point E2 have negative real parts. Thus, due to the Routh–Hurwitz criterion, the equilibrium point E2 is locally asymptotically stable.The local stability of the coexistence equilibrium pointE3=P3,L3,F3,M3,W3 is investigated as follows. The Jacobian matrix of model (1) at E3 is(16)JE3=β1L3P3a+P32−rP3k−β1P3a+P3000ae1β1L3a+P32000−L3β2b+L30ɛα1−μ30001−ɛα1γηM3η+F32−γηη+F3−μ400be2β2W3b+L32000.The first two eigenvalues ofJE2 are λ1=−μ3 and λ2=−γη/η+F2−μ4. The other eigenvalues are determined by(17)λ3+c1λ2+c2λ+c3=0,where(18)c1=β1L3P3a+P32−β2L3W3b+L32−rP3k,c2=L3aβ12e1P3a+P33+bβ22e2W3b+L33+β2P3rW3kb+L32−β1β2L3P3W3a+P32b+L32,c3=bβ22e2L3P3W3β1kL3−ra+P32ka+P32b+L33,when c1>0,c3>0 and c1c2>c3; then, all the eigenvalues of the Jacobian matrix J3 near the coexistence equilibrium point E3 have negative real parts. Hence, due to the Routh–Hurwitz criterion, the coexistence equilibrium point E3 is locally asymptotically stable. The local stability of the coexistence equilibrium point E3 is investigated in the following theorem.Theorem 7.
Ifc1>0,c3>0 and c1c2>c3, then the coexistence equilibrium point E3 is locally asymptotically stable.


## 4. Bifurcation Analysis

In this section, the local bifurcations near the equilibrium points of lesser date moth model (1) are investigated with the help of Sotomayor’s theorem [25] and the Hopf bifurcation theorem [26] to discuss the bifurcation analysis of the underlying system. LDM model (1) can be rewritten in a vector form dX/dt=GX, where X=P,L,F,M,WT and G=f1,f2,f3,f4,f5T with fi,i=1,2,3,4,5 are given in the right hand side of LDM model (1). Following [27, 28], according to Jacobian matrix of LDM system, one can verify that for any nonzero vector U=ζ1,ζ2,ζ3,ζ4,ζ5T and φ is any bifurcation parameter, and one obtains the following:(19)D2GX,φU,U=∂2G∂P2ζ12+∂2G∂L2ζ22+∂2G∂f2ζ32+∂2G∂M2ζ42+∂2G∂W2ζ52+∂2G∂P∂Lζ1ζ2+∂2G∂L∂Pζ2ζ1+∂2G∂P∂Fζ1ζ3+∂2G∂F∂Pζ3ζ1+∂2G∂P∂Mζ1ζ4+∂2G∂M∂Pζ4ζ1+∂2G∂P∂Wζ1ζ5+∂2G∂W∂Pζ5ζ1+∂2G∂L∂Fζ2ζ3+∂2G∂F∂Lζ3ζ2+∂2G∂L∂Mζ2ζ4+∂2G∂M∂Lζ4ζ2+∂2G∂L∂Wζ2ζ5+∂2G∂W∂Lζ5ζ2+∂2G∂F∂Mζ3ζ4+∂2G∂M∂fζ4ζ3+∂2G∂F∂Wζ3ζ5+∂2G∂W∂Fζ5ζ3+∂2G∂M∂Wζ4ζ5+∂2G∂W∂Mζ5ζ4.Consequently, we obtain that(20)D2GX,φU,U=2aβ1La+P3−2rkζ12−2aLe1β1ζ12a+P3+2ae1β1ζ2ζ1a+P2+2bWβ2ζ22b+L3−2bβ2ζ2ζ5b+L202γηζ3ζ4F+η2−2γηMζ32F+η3−2be2β2Wζ22b+L3+2be2β2ζ5ζ2b+L2.Theorem 8.
LDM system (1) undergoes a transcritical bifurcation with respect to the bifurcation parameter r around E0 if r=μ1.Proof.
LetV1=ν1,ν2,ν3,ν4,ν5T be the eigenvector corresponding to eigenvalue λ=0 of JE0; hence, JE0V1=0 gives V1=ν1,0,0,0,0,T, where ν1 is any nonzero real number. Similarly, suppose V2=τ1,τ2,τ3,τ4,τ5T be the eigenvector corresponding to eigenvalue λ=0 of JE0T, thus JE0TV2=0 gives V2=τ1,0,0,0,0, where τ1 is any nonzero real number. Consider ∂G/∂r=GrX,r=P1−P/k,0,0,0,0T, thus V2TGrE0,r∗=0. Therefore, according to Sotomayor’s theorem for local bifurcation, LDM model (1) has no saddle-node bifurcation near E0 at r∗=μ1.
Now,(21)DGrE0,r∗=1000000000000000000000000,then V2TDGrE0,r∗V1=ν1τ1≠0. Using (20), one obtains(22)V2TD2GX,rV1,V1=−β1ζ5ζ1a−2rζ12k≠0.
Thus, according to Sotomayor’s theorem, LDM system (1) has a transcritical bifurcation at r∗=μ1 as the parameter r passes through the value r∗, thus the proof is complete.Theorem 9.
LDM system (1) undergoes a transcritical bifurcation with respect to the bifurcation parameter β1 around E1=P1,0,0,0,0 if R0=1.Proof.
The Jacobian matrix of LDM system (1) at the free lesser date moth equilibrium point E1 with β1=β1∗=a+P1α+μ2/e1P1 has zero eigenvalue taking the form(23)JE1,β1∗=−r−μ1−β1P1a+P1000000000ɛα−μ30001−ɛα0−γ−μ400000−μ5.
The eigenvector corresponding toJE1V3=0 is(24)V3=−β1P1ν2a+P1r−μ1ν2ɛαν2μ31−ɛαν2γ+μ40,where ν2 is any nonzero real number. Similarly, the eigenvector corresponding to JE1TV4=0 is given by V4=0,τ2,0,0,0T, where τ2 is any nonzero real number. Consider ∂G/∂β1=Gβ1X,β1=−PL/a+P,e1PL/a+P,0,0,0T, thus V4TGβ1E1,β1∗=0. Therefore, according to Sotomayor’s theorem for local bifurcation, LDM model (1) has no saddle-node bifurcation near E1 at β1∗=a+P1α+μ2/e1P1. Now,(25)DGβ1E1,β1∗=0−β1P1a+P10000e1β1P1a+P1000000000000000000,then V4TDGβ1E1,β1∗V3=e1P1ν1τ1/a+P1≠0. Using (20), one obtains(26)V4TD2GX,β1V3,V3=2aβ1e1a+P12ν1ν2−2β2bν2ν5≠0.
Thus, according to Sotomayor’s theorem, LDM system (1) has a transcritical bifurcation at β1∗=a+P1α+μ2/e1P1, as the parameter β1 passes through the value β1∗, thus the proof is complete. It is interesting to note that R0=1 is equivalent to r∗=kμ1α−β1e1+μ2/a+kα+μ2−β1e1k.Theorem 10.
LDM system (1) undergoes a transcritical bifurcation with respect to the bifurcation parameter β2 around E2=P1,L1,F1,M1,0 if β2=μ5b+L2/e2L2.Proof.
The Jacobian matrix of LDM system (1) at the free lesser date moth equilibrium point E2 with β2=β2∗=μ5b+L2/e2L2 has zero eigenvalue taking the form(27)JE2,β2∗=β1P2L2a+P22−rP2k−β1P2a+P2000e1aβ1L2a+P22000−L2β2b+L20ɛα−μ30001−ɛαγηM2η+F22−γηη+F2−μ4000000.
The eigenvector corresponding toJE1V5=0 is(28)V5=ν1L2a+P2−ra+P2kβ1ν1ɛα1kL2β1−ra+P22ν1kβ1μ3a+P2α1kL2β1−ra+P221−ɛμ3η+F22+γεηM2ν1kη+F2a+P2β1μ3γη+η+F2μ4ae1β1b+L2ν1β2a+P22,where ν1 is any nonzero real number. Similarly, the eigenvector corresponding to JE2TV6=0 is given by V6=0,0,0,0,τ5T, where τ5 is any nonzero real number. Consider ∂G/∂β2=Gβ2X,β2=0,−LW/b+L,0,0,0,e2LW/b+LT, thus V6TGβ2E2,β2∗=0. Therefore, according to Sotomayor’s theorem for local bifurcation, LDM model (1) has no saddle-node bifurcation near E2 at β2∗=μ5b+L2/e2L2. Now,(29)DGβ2E2,β2∗=000000000−L2b+L200000000000000e2L2b+L2,then V6TDGβ2E2,β2∗V5=aβ1e1e2L2ν1τ5/β2a+P22≠0. Using (20), one obtains(30)V6TD2GX,β2V5,V5=2bβ2e2ν2ν5τ5b+L22≠0.
Thus, according to Sotomayor’s theorem, LDM system (1) has a transcritical bifurcation at β2∗=a+P1α+μ2/e1P1 as the parameter β2 passes through the value β2∗=μ5b+L2/e2L2, thus the proof is complete.In this part, we shall show that as the coexistence equilibriumE3 loses stability, periodic solutions can bifurcate from the positive equilibrium. We choose the parameter r as the bifurcation parameter as it plays a crucial role in the growth of date palm tree. We shall apply Liu’s criteria [26] to obtain the conditions for periodic solution arising from Hopf bifurcation.Theorem 11.
For the coexistence equilibrium pointE3 of lesser date moth (1), the system around E3 enters into the Hopf bifurcation when r passes r∗ if the coefficients cjrj=1,2,3 at r=r∗ satisfy the following condition:(1)
Φr∗=c1r∗c2r∗−c3r∗=0(2)
dΦr/dφ|r=r∗≠0The characteristic equation (15) of the coexistence equilibrium point E3 has two purely imaginary roots if and only if c1c2−c3=0. The value of Φr=c1c2−c3 is given by(31)Φr=L3B11−β2L3W3b+L32aβ12e1P3a+P33+β2W3bβ2e2−B11b+L3b+L33−bβ22B11e2W3b+L33,where B11=P3r/k−β1L3P3/a+P32. For r=r∗, the characteristic equation (15) turns into(32)λ+c1λ2+c2=0.The roots of equation (32) are λ1=ic2,λ2=−ic2, and λ3=c1. One can rewrite the roots as follows: λ1=mr+inr,λ2=mr−inr, and λ3=−c1. To apply Liu’s Hopf bifurcation criterion [26], we need to verify the transversality condition. Substituting λ=mr+inr into (15) and separating the real and imaginary, we get(33)m3−3mn2+c1m2−c1n2+c2m+c3=0,(34)n2=3m2+2c1m+c2.Substituting the value ofn2 from (34) into (33), differentiating with respect to r, and utilizing mr∗=0 and nr∗≠0, we have(35)dmdr|r=r∗=d/drc1c2−c32c12+c2r=r∗=dΦr/dr2c12+c2r=r∗≠0.Thus, it can be concluded that the coexistence equilibrium pointE3 is locally asymptotically stable for r<r∗. Furthermore, according to Liu’s criterion, a simple Hopf bifurcation occurs at r=r∗, and for r>r∗, the interior equilibrium point E3 approaches to a periodic solution. According to Theorem 11, there exists a Hopf bifurcation in lesser date moth model (1) where the Hopf bifurcation is controlled by r.


## 5. Numerical Simulations

In this part, numerical simulations of lesser date moth system (1) are conducted to illustrate the theoretical results obtained before. The interactions between date palm, lesser date moth, and natural enemy will be simulated by the following parameters:(36)r=3,k=3,μ1=0.01,μ2=0.03,μ3=0.01,μ4=0.01,μ5=0.01,α=0.5,β1=2,β2=0.5,γ=0.5,η=0.2,ɛ=0.5,a=3,b=10.From Figure1, it can be seen that the sex pheromone trap parameter η is important in that it affects the population density of the lesser date moth male. One can observe from Figure 1 that the population density of lesser date moth male decreases with increasing η. We conclude that the dynamics of lesser date moth can be controlled by sex pheromone trap parameters η. In order to show the effect of the intrinsic growth rate of the date palm tree r, we draw the bifurcation diagram with respect to r as a bifurcation parameter. If we increase the value of growth rate r and keeping all other parameter value fixes, it can be seen that a transcritical bifurcation occurs at r=0.01, as shown in Figure 2 and stated in Theorem 8. It can also be observed that when 0.01<r<0.0198148, the lesser date moth free equilibrium point E1=P1,0,0,0,0 is locally asymptotically stable. It can be seen that a transcritical bifurcation occurs at r=0.0198148, as shown in Figure 2 and stated in Theorem 9. To understand the change in the dynamical behavior of the lesser date moth system model system (1) due to variation of the predation intensity between date palm and larvae β1, we increase the value of β1 and keep all other parameter value fixes. It can be seen that a transcritical bifurcation occurs at β1=1.32722, as shown in Figure 3 and stated in Theorem 9. In order to show the effect of predation intensity between larvae and natural enemy β2, we draw the bifurcation diagram with respect to β2 as a bifurcation parameter. If we increase the value of predation rate β2 and keeping all other parameter value fixes, it can be seen that a transcritical bifurcation occurs at β2=0.132143, as shown in Figure 4 and stated in Theorem 10. The bifurcation diagram with respect to r as a bifurcation parameter is indicated in Figure 5. According to Theorem 11, it can be seen that the supercritical Hopf bifurcation value for F and W is localized at r=6.13149 and one can verify the results c16.13149=5.74251>0,c36.13149=0.0136073>0, and Φ6.13149=0. Furthermore, dΦr/dr=−0.105178, thus the transversality condition for Hopf bifurcation is also satisfied. Figure 6 shows stable steady distribution for r=0.5. Now, if we increase the values of r from 5 to 7, we observe that the system becomes unstable, and the lesser date moth model shows limit cycle oscillations. It is clear that if we increase the increase date palm tree intrinsic growth rate, the system enters into Hopf bifurcation. In order to show the effect of carrying capacity around the coexistence equilibrium points, we draw the bifurcation diagram with respect to k as a bifurcation parameter. It can be seen that the supercritical Hopf bifurcation value for F localized at k=4.68754, as shown in Figure 7. It can also be observed that when k<4.68754, the coexistence equilibrium point E3 is locally asymptotically stable, as shown in Figure 8(a). For k>4.68754, the system undergoes limit cycle behavior, as shown in Figure 8(b).Figure 1
Time series of model (1) with.η=0.05,0.2,0.5.Figure 2
Bifurcation diagram of lesser date moth model (1) with respect to r,0≤r≤0.04Figure 3
Bifurcation diagram of lesser date moth model (1) with respect to β1.Figure 4
Bifurcation diagram of lesser date moth model (1) with respect to β2.Figure 5
Bifurcation diagram of lesser date moth model (1) with respect to r,1≤r≤10.Figure 6
Time series of model (1) with r=5,7.
(a)(b)Figure 7
Bifurcation diagram of lesser date moth model (1) with respect to k.Figure 8
Time series of model (1) with k=4,5. (a) k = 4. (b) k = 5.
(a)(b)


## 6. Discussion and Conclusion

In this paper, we consider a mathematical model for lesser date moth using sex pheromone traps. We have investigated the sex pheromone trap impact on the spread of lesser date moth in date palm trees. The interaction between date palm tree, lesser date moth, and the natural enemy was investigated. We obtain some sufficient conditions that ensure the local and global stability of equilibrium points. It has been shown that forR0<1, the equilibrium E1 is locally asymptotically stable. According to Theorem 9, we may conclude that for R0=1, the LDM model undergoes a transcritical bifurcation. According to Theorem 10, there exists a Hopf bifurcation in the lesser date moth model where the Hopf bifurcation is controlled by the date palm tree intrinsic growth rate r. We conclude that sex pheromone trap parameters can control the dynamics of lesser date moth. The occurrence of local bifurcation near the equilibrium point is performed using Sotomayor’s theorem. Analytical studies can never be completed without numerical illustrations of the analytical results. So, all analytical findings are numerically verified. The numerical studies in this paper show the system’s dynamical behavior at equilibrium points, which is in good agreement with the analytical studies. We have noticed that the sex pheromone trap parameter η plays a key role in controlling the population density of the lesser date moth male. It is worth noting that although we presented a realistic mathematical model on the spread of the lesser date moth and its control using pheromone traps and natural enemy, our study is only limited to a comprehensive theoretical analysis of such scenarios. Our theoretical study can capture the main mechanisms associated with lesser date moth control using pheromone traps with no chemical pesticides or mechanical resistance to the insect. A mathematical model representing the integrated lesser date moth control using pheromone traps, insecticides, and mechanical resistance would benefit significantly. As part of future work to improve the model of this paper, the model studied here could be refined to describe and explain the role of pesticides in LDM control in the presence of pheromone traps. Additional experimental or field investigations in this direction may be useful to verify whether or not the processes defined by the model are actually operating similarly in the real world. We assumed the environmental effect and temperature, and all other situations are homogeneous in the present work, but it is inhomogeneous in reality. One may consider a heterogeneous lesser date moth system for future work.



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*Source: 8835321-2021-01-18.xml*