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

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 dem-onstrated. 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.


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][2][3][4][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. e sex pheromone trap is one of the most important methods used to combat lesser date moth. is 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][8][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. e lesser date moth is attacked by its natural enemy such as Goniozus swirskiana [10,11]. e 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. e focus will be on the effect of sex pheromone traps parameter on lesser date moth. e 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. e local and global stability of the lesser date moth system is analyzed in Section 3. e 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.

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) Let P(t) 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. e 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. Let L(t) represent the population density of the lesser date moth larvae. e adult stage is split into two compartments: the lesser date moth female F(t) and lesser date moth male M(t).
(iii) Let W(t) represent the population density of natural enemy at time t. (iv) e lesser date moth larvae emerge to female or male at the rate of α. Among lesser date moth larvae which emerge, we suppose that a proportion ϵ becomes female, while complimentary part (1 − ε) becomes male.
(v) We consider the lesser date moth response on date palm as Holling type-II response function ((β 1 P)/(a + P)), which denotes the rate of consumption of date palm per LDM. β 1 is the predation rate of a date palm tree by lesser date moth. e parameter a is the half-saturation point, i.e., the value of P at which the predation rate is half the maximum value. e 1 is the conversion rate of date palm biomass to LDM biomass, and we assume 0 < e 1 < 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 β 2 , following Holling type-II functional response and b is the value of L at which the predation rate is half the maximum value. e 2 is the conversion rate of lesser date moth biomass to natural enemy biomass and 0 < e 2 < 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. is is done using traps that are releasing a female pheromone lure to which males are attracted. is 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]. at is, the strength of the trap is represented by the amount of pheromones released by an equivalent number of wild females. Following [14][15][16][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. e parameter c represents the mortality rate of males attracted to pheromones traps.
e model of lesser date moth with sex pheromone trap is described by the following system: with initial values e variables and parameters of system (1) are presented in Table 1.

Nonnegativity and Boundedness.
e boundedness of the solutions of model (1) is given in the following.

Proposition 1.
e population density of date palm tree P is always bounded from above by P 1 � (k/r)(r − μ 1 ).
e locally asymptotically stable equilibrium points of lesser date moth system (1) are now investigated. Following [23,24], the Jacobian matrix is given as follows: Table 1: Biological description of parameters used in system (1).

Parameters
Description r e date palm tree intrinsic growth rate k e carrying capacity α Transfer rate from larvae to adult lesser date moth β 1 Predation intensity between date palm and larvae β 2 Predation intensity between larvae and natural enemy e 1 Conversion coefficient for predation term between P and L e 2 Conversion coefficient for predation term between L and W μ 1 Mortality rate of date palm tree μ 2 Mortality rate of lesser date moth larvae μ 3 Mortality rate of lesser date moth female μ 4 Mortality rate of lesser date moth male μ 5 Mortality rate of natural enemy a, b Half-saturation constants η Maximal death rate by sex pheromone trap c e capture rate for male by sex pheromone trap e stability of lesser date moth extinction equilibrium point E 0 � (0, 0, 0, 0, 0) is investigated as follows. Proof.
e Jacobian matrix of LDM system (1) at E 0 is , and λ 5 � μ 5 . When r < μ 1 , all the eigenvalues of the Jacobian matrix J(E 0 ) near the equilibrium point E 0 have negative real parts. us, due to the Routh-Hurwitz criterion, E 0 is locally asymptotically stable if r < μ 1 .
e following positive definite Lyapunov function is considered: By calculating the time derivative of Ψ 2 along the solution of system (1), one obtains 4 Mathematical Problems in Engineering Choosing e 1 β 1 P 1 < aμ 2 , one obtains (dΨ 2 /dt) ≤ 0, and hence, the free lesser date moth equilibrium point E 1 is globally asymptotically stable. e stability of the free natural enemy equilibrium point Proof.
e Jacobian matrix of LDM model (1) at E 2 is e first three eigenvalues of J( and 5 . e other roots are determined by When ((β 1 L 2 )/(a + P 2 ) 2 ) < (r/k) and ((e 2 β 2 L 2 )/ (b + L 2 )) < μ 5 , then all the eigenvalues of the Jacobian matrix J(E 2 ) near the equilibrium point E 2 have negative real parts.
us, due to the Routh-Hurwitz criterion, the equilibrium point E 2 is locally asymptotically stable.
Mathematical Problems in Engineering e local stability of the coexistence equilibrium point e first two eigenvalues of J(E 2 ) are λ 1 � − μ 3 and e other eigenvalues are determined by where when c 1 > 0, c 3 > 0 and c 1 c 2 > c 3 ; then, all the eigenvalues of the Jacobian matrix J 3 near the coexistence equilibrium point E 3 have negative real parts. Hence, due to the Routh-Hurwitz criterion, the coexistence equilibrium point E 3 is locally asymptotically stable. e local stability of the coexistence equilibrium point E 3 is investigated in the following theorem.

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) � G(X), where X � (P, L, F, M, W) T and G � (f 1 , f 2 , f 3 , f 4 , f 5 ) T with f i , 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 � (ζ 1 , ζ 2 , ζ 3 , ζ 4 , ζ 5 ) T and φ is any bifurcation parameter, and one obtains the following: Consequently, we obtain that
In this part, we shall show that as the coexistence equilibrium E 3 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 point E 3 of lesser date moth (1), the system around E 3 enters into the Hopf bifurcation when r passes r * if the coefficients c j (r) (j � 1, 2, 3) at r � r * satisfy the following condition: (1) Φ(r * ) � [c 1 (r * )c 2 (r * ) − c 3 (r * )] � 0 (2) (dΦ(r)/dφ)|r � r * ≠ 0 e characteristic equation (15) of the coexistence equilibrium point E 3 has two purely imaginary roots if and only if c 1 c 2 − c 3 � 0. e value of Φ(r) � c 1 c 2 − c 3 is given by where B 11 � P 3 ((r/k) − ((β 1 L 3 P 3 )/(a + P 3 ) 2 )). For r � r * , the characteristic equation (15) turns into e roots of equation (32) are λ 1 � ic 2 , λ 2 � − ic 2 , and λ 3 � c 1 . One can rewrite the roots as follows: To apply Liu's Hopf bifurcation criterion [26], we need to verify the transversality condition. Substituting λ � m(r) + in(r) into (15) and separating the real and imaginary, we get Substituting the value of n 2 from (34) into (33), differentiating with respect to r, and utilizing m(r * ) � 0 and n(r * ) ≠ 0, we have us, it can be concluded that the coexistence equilibrium point E 3 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 E 3 approaches to a periodic solution. According to eorem 11, there exists a Hopf bifurcation in lesser date moth model (1) where the Hopf bifurcation is controlled by r.

Numerical Simulations
In this part, numerical simulations of lesser date moth system (1) are conducted to illustrate the theoretical results obtained before. e interactions between date palm, lesser date moth, and natural enemy will be simulated by the following parameters: (36) From Figure 1, 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 eorem 8. It can also be observed that when 0.01 < r < 0.0198148, the lesser date moth free equilibrium point E1 � (P 1 , 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 eorem 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 β 1 , we increase the value of β 1 and keep all other parameter value fixes. It can be seen that a transcritical bifurcation occurs at β 1 � 1.32722, as shown in Figure 3 and stated in eorem 9. In order to show the effect of predation intensity between larvae and natural enemy β 2 , we draw the bifurcation diagram with respect to β 2 as a bifurcation parameter. If we increase the value of predation rate β 2 and keeping all other parameter value fixes, it can be seen that a transcritical bifurcation occurs at β 2 � 0.132143, as shown in Figure 4 and stated in eorem 10. e bifurcation diagram with respect to r as a bifurcation parameter is indicated in Figure 5. According to eorem 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 c 1 (6.13149) � 5.74251 > 0, c 3 (6.13149) � 0.0136073 > 0, and Φ(6.13149)

Mathematical Problems in Engineering
� 0. Furthermore, (dΦ(r)/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 E 3 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

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. e 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 for R 0 < 1, the equilibrium E 1 is locally asymptotically stable. According to eorem 9, we may conclude that for R 0 � 1, the LDM model undergoes a transcritical bifurcation. According to eorem 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. e 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. e 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.

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.