Impulsive Biological Pest Control Strategies of the Sugarcane Borer

We propose an impulsive biological pest control of the sugarcane borer Diatraea saccharalis by its egg parasitoid Trichogramma galloi based on a mathematical model in which the sugarcane borer is represented by the egg and larval stages, and the parasitoid is considered in terms of the parasitized eggs. By using the Floquet theory and the small amplitude perturbation method, we show that there exists a globally asymptotically stable pest-eradication periodic solution when some conditions hold. The numerical simulations show that the impulsive release of parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.


Introduction
One of the challenges for the improvements in the farming and harvesting of cane is the biological pest control.Biological control is defined as the reduction of pest populations by using their natural enemies: predators, parasitoids, and pathogens 1 .Parasitoids are species which develop within or on the host and ultimately kill it.Thus, parasitoids are commonly reared in laboratories and periodically released in high-density populations as biological control agents of crop pests 2 .
The sugarcane borer Diatraea saccharalis is reported to be the most important sugarcane pest in the southeast region of Brazil 3 .The sugarcane borer builds internal galleries in the sugarcane plants causing direct damage that results in apical bud death, weight loss, and atrophy.Indirect damage occurs when there is contamination by yeasts that cause red rot in the stalks, either causing contamination or inverting the sugar, increasing yield loss in both sugar and alcohol 4 .
There is an important larvae parasitoid of the sugarcane borer, a wasp named Cotesia flavipes which is widely used in biological control in Brazil 3 .In spite of this control being considered successful in Brazil, there are some areas where Cotesia flavipes does not control the sugarcane borer efficiently.The using of the egg parasitoid Trichogramma galloi is considered an interesting option in this case 5 .
Mathematical modelling is an important tool used in studying agricultural problems.Thus, a good strategy of biological pest control, based on mathematical modelling, can increase the ethanol production.The applications of host-parasitoid models for biological control were reviewed in 6 .
In 7 , a mathematical model of interaction between the sugarcane borer (Diatraea saccharalis) and its egg parasitoid Trichogramma galloi was proposed which consists of three differential equations where x 1 is the egg density of the sugarcane borer, x 2 is the density of eggs parasitized by Trichogramma galloi, and x 3 is the larvae density of the sugarcane borer; r is the net reproduction rate; K is the carrying capacity of the environment; m 1 , m 2 , and m 3 are mortality rates of the egg, parasitized egg, and larvae populations; n 1 is the fraction of the eggs from which the larvae emerge at time t; n 2 is the fraction of the parasitized eggs from which the adult parasitoids emerge at time t; n 3 is the fraction of the larvae population which moults into pupal stage at time t; β is the rate of parasitism.The dynamics of this model without control was considered in 7 .
The proposed model 1.1 is a simplified compartmental one which considers only three easy monitoring stages of the sugarcane borer species: egg stage, parasitized egg stage, and larvae stage.Then, a reducing effect related to the searching efficiency of the adult parasitoid may be ignored by this model.According to 5 , the searching time of the parasitoid Trichogramma galloi is 1-2 day, and it can cause some augmentation of the parasitoid egg numbers when biological control measures are implemented.
Meanwhile, many authors have investigated the different population models concerning the impulsive pest control 8-15 .The impulsive pest control strategies based on preypredator models were presented in 8, 11, 13 .The host-parasitoid model with impulsive control was considered in 10 .Impulsive strategies of a pest management for SI epidemic models were proposed in 9, 12 .Pulse vaccination strategies for SIR epidemic models were considered in 14, 15 .In this paper, we suggest impulsive differential equations 16 to model the process of the biological pest control of the sugarcane borer.So we develop 1.1 introducing a periodic releasing of the parasitoids at fixed times

Preliminary
In this section, we will give some definitions, notations, and some lemmas which will be useful for our main results.
Let R 0, T , the map defined by the right-hand side of the first three equations of the system 1.2 .Let V 0 {V : R × R 3 -→R } be continuous on nτ, n 1 τ × R 3 , lim t,y → nτ ,x V t, y V nτ , x exist and V is locally Lipschitzian in x.
Definition 2.1.V ∈ V 0 , then for t, x ∈ nτ, n 1 τ × R 3 , the upper right derivative of V t, x with respect to the impulsive differential system 1.2 is defined as The solution of system 1.2 , denoted by x t : R -→R 3 , is continuously differentiable on nτ, n 1 τ × R 3 .Obviously, the global existence and uniqueness of solution of system 1.2 is guaranteed by the smoothness properties of f, for details see 16 .
We will use a basic comparison result from impulsive differential equations.
where g : R × R 3 -→R is continuous on nτ, n 1 τ × R and ψ n : R -→R is nondecreasing.Let R t be the maximal solution of the scalar impulsive differential equation 2 , similar results can be obtained when all the directions of the inequalities in the lemma are reversed and ψ n is nonincreasing.Note that if one has some smoothness conditions of g to guarantee the existence and uniqueness of solutions for 2.3 , then R(t) is exactly the unique solution of 2.3 .
Next, we consider the following system:

2.4
Lemma 2.3.System 2.4 has a unique positive periodic solution u 2 t with period τ and for every solution

2.5
Proof.Integrating and solving the first equation of 2.4 between pulses, we get After each successive pulse, we can deduce the following map of system 2.6 : , it corresponds to the unique positive periodic solution u 2 t of system 2.4 with the initial value u 2 0 a/ m 2 n 2 p/ 1 − e − m 2 n 2 τ .The fixed point u * 2 of map 2.7 implies that there is a corresponding cycle of period τ in u 2 t , that is, thus, u 2 nτ → u * 2 as t → ∞, so u 2 t is globally asymptotically stable.Thus, we have If a 0, the system 2.4 has a unique positive periodic solution and u 2 t is globally asymptotically stable.This completes the proof.
Therefore, system 1.2 has a pest-eradication periodic solution 0, x 2 t , 0 , where To study the stability of the pest-eradication periodic solution of 1.2 , we present the Floquet theory for a linear τ periodic impulsive equation 2.9 Then, we introduce the following conditions: , where PC R, C n×n is the set of all piecewise continuous matrix functions which is left continuous at t τ k , and C n×n is the set of all n × n matrices.
Let Φ t be the fundamental matrix of 2.9 , then there exists a unique nonsingular matrix M ∈ C n×n such that By equality 2.10 there correspondents to the fundamental matrix Φ t the constant matrix M which is called monodromy matrix of 2.9 .All monodromy matrices of 2.9 are similar and have the same eigenvalues.The eigenvalues λ 1 , λ 2 , . . ., λ n of the monodromy matrices are called the Floquet multipliers of 2.9 .

Stability of the Pest-Eradication Periodic Solution
In this section, we study the stability of the pest-eradication periodic solution 0, x 2 t , 0 of the system 1.2 .Next, we present an important result, concerning a condition that guarantees the global stability of this solution.
Theorem 3.1.The pest-eradication periodic solution 0, x 2 t , 0 of the system 1.2 is globally asymptotically stable provided that inequality Proof.The local stability of a periodic solution 0, x 2 t , 0 of system 1.2 may be determined by considering the behavior of small-amplitude perturbations y 1 t , y 2 t , y 3 t of the solution.Define where y 1 t , y 2 t , y 3 t are small perturbations.
Linearizing the system 1.2 , we have the following linear τ periodic impulsive system:

3.3
Let Φ t be the fundamental matrix of 3.3 .Then we have where Φ t must satisfy the following equation: and initial condition where I is the identity matrix.The solution of 3.5 is

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There is no need to calculate the exact form of * as it is not required in the analysis that follows.The resetting impulsive condition of 3.3 becomes Hence, if absolute values of all eigenvalues of are less than one, the τ periodic solution is locally stable.Then the eigenvalues of M are the following:

3.10
From 3.10 , one can see that |λ 1 | < 1 if and only if condition 3.1 holds true.According to Lemma 2.4, the pest-eradication periodic solution 0, x 2 t , 0 is locally asymptotically stable.
In the following, we prove the global attractivity.Choose sufficiently small ε > 0 such that

3.11
From the second equation of system 1.2 , noting that dx 2 /dt ≥ − m 2 n 2 x 2 , we consider the following impulsive differential equation

3.12
From Lemma 2.3, system 3.12 has a globally asymptotically stable positive periodic solution

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So by Lemma 2.2, we get for all t large enough.
From system, 1.2 and 3.14 , we obtain that

3.15
Integrating 3.15 on nτ, n 1 τ , we get Next, we prove that x 2 t → x 2 t as t → ∞.For 0 < ε 1 ≤ m 2 n 2 , there must exist a t 0 > 0 such that 0 < x 1 t ≤ ε 1 for all t ≥ t 0 .Without loss of generality, we may assume that 0 < x 1 t ≤ ε 1 for all t ≥ 0, from system 1.2 we have 3.17 Then, we have

3.18
By Lemma 2.3, system 3.18 has a positive periodic solution

3.19
Therefore, for any ε 2 > 0, there exists a t 1 , t > t 1 such that

3.20
Combining 3.14 and 3.20 , we obtain Assuming that 0 < x 1 t ≤ ε 1 for all t ≥ 0, from system 1.2 we have

3.22
Then, we have x 3 t ≤ v 3 t , while v 3 t is the solution of the following system:

3.23
By Lemma 2.3, system 3.23 has a positive solution

3.24
Thus, for any ε 3 > 0, there exists a t 1 , t > t 1 such that

Numerical Simulations of the Impulsive Biological Control
For numerical simulations of interactions between the sugarcane borer and its parasitoid the following values of model coefficients were used: n 1 0.1, n 2 0.1, n 3 0.02439, m 1 0.03566, m 2 0.03566, m 3 0.00256, K 25000.These values were obtained based on data published about the use of the egg parasitoid Trichogramma galloi against the sugarcane borer Diatraea saccharalis 3, 5, 7 .Figure 1 shows the population oscillations for r 0.1908 and β 0.0001723 without control.
According to 10 , economic injury level EIL cause economic damage.Economic threshold ET is population density at which control measures should be determined to prevent an increasing pest population from reaching the economic injury level.One can see from Figure 1 that the sugarcane borer larvae density x 3 takes on values more than the EIL for this pest x EIL 2500 numbers/ha 3 .In this case, it is necessary to apply the biological control.From Theorem 3.1, we have shown that the pest-eradication periodic solution 0, x 2 t , 0 of the system 1.2 is globally asymptotically stable if the condition 3.1 holds

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Choosing τ 70 days from 4.1 , we derive that when p > p min 3027 parasitoids/ha, the pest-eradication periodic solution of the host-parasitoid system is asymptotically stable.Dynamical behavior of the system with impulsive control p 3500 parasitoids/ha and with economic threshold x ET 2000 is shown in Figure 2. We can conclude that this control strategy seems successful because the larvae population of the sugarcane borer goes to extinction.But the aim of the biological control is not to eliminate all larvae population.The aim of the biological control of the sugarcane borer is to keep the larvae population at an acceptable low level below the EIL that indicates the pest densities at which applied biological control is economically justified.Choosing the release amount p 1500 parasitoids/ha, we can control the larvae population and keep it below the EIL see Figure 3 .It is obvious that the cost of the control strategy p 1500 is less than the cost of p 3500.
Applying the control strategy p 1000, we can see that the number of larvae individuals exceed x EIL at some time see Figure 4 .

Discussion and Conclusion
In this paper, we suggest a system of impulsive differential equations to model the process of the biological control of the sugarcane borer by periodically releasing its parasitoids.By using the Floquet theory and small amplitude perturbation method, we have proved that for any fixed period τ there exists a globally asymptotically stable pest-eradication periodic solution 0, x 2 t , 0 of the system 1.2 if the number of the parasitoids in periodic releases is greater than some critical value p min .When the stability of the pest-eradication periodic solution is lost, the numerical results show that the system 1.2 has rich dynamics.
If we choose the biological control strategy by periodical releases of the constant amount of parasitoids, the results of Theorem 3.1 can help in designing the control strategy by informing decisions on the timing of parasitoid releases.In this case, from 3.1 we have From 5.1 we can conclude that there exists a globally asymptotically stable pest-eradication periodic solution 0, x 2 t , 0 of the system 1.2 if the impulsive period is less than some critical value τ max .
It is interesting to discuss the result of Theorem 3.1, comparing the condition of the pest-free globally stable solution 3.1 with the similar result presented in 10 .In their paper, Mathematical Problems in Engineering the authors considered integrated pest management strategies based on discrete-time hostparasitoid models.Integrated pest management IPM is a long-term control strategy that combines biological, cultural, and chemical tactics to reduce pest populations to tolerable levels when the pests reach the ET 17 .IPM control strategy is not used in sugarcane crops because it is impossible to kill the sugarcane borer larvae by insecticides when it builds internal galleries in the sugarcane plants.The biological control is unique strategy of the pest control in this case.The resetting impulsive condition 3.1 which guarantees the global stability of the host-eradication periodic solution becomes which means that if the intrinsic growth rate is less than the mean parasitization rate over period τ, then the host population will become extinct eventually.This is the same conclusion which Tang et al. presented in 10 based on the inequality 3.9 from 10 .Then, the inequality 5.2 for the continuous-time model 1.2 and the inequality 3.9 for the discrete-time host-parasitoid model from 10 lead to similar results.Thus, the results of the present study show that the impulsive release of the parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.

3 dFigure 1 :
Figure 1: Evolution of the egg a , parasitized egg b , larvae populations c , and phase portraits d of system 1.2 without control.

1 dFigure 2 :
Figure 2: Evolution of the egg a , parasitized egg b , larvae populations c , and phase portraits d of system 1.2 for p 3500 parasitoids/ha.

1 dFigure 3 :
Figure 3: Evolution of the egg a , parasitized egg b , larvae populations c , and phase portraits d of system 1.2 for p 1500 parasitoids/ha.

1 dFigure 4 :
Figure 4: Evolution of the egg a , parasitized egg b , larvae populations c , and phase portraits d of system 1.2 for p 1000 parasitoids/ha.
3 . . ., }, τ is the period of the impulsive effect.Δx i x i t − x i t , x i t lim t → t x i t , i 1, 2, 3.That is, we can use releasing parasitized eggs to eradicate pests or keep the pest population below the economic damage level.