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Mathematical Problems in Engineering
Volume 2012, Article ID 726783, 14 pages
http://dx.doi.org/10.1155/2012/726783
Research Article

Impulsive Biological Pest Control Strategies of the Sugarcane Borer

Centro de Engenharia, Modelagem e Ciências Sociais Aplicadas, Universidade Federal do ABC, 09.210-110 Santo André, SP, Brazil

Received 7 April 2012; Accepted 11 June 2012

Academic Editor: Alexander P. Seyranian

Copyright © 2012 Marat Rafikov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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 𝑑𝑥1𝑥𝑑𝑡=𝑟11𝐾𝑥1𝑚1𝑥1𝑛1𝑥1𝛽𝑥1𝑥2,𝑑𝑥2𝑑𝑡=𝛽𝑥1𝑥2𝑚2𝑥2𝑛2𝑥2,𝑑𝑥3𝑑𝑡=𝑛1𝑥1𝑚3𝑥3𝑛3𝑥3,(1.1) where 𝑥1 is the egg density of the sugarcane borer, 𝑥2 is the density of eggs parasitized by Trichogramma galloi, and 𝑥3 is the larvae density of the sugarcane borer; 𝑟 is the net reproduction rate; 𝐾 is the carrying capacity of the environment; 𝑚1, 𝑚2, and 𝑚3 are mortality rates of the egg, parasitized egg, and larvae populations; 𝑛1 is the fraction of the eggs from which the larvae emerge at time 𝑡; 𝑛2 is the fraction of the parasitized eggs from which the adult parasitoids emerge at time 𝑡; 𝑛3 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 [815]. The impulsive pest control strategies based on prey-predator 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 𝑑𝑥1𝑥𝑑𝑡=𝑟11𝐾𝑥1𝑚1𝑥1𝑛1𝑥1𝛽𝑥1𝑥2𝑑𝑥2𝑑𝑡=𝛽𝑥1𝑥2𝑚2𝑥2𝑛2𝑥2𝑡𝑛𝜏,𝑛𝑍+,𝑑𝑥3𝑑𝑡=𝑛1𝑥1𝑚3𝑥3𝑛3𝑥3Δ𝑥1(𝑡)=0Δ𝑥2(𝑡)=𝑝𝑡=𝑛𝜏,𝑛𝑍+,Δ𝑥3(𝑡)=0,(1.2) where 𝑝 is the release amount of the parasitized eggs at 𝑡=𝑛𝜏,𝑛𝑍+,𝑍+={0,1,2,,},𝜏 is the period of the impulsive effect. Δ𝑥𝑖=𝑥𝑖(𝑡+)𝑥𝑖(𝑡),𝑥𝑖(𝑡+)=lim𝑡𝑡+𝑥𝑖(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.

2. Preliminary

In this section, we will give some definitions, notations, and some lemmas which will be useful for our main results.

Let 𝑅+=[0,),𝑅3+={𝑥𝑅3𝑥>0}. Denote 𝑓=(𝑓1,𝑓2,𝑓3)𝑇, the map defined by the right-hand side of the first three equations of the system (1.2). Let 𝑉0={𝑉𝑅+×𝑅3+𝑅+} be continuous on (𝑛𝜏,(𝑛+1)𝜏]×𝑅3+, lim(𝑡,𝑦)(𝑛𝜏+,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑛𝜏+,𝑥)  exist and 𝑉 is locally Lipschitzian in 𝑥.

Definition 2.1. 𝑉𝑉0, then for (𝑡,𝑥)(𝑛𝜏,(𝑛+1)𝜏]×𝑅3+, the upper right derivative of 𝑉(𝑡,𝑥) with respect to the impulsive differential system (1.2) is defined as 𝐷+𝑉(𝑡,𝑥)=lim01sup[].𝑉(𝑡+,𝑥+𝑓(𝑡,𝑥))𝑉(𝑡,𝑥)(2.1)

The solution of system (1.2), denoted by 𝑥(𝑡)𝑅+𝑅3+, is continuously differentiable on (𝑛𝜏,(𝑛+1)𝜏]×𝑅3+. 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.

Lemma 2.2 (see [16]). Let 𝑉𝑉0, assume that 𝐷+𝑉𝑡𝑉(𝑡,𝑥)𝑔(𝑡,𝑉(𝑡,𝑥)),𝑡𝑛𝜏,𝑡,𝑥+𝜓𝑛(𝑉(𝑡,𝑥(𝑡))),𝑡=𝑛𝜏,(2.2) where 𝑔𝑅+×𝑅3+𝑅+ is continuous on (𝑛𝜏,(𝑛+1)𝜏]×𝑅+ and 𝜓𝑛𝑅+𝑅+ is nondecreasing. Let 𝑅(𝑡) be the maximal solution of the scalar impulsive differential equation 𝑢𝑡̇𝑢(𝑡,𝑥)=𝑔(𝑡,𝑢(𝑡)),𝑡𝑛𝜏,+=𝜓𝑛𝑢0(𝑢(𝑡)),𝑡=𝑛𝜏,+=𝑢0(2.3) existing on [0,). Then 𝑉(0+,𝑥0)𝑢0 implies that 𝑉(𝑡,𝑥(𝑡))𝑅(𝑡),𝑡0, where x(t) is any solution of (1.2), similar results can be obtained when all the directions of the inequalities in the lemma are reversed and 𝜓𝑛 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: 𝑑𝑢2𝑑𝑡=𝑎𝑚2𝑢2𝑛2𝑢2,𝑡𝑛𝜏,Δ𝑢2𝑢(𝑡)=𝑏,𝑡=𝑛𝜏20+=𝑢200.(2.4)

Lemma 2.3. System (2.4) has a unique positive periodic solution ̃𝑢2(𝑡) with period 𝜏 and for every solution 𝑢2(𝑡) of (2.4) |𝑢2(𝑡)̃𝑢2(𝑡)(𝑡)|0 as 𝑡, where ̃𝑢2𝑎(𝑡)=𝑚2+𝑛2+𝑏𝑒(𝑚2+𝑛2)(𝑡𝑛𝜏)1𝑒(𝑚2+𝑛2)𝜏],𝑡(𝑛𝜏,(𝑛+1)𝜏,𝑛𝑍+,̃𝑢20+=𝑎𝑚2+𝑛2+𝑝1𝑒(𝑚2+𝑛2)𝜏.(2.5)

Proof. Integrating and solving the first equation of (2.4) between pulses, we get 𝑢2𝑎(𝑡)=𝑚2+𝑛2+𝑢2𝑛𝜏+𝑒(𝑚2+𝑛2)(𝑡𝑛𝜏)].,𝑡(𝑛𝜏,(𝑛+1)𝜏(2.6)
After each successive pulse, we can deduce the following map of system (2.6): 𝑢2(𝑛+1)𝜏+=𝑎𝑚2+𝑛2+𝑢2𝑛𝜏+𝑎𝑚2+𝑛2𝑒(𝑚2+𝑛2)𝜏]+𝑝,𝑡(𝑛𝜏,(𝑛+1)𝜏.(2.7) Equation (2.7) has a unique fixed point 𝑢2=𝑎/(𝑚2+𝑛2)+𝑝/(1𝑒(𝑚2+𝑛2)𝜏), it corresponds to the unique positive periodic solution ̃𝑢2(𝑡) of system (2.4) with the initial value ̃𝑢2(0+)=𝑎/(𝑚2+𝑛2) + 𝑝/(1𝑒(𝑚_2+𝑛_2)𝜏). The fixed point 𝑢2 of map (2.7) implies that there is a corresponding cycle of period 𝜏 in 𝑢2(𝑡), that is, ̃𝑢2(𝑡)=(𝑎/𝑚2+𝑛2)+(𝑝𝑒(𝑚2+𝑛2)(𝑡𝑛𝜏)/(1𝑒(𝑚2+𝑛2)𝜏)),𝑡(𝑛𝜏,(𝑛+1)𝜏],𝑛𝑍+. From (2.7) we obtain 𝑢2𝑛𝜏+=𝑢20+𝑒𝑛(𝑚2+𝑛2)𝜏+𝑎𝑝+𝑚2+𝑛21𝑒(𝑚2+𝑛2)𝜏1𝑒𝑛(𝑚2+𝑛2)𝜏1𝑒(𝑚2+𝑛2)𝜏,(2.8) thus, 𝑢2(𝑛𝜏+)𝑢2 as 𝑡, so ̃𝑢2(𝑡) is globally asymptotically stable. Thus, we have 𝑢2(𝑡) = (𝑢2(0+)̃𝑢2(0+))𝑒(𝑚2+𝑛2)𝑡+̃𝑢2(𝑡).

Consequently, 𝑢2(𝑡)̃𝑢2(𝑡) as 𝑡, that is, |𝑢2(𝑡)̃𝑢2(𝑡)|0 as 𝑡.

If 𝑎=0, the system (2.4) has a unique positive periodic solution ̃𝑢2(𝑡)=(𝑝𝑒(𝑚2+𝑛2)(𝑡𝑛𝜏)/(1𝑒(𝑚2+𝑛2)𝜏) with initial condition ̃𝑢2(0+)=𝑝/(1𝑒(𝑚2+𝑛2)𝜏) and ̃𝑢2(𝑡) is globally asymptotically stable. This completes the proof.

Therefore, system (1.2) has a pest-eradication periodic solution (0,̃𝑥2(𝑡),0), where ̃𝑥2(𝑡)=𝑝𝑒((𝑚2+𝑛2)(𝑡𝑛𝜏))/(1𝑒(𝑚2+𝑛2)𝜏).

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 𝑑𝑥𝑑𝑡=𝐴(𝑡)𝑥,𝑡𝜏𝑘𝑥𝑡,𝑡𝑅,+=𝑥(𝑡)+𝐵𝑘𝑥(𝑡),𝑡=𝜏𝑘,𝑘𝑍+.(2.9) Then, we introduce the following conditions:(𝐻1)𝐴()𝑃𝐶(𝑅,𝐶𝑛×𝑛) and 𝐴(𝑡+𝜏)=𝐴(𝑡)(𝑡𝑅), where 𝑃𝐶(𝑅,𝐶𝑛×𝑛) is the set of all piecewise continuous matrix functions which is left continuous at 𝑡=𝜏𝑘, and 𝐶𝑛×𝑛 is the set of all 𝑛×𝑛 matrices.(𝐻2)𝐵𝑘𝐶𝑛×𝑛,det(𝐸+𝐵𝑘)0,𝜏𝑘<𝜏𝑘+1(𝑘𝑍+).(𝐻3) There exist a 𝑍+, such that 𝐵𝑘+=𝐵𝑘, 𝜏𝑘+=𝜏𝑘+𝜏(𝑘𝑍+).

Let Φ(𝑡) be the fundamental matrix of (2.9), then there exists a unique nonsingular matrix 𝑀𝐶𝑛×𝑛 such that Φ(𝑡+𝜏)=Φ(𝑡)𝑀.(2.10)

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 𝜆1,𝜆2,,𝜆𝑛 of the monodromy matrices are called the Floquet multipliers of (2.9).

Lemma 2.4 (Floquet theory [16]). Let conditions (𝐻1-𝐻3) hold. Then the linear 𝜏 periodic impulsive system (2.9) is as follows:(a)stable if and only if all multipliers 𝜆𝑖 (𝑖=1,2,,𝑛) of equation (2.9) satisfy the inequality |𝜆𝑖|1,(b)asymptotically stable if and only if all multipliers 𝜆𝑖 (𝑖=1,2,,𝑛) of equation (2.9) satisfy the inequality |𝜆𝑖|<1,(c)unstable if |𝜆𝑖|>1 for some 𝑖=1,2,,𝑛.

3. Stability of the Pest-Eradication Periodic Solution

In this section, we study the stability of the pest-eradication periodic solution (0,̃𝑥2(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,̃𝑥2(𝑡),0) of the system (1.2) is globally asymptotically stable provided that inequality 𝑝>𝑟𝑚1𝑛1𝑚2+𝑛2𝜏𝛽(3.1) holds.

Proof. The local stability of a periodic solution (0,̃𝑥2(𝑡),0) of system (1.2) may be determined by considering the behavior of small-amplitude perturbations (𝑦1(𝑡),𝑦2(𝑡),𝑦3(𝑡)) of the solution.
Define 𝑥1(𝑡)=𝑦1(𝑡),𝑥2(𝑡)=̃𝑥2(𝑡)+𝑦2(𝑡),𝑥3(𝑡)=𝑦3(𝑡),(3.2) where 𝑦1(𝑡),𝑦2(𝑡),𝑦3(𝑡) are small perturbations.
Linearizing the system (1.2), we have the following linear 𝜏 periodic impulsive system: 𝑑𝑦1𝑑𝑡=𝑟𝑦1𝑚1𝑦1𝑛1𝑦1𝛽̃𝑥2𝑦1𝑑𝑦2𝑑𝑡=𝛽̃𝑥2𝑦1𝑚2𝑦2𝑛2𝑦2𝑡𝑛𝜏,𝑛𝑍+,𝑑𝑦3𝑑𝑡=𝑛1𝑦1𝑚3𝑦3𝑛3𝑦3𝑦1𝑡+=𝑦1𝑦(𝑡)2𝑡+=𝑦2(𝑡)𝑡=𝑛𝜏,𝑛𝑍+,𝑦3𝑡+=𝑦3(𝑡).(3.3)
Let Φ(𝑡) be the fundamental matrix of (3.3). Then we have 𝑦1𝑦(𝑡)2(𝑦𝑡)3𝑦(𝑡)=Φ(𝑡)1𝑦(0)2(𝑦0)3(0),(3.4) where Φ(𝑡) must satisfy the following equation: 𝑑Φ(𝑡)=𝑑𝑡𝑟𝑚1𝑛1𝛽̃𝑥200𝛽̃𝑥2𝑚2𝑛20𝑛10𝑚3𝑛3Φ(𝑡),(3.5) and initial condition Φ(𝑡)=𝐼,(3.6) where 𝐼 is the identity matrix.
The solution of (3.5) is Φ(𝑡)=exp𝑡0𝑟𝑚1𝑛1𝛽̃𝑥2𝑚(𝑠)𝑑𝑠00exp2+𝑛2𝜏0𝑚exp3+𝑛3𝜏.(3.7)
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 𝑦1𝑛𝜏+𝑦2𝑛𝜏+𝑦3𝑛𝜏+=𝑦1000100011𝑦(𝑛𝜏)2(𝑦𝑛𝜏)3(𝑛𝜏).(3.8)
Hence, if absolute values of all eigenvalues of 𝑀=100010001Φ(𝜏)=Φ(𝜏)(3.9) are less than one, the 𝜏 periodic solution is locally stable. Then the eigenvalues of 𝑀 are the following: 𝜆1=exp𝑡0𝑟𝑚1𝑛1𝛽̃𝑥2(𝜆𝑠)𝑑𝑠2𝑚=exp2+𝑛2𝜏𝜆<13𝑚=exp3+𝑛3𝜏<1.(3.10) From (3.10), one can see that |𝜆1|<1 if and only if condition (3.1) holds true. According to Lemma 2.4, the pest-eradication periodic solution (0,̃𝑥2(𝑡),0) is locally asymptotically stable.
In the following, we prove the global attractivity. Choose sufficiently small 𝜀>0 such that 𝛿=exp𝜏0𝑟𝑚1𝑛1𝛽̃𝑥2(𝑡)𝜀𝑑𝑡<1.(3.11) From the second equation of system (1.2), noting that ((𝑑𝑥2)/𝑑𝑡)(𝑚2+𝑛2)𝑥2, we consider the following impulsive differential equation 𝑑𝑢2𝑚𝑑𝑡=2+𝑛2𝑢2,𝑡𝑛𝜏Δ𝑢2𝑢(𝑡)=𝑝,𝑡=𝑛𝜏,20+=𝑥20+.(3.12) From Lemma 2.3, system (3.12) has a globally asymptotically stable positive periodic solution ̃𝑢2(𝑡)=𝑝𝑒(𝑚2+𝑛2)(𝑡𝑛𝜏)1𝑒(𝑚2+𝑛2)𝜏=̃𝑥2],𝑡(𝑛𝜏,(𝑛+1)𝜏,𝑛𝑍+.(3.13) So by Lemma 2.2, we get 𝑥2(𝑡)𝑢2(𝑡)̃𝑥2(𝑡)𝜀,(3.14) for all 𝑡 large enough.
From system, (1.2) and (3.14), we obtain that 𝑑𝑥1𝑥𝑑𝑡𝑟11𝐾𝑥1𝑚1𝑥1𝑛1𝑥1𝛽𝑥1̃𝑥2𝜀,𝑡𝑛𝜏,Δ𝑥1(𝑡)=0,𝑡=𝑛𝜏.(3.15) Integrating (3.15) on (𝑛𝜏,(𝑛+1)𝜏], we get 𝑥1(𝑛+1)𝜏+𝑥1(𝑛𝜏)exp𝜏0𝑟𝑚1𝑛1𝛽̃𝑥2(𝑡)𝜀𝑑𝑡=𝑥1(𝑛𝜏)𝛿.(3.16) Thus, 𝑥1(𝑛𝜏)𝑥1(0+)𝛿𝑛 and 𝑥1(𝑛𝜏)0 as 𝑛. Therefore, 𝑥1(𝑡)0 as 𝑛, since 0<𝑥1(𝑡)𝑥1(𝑛𝜏) for 𝑡(𝑛𝜏,(𝑛+1)𝜏],𝑛𝑍+.
Next, we prove that 𝑥2(𝑡)̃𝑥2(𝑡) as 𝑡. For 0<𝜀1𝑚2+𝑛2, there must exist a 𝑡0>0 such that 0<𝑥1(𝑡)𝜀1 for all 𝑡𝑡0. Without loss of generality, we may assume that 0<𝑥1(𝑡)𝜀1 for all 𝑡0, from system (1.2) we have 𝑑𝑥2𝜀𝑑𝑡1𝑚2+𝑛2𝑥2.(3.17)
Then, we have 𝑥2(𝑡)𝑣2(𝑡), while 𝑣2(𝑡)is the solution of 𝑑𝑣2=𝜀𝑑𝑡1𝑚2+𝑛2𝑣2,𝑡𝑛𝜏,Δ𝑣2𝑣(𝑡)=𝑝,𝑡=𝑛𝜏,20+=𝑥20+.(3.18) By Lemma 2.3, system (3.18) has a positive periodic solution ̃𝑣2(𝑡)=𝑝𝑒[𝜀1(𝑚2+𝑛2)](𝑡𝑛𝜏)1𝑒[𝜀1(𝑚2+𝑛2)]𝜏],𝑡(𝑛𝜏,(𝑛+1)𝜏,𝑛𝑍+.(3.19) Therefore, for any 𝜀2>0, there exists a 𝑡1,𝑡>𝑡1 such that 𝑥2(𝑡)𝑣2̃𝑣(𝑡)<2(𝑡)+𝜀2.(3.20) Combining (3.14) and (3.20), we obtain ̃𝑥2(𝑡)𝜀𝑥2̃𝑣(𝑡)<2(𝑡)+𝜀2,(3.21) for 𝑡 large enough. Let 𝜀1,𝜀20, we get ̃𝑣2(𝑡)̃𝑥2(𝑡), then 𝑥2(𝑡)̃𝑥2(𝑡) as 𝑡.
Assuming that 0<𝑥1(𝑡)𝜀1 for all 𝑡0, from system (1.2) we have 𝑑𝑥3𝑑𝑡𝑛1𝜀1𝑚3+𝑛3𝑥3.(3.22) Then, we have 𝑥3(𝑡)𝑣3(𝑡), while 𝑣3(𝑡) is the solution of the following system: 𝑑𝑣3𝑑𝑡=𝑛1𝜀1𝑚3𝑣3𝑛3𝑣3,𝑡𝑛𝜏,Δ𝑣3𝑣(𝑡)=0,𝑡=𝑛𝜏,30+=𝑣300.(3.23) By Lemma 2.3, system (3.23) has a positive solution ̃𝑣3=𝑛1𝜀1𝑚3+𝑛3.(3.24) Thus, for any 𝜀3>0, there exists a 𝑡1,𝑡>𝑡1 such that 𝑥3(𝑡)𝑣3̃𝑣(𝑡)<3(𝑡)+𝜀3.(3.25) Let 𝜀1,𝜀30, we get ̃𝑣3(𝑡)0, then 𝑥3(𝑡)0 as 𝑡. This completes the proof.

4. 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: 𝑛1=0.1, 𝑛2=0.1, 𝑛3=0.02439, 𝑚1=0.03566, 𝑚2=0.03566, 𝑚3=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.

fig1
Figure 1: Evolution of the egg (a), parasitized egg (b), larvae populations (c), and phase portraits (d) of system (1.2) 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 𝑥3 takes on values more than the EIL for this pest 𝑥EIL=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,̃𝑥2(𝑡),0) of the system (1.2) is globally asymptotically stable if the condition (3.1) holds 𝑝>𝑝min=𝑟𝑚1𝑛1𝑚2+𝑛2𝜏𝛽.(4.1) Choosing 𝜏=70 days from (4.1), we derive that when 𝑝>𝑝min=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 𝑥ET=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.

fig2
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.

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.

fig3
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.

Applying the control strategy 𝑝=1000, we can see that the number of larvae individuals exceed 𝑥EIL at some time (see Figure 4).

fig4
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.

5. 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,̃𝑥2(𝑡),0) of the system (1.2) if the number of the parasitoids in periodic releases is greater than some critical value 𝑝min.

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 𝜏<𝜏max=𝛽𝑝𝑟𝑚1𝑛1𝑚2+𝑛2.(5.1) From (5.1) we can conclude that there exists a globally asymptotically stable pest-eradication periodic solution (0,̃𝑥2(𝑡),0) of the system (1.2) if the impulsive period is less than some critical value 𝜏max.

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, the authors considered integrated pest management strategies based on discrete-time host-parasitoid 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 𝑟𝑚1𝑛1<𝛽𝑝𝜏𝑚2+𝑛2,(5.2) 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.

Acknowledgments

The authors would like to thank the referees for their careful reading of the original paper and their valuable comments and suggestions that improved the presentation of this paper. The first author thanks Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Pesquisas (CNPq) for the financial supports on this research.

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