Impulsive Biological Pest Control Strategies of the Sugarcane Borer

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 𝑑𝑥₁𝑑𝑡=𝑟1−𝑥𝐾¹𝑥¹𝑑𝑥₂−𝑚₁𝑥₁−𝑛₁𝑥₁−𝛽𝑥₁𝑥₂,𝑑𝑡=𝛽𝑥¹𝑑𝑥₃𝑥₂−𝑚₂𝑥₂−𝑛₂𝑥₂,𝑑𝑡=𝑛¹𝑥₁−𝑚₃𝑥₃−𝑛₃𝑥₃,(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 [8–15]. 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 𝑑𝑥₁𝑑𝑡=𝑟1−𝑥𝐾¹𝑥¹−𝑚₁𝑥₁−𝑛₁𝑥₁−𝛽𝑥₁𝑥₂𝑑𝑥₂𝑑𝑡=𝛽𝑥¹𝑑𝑥₃𝑥₂−𝑚₂𝑥₂−𝑛₂𝑥₂𝑡≠𝑛𝜏,𝑛∈𝑍₊,𝑑𝑡=𝑛¹𝑥₁−𝑚₃𝑥₃−𝑛₃𝑥₃ΔΔ𝑥𝑥₁₂(𝑡)=0Δ𝑥₃(𝑡)=𝑝𝑡=𝑛𝜏,𝑛∈𝑍₊,(𝑡)=0,(1.2) where 𝑝 is the release amount of the parasitized eggs at 𝑡=𝑛𝜏,𝑛∈𝑍₊,𝑍₊={0,1,2,…,},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.

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⁺={𝑥∈𝑅³∶𝑥>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={𝑉∶𝑅₊×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. 𝑉∈𝑉₀, 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)=lim_ℎ→0sup1ℎ[𝑉(𝑡+ℎ,𝑥+ℎ𝑓(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 𝑉∈𝑉₀, assume that 𝑉𝐷𝑡⁺,𝑥𝑉(𝑡𝑡₊,𝑥)≤𝑔(𝑡,𝑉(𝑡,𝑥)),𝑡≠𝑛𝜏,≤𝜓_𝑛(𝑉(𝑡,𝑥(𝑡))),𝑡=𝑛𝜏,(2.2) where 𝑔∶𝑅₊×𝑅³⁺↦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 𝑢𝑢̇(𝑡𝑡,𝑥)=𝑔(𝑡,𝑢(𝑡)),𝑡≠𝑛𝜏,₊=𝑢𝜓_𝑛0(𝑢(𝑡)),𝑡=𝑛𝜏,₊=𝑢₀(2.3) existing on [0,∞). Then 𝑉(0⁺,𝑥₀)≤𝑢₀ implies that 𝑉(𝑡,𝑥(𝑡))≤𝑅(𝑡),𝑡≥0, 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 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: 𝑑𝑢₂𝑑𝑡=𝑎−𝑚²Δ𝑢₂𝑢₂−𝑛₂𝑢₂,𝑡≠𝑛𝜏,𝑢(𝑡)=𝑏,𝑡=𝑛𝜏₂0⁺=𝑢₂₀≥0.(2.4)

Lemma 2.3. System (2.4) has a unique positive periodic solution 𝑢₂̃(𝑡) with period 𝜏 and for every solution 𝑢₂(𝑡) of (2.4) |𝑢₂(𝑡)−𝑢₂̃(𝑡)(𝑡)|→0 as 𝑡→∞, where 𝑢₂̃(𝑡)=𝑎𝑚₂+𝑛₂+𝑏𝑒^{−(𝑚2+𝑛2)(𝑡−𝑛𝜏)}1−𝑒^{−(𝑚2+𝑛2)𝜏}𝑢₂̃,𝑡∈(𝑛𝜏,(𝑛+1)𝜏],𝑛∈𝑍₊,0⁺=𝑚₂𝑎+𝑛₂+𝑝1−𝑒^{−(𝑚2+𝑛2)𝜏}.(2.5)

Proof. Integrating and solving the first equation of (2.4) between pulses, we get 𝑢₂(𝑡)=𝑎𝑚₂+𝑛₂+𝑢₂𝑛𝜏⁺𝑒^{−(𝑚2+𝑛2)(𝑡−𝑛𝜏)},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+𝑢₂𝑛𝜏⁺−𝑎𝑚₂+𝑛₂𝑒^{−(𝑚2+𝑛2)𝜏}+𝑝,𝑡∈(𝑛𝜏,(𝑛+1)𝜏].(2.7) Equation (2.7) has a unique fixed point 𝑢^∗2=𝑎/(𝑚₂+𝑛₂)+𝑝/(1−𝑒^{−(𝑚2+𝑛2)𝜏}), it corresponds to the unique positive periodic solution 𝑢₂̃(𝑡) of system (2.4) with the initial value 𝑢₂̃(0⁺)=𝑎/(𝑚₂+𝑛₂) + 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 𝑢₂(𝑡), that is, 𝑢₂̃(𝑡)=(𝑎/𝑚₂+𝑛₂)+(𝑝𝑒^{−(𝑚2+𝑛2)(𝑡−𝑛𝜏)}/(1−𝑒^{−(𝑚2+𝑛2)𝜏})),𝑡∈(𝑛𝜏,(𝑛+1)𝜏],𝑛∈𝑍₊. From (2.7) we obtain 𝑢₂𝑛𝜏⁺=𝑢₂0⁺𝑒^{−𝑛(𝑚2+𝑛2)𝜏}+𝑝+𝑎𝑚₂+𝑛₂1−𝑒^{−(𝑚2+𝑛2)𝜏}1−𝑒^{−𝑛(𝑚2+𝑛2)𝜏}1−𝑒^{−(𝑚2+𝑛2)𝜏},(2.8) thus, 𝑢₂(𝑛𝜏⁺)→𝑢^∗2 as 𝑡→∞, so 𝑢₂̃(𝑡) is globally asymptotically stable. Thus, we have 𝑢₂(𝑡) = (𝑢₂(0⁺)−𝑢₂̃(0⁺))𝑒^{−(𝑚2+𝑛2)𝑡}+𝑢₂̃(𝑡).

Consequently, 𝑢₂(𝑡)→𝑢₂̃(𝑡) as 𝑡→∞, that is, |𝑢₂(𝑡)−𝑢₂̃(𝑡)|→0 as 𝑡→∞.

If 𝑎=0, the system (2.4) has a unique positive periodic solution 𝑢₂̃(𝑡)=(𝑝𝑒^{−(𝑚2+𝑛2)(𝑡−𝑛𝜏)}/(1−𝑒^{−(𝑚2+𝑛2)𝜏}) with initial condition 𝑢₂̃(0⁺)=𝑝/(1−𝑒^{−(𝑚2+𝑛2)𝜏}) and 𝑢₂̃(𝑡) is globally asymptotically stable. This completes the proof.

Therefore, system (1.2) has a pest-eradication periodic solution (0,𝑥̃₂(𝑡),0), where 𝑥̃₂(𝑡)=𝑝𝑒^{(−(𝑚2+𝑛2)(𝑡−𝑛𝜏))}/(1−𝑒^{−(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,𝑡≠𝜏^𝑘𝑥𝑡,𝑡∈𝑅,₊=𝑥(𝑡)+𝐵_𝑘𝑥(𝑡),𝑡=𝜏_𝑘,𝑘∈𝑍₊.(2.9) Then, we introduce the following conditions:(𝐻₁)𝐴(⋅)∈𝑃𝐶(𝑅,𝐶^𝑛×𝑛) and 𝐴(𝑡+𝜏)=𝐴(𝑡)(𝑡∈𝑅), 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.(𝐻₂)𝐵_𝑘∈𝐶^𝑛×𝑛,det(𝐸+𝐵_𝑘)≠0,𝜏_𝑘<𝜏_𝑘+1(𝑘∈𝑍₊).(𝐻₃) There exist a ℎ∈𝑍₊, such that 𝐵_𝑘+ℎ=𝐵_𝑘, 𝜏_𝑘+ℎ=𝜏_𝑘+𝜏(𝑘∈𝑍₊).

Let Φ(𝑡) be the fundamental matrix of (2.9), then there exists a unique nonsingular matrix 𝑀∈𝐶^𝑛×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,…,𝜆_𝑛 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 𝜆_𝑖 (𝑖=1,2,…,𝑛) of equation (2.9) satisfy the inequality |𝜆_𝑖|≤1,(b)asymptotically stable if and only if all multipliers 𝜆_𝑖 (𝑖=1,2,…,𝑛) of equation (2.9) satisfy the inequality |𝜆_𝑖|<1,(c)unstable if |𝜆_𝑖|>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,𝑥̃₂(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,𝑥̃₂(𝑡),0) of the system (1.2) is globally asymptotically stable provided that inequality 𝑝>𝑟−𝑚¹−𝑛₁𝑚₂+𝑛₂𝜏𝛽(3.1) holds.

Proof. The local stability of a periodic solution (0,𝑥̃₂(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)=𝑥̃₂(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=𝑟𝑦¹−𝑚₁𝑦₁−𝑛₁𝑦₁−𝛽𝑥̃₂𝑦₁𝑑𝑦₂𝑑𝑡=𝛽𝑥̃²𝑑𝑦₃𝑦₁−𝑚₂𝑦₂−𝑛₂𝑦₂𝑡≠𝑛𝜏,𝑛∈𝑍₊,𝑑𝑡=𝑛¹𝑦₁−𝑚₃𝑦₃−𝑛₃𝑦₃𝑦𝑦₁𝑡⁺=𝑦₁(𝑡)₂𝑦₃𝑡⁺=𝑦₂(𝑡)𝑡=𝑛𝜏,𝑛∈𝑍₊,𝑡⁺=𝑦₃(𝑡).(3.3)
Let Φ(𝑡) be the fundamental matrix of (3.3). Then we have ⎡⎢⎢⎢⎢⎣𝑦𝑦₁(𝑡)₂𝑦(𝑡)₃(𝑡)⎤⎥⎥⎥⎥⎦=Φ(𝑡)⎡⎢⎢⎢⎢⎣𝑦¹𝑦(0)₂𝑦(0)₃(0)⎤⎥⎥⎥⎥⎦,(3.4) where Φ(𝑡) must satisfy the following equation: 𝑑Φ(𝑡)𝑑𝑡=⎡⎢⎢⎢⎢⎣𝑟−𝑚¹−𝛽𝑛𝑥₁̃₂−𝛽𝑥̃₂00𝑛₁−𝑚₂−𝑛₂00−𝑚₃−𝑛₃⎤⎥⎥⎥⎥⎦Φ(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)=⎡⎢⎢⎢⎢⎣exp∫^𝑡0𝑟−𝑚₁−∗𝑛₁−𝛽𝑥̃₂(𝑠)𝑑𝑠exp−𝑚0₂0∗∗+𝑛₂𝜏exp−𝑚0₃+𝑛₃𝜏⎤⎥⎥⎥⎥⎦.(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 ⎡⎢⎢⎢⎢⎣𝑦𝑦₁₂𝑛𝜏⁺𝑦₃𝑛𝜏⁺𝑛𝜏⁺⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣100010001⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑦¹𝑦(𝑛𝜏)₂𝑦(𝑛𝜏)₃(𝑛𝜏)⎤⎥⎥⎥⎥⎦.(3.8)
Hence, if absolute values of all eigenvalues of 𝑀=⎡⎢⎢⎢⎢⎣100010001⎤⎥⎥⎥⎥⎦Φ(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=e𝜆xp^𝑡0𝑟−𝑚₁−𝑛₁−𝛽𝑥̃₂(𝑠)𝑑𝑠₂𝜆=exp−𝑚₂+𝑛₂𝜏<1₃=exp−𝑚₃+𝑛₃𝜏<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,𝑥̃₂(𝑡),0) is locally asymptotically stable.
In the following, we prove the global attractivity. Choose sufficiently small 𝜀>0 such that 𝛿=exp^𝜏0𝑟−𝑚₁−𝑛₁−𝛽𝑥̃₂(𝑡)−𝜀𝑑𝑡<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 𝑑𝑢₂𝑑𝑡=−𝑚²Δ𝑢₂+𝑛₂𝑢₂,𝑡≠𝑛𝜏𝑢(𝑡)=𝑝,𝑡=𝑛𝜏,₂0⁺=𝑥₂0⁺.(3.12) From Lemma 2.3, system (3.12) has a globally asymptotically stable positive periodic solution 𝑢₂̃(𝑡)=𝑝𝑒^{−(𝑚2+𝑛2)(𝑡−𝑛𝜏)}1−𝑒^{−(𝑚2+𝑛2)𝜏}=𝑥̃₂,𝑡∈(𝑛𝜏,(𝑛+1)𝜏],𝑛∈𝑍₊.(3.13) So by Lemma 2.2, we get 𝑥₂(𝑡)≥𝑢₂(𝑡)≥𝑥̃₂(𝑡)−𝜀,(3.14) for all 𝑡 large enough.
From system, (1.2) and (3.14), we obtain that 𝑑𝑥₁𝑑𝑡≤𝑟1−𝑥𝐾¹𝑥¹Δ−𝑥₁𝑚₁𝑥₁−𝑛₁𝑥₁−𝛽𝑥₁𝑥̃₂−𝜀,𝑡≠𝑛𝜏,(𝑡)=0,𝑡=𝑛𝜏.(3.15) Integrating (3.15) on (𝑛𝜏,(𝑛+1)𝜏], we get 𝑥₁(𝑛+1)𝜏⁺≤𝑥₁(𝑛𝜏)exp^𝜏0𝑟−𝑚₁−𝑛₁−𝛽𝑥̃₂(𝑡)−𝜀𝑑𝑡=𝑥₁(𝑛𝜏)𝛿.(3.16) Thus, 𝑥₁(𝑛𝜏)≤𝑥₁(0⁺)𝛿^𝑛 and 𝑥₁(𝑛𝜏)→0 as 𝑛→∞. Therefore, 𝑥₁(𝑡)→0 as 𝑛→∞, since 0<𝑥₁(𝑡)≤𝑥₁(𝑛𝜏) for 𝑡∈(𝑛𝜏,(𝑛+1)𝜏],𝑛∈𝑍₊.
Next, we prove that 𝑥₂(𝑡)→𝑥̃₂(𝑡) as 𝑡→∞. For 0<𝜀₁≤𝑚₂+𝑛₂, there must exist a 𝑡₀>0 such that 0<𝑥₁(𝑡)≤𝜀₁ for all 𝑡≥𝑡₀. Without loss of generality, we may assume that 0<𝑥₁(𝑡)≤𝜀₁ for all 𝑡≥0, from system (1.2) we have 𝑑𝑥₂𝑑𝑡≤𝜀¹−𝑚₂+𝑛₂𝑥₂.(3.17)
Then, we have 𝑥₂(𝑡)≤𝑣₂(𝑡), while 𝑣₂(𝑡)is the solution of 𝑑𝑣₂𝑑𝑡=𝜀¹Δ𝑣₂−𝑚₂+𝑛₂𝑣₂,𝑡≠𝑛𝜏,𝑣(𝑡)=𝑝,𝑡=𝑛𝜏,₂0⁺=𝑥₂0⁺.(3.18) By Lemma 2.3, system (3.18) has a positive periodic solution 𝑣₂̃(𝑡)=𝑝𝑒^{[𝜀1−(𝑚2+𝑛2)](𝑡−𝑛𝜏)}1−𝑒^{[𝜀1−(𝑚2+𝑛2)]𝜏},𝑡∈(𝑛𝜏,(𝑛+1)𝜏],𝑛∈𝑍₊.(3.19) Therefore, for any 𝜀₂>0, there exists a 𝑡₁,𝑡>𝑡₁ such that 𝑥₂(𝑡)≤𝑣₂(𝑡)<𝑣₂̃(𝑡)+𝜀₂.(3.20) Combining (3.14) and (3.20), we obtain 𝑥̃₂(𝑡)−𝜀≤𝑥₂(𝑡)<𝑣₂̃(𝑡)+𝜀₂,(3.21) for 𝑡 large enough. Let 𝜀₁,𝜀₂→0, we get 𝑣₂̃(𝑡)→𝑥̃₂(𝑡), then 𝑥₂(𝑡)→𝑥̃₂(𝑡) as 𝑡→∞.
Assuming that 0<𝑥₁(𝑡)≤𝜀₁ for all 𝑡≥0, from system (1.2) we have 𝑑𝑥₃𝑑𝑡≤𝑛¹𝜀₁−𝑚₃+𝑛₃𝑥₃.(3.22) Then, we have 𝑥₃(𝑡)≤𝑣₃(𝑡), while 𝑣₃(𝑡) is the solution of the following system: 𝑑𝑣₃𝑑𝑡=𝑛¹Δ𝑣𝜀₁₃−𝑚₃𝑣₃−𝑛₃𝑣₃,𝑡≠𝑛𝜏,𝑣(𝑡)=0,𝑡=𝑛𝜏,₃0⁺=𝑣₃₀≥0.(3.23) By Lemma 2.3, system (3.23) has a positive solution 𝑣₃̃=𝑛¹𝜀₁𝑚₃+𝑛₃.(3.24) Thus, for any 𝜀₃>0, there exists a 𝑡₁,𝑡>𝑡₁ such that 𝑥₃(𝑡)≤𝑣₃(𝑡)<𝑣₃̃(𝑡)+𝜀₃.(3.25) Let 𝜀₁,𝜀₃→0, we get 𝑣₃̃(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.

(a)

(b)

(c)

(d)

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 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,𝑥̃₂(𝑡),0) of the system (1.2) is globally asymptotically stable if the condition (3.1) holds 𝑝>𝑝_min=𝑟−𝑚¹−𝑛₁𝑚₂+𝑛₂𝜏𝛽.(4.1) 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.

(a)

(b)

(c)

(d)

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.

(a)

(b)

(c)

(d)

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 𝑥_EIL at some time (see Figure 4).

(a)

(b)

(c)

(d)

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,𝑥̃₂(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 𝜏<𝜏_max=𝛽𝑝𝑟−𝑚₁−𝑛₁𝑚₂+𝑛₂.(5.1) From (5.1) we can conclude that there exists a globally asymptotically stable pest-eradication periodic solution (0,𝑥̃₂(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, 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 𝑟−𝑚₁−𝑛₁<𝛽𝑝𝜏𝑚₂+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.