Modelling and analysis of periodic impulsive releases of the Nilaparvata lugens infected with wStri-Wolbachia

In this paper, we formulate a population suppression model and a population replacement model with periodic impulsive releases of Nilaparvata lugens infected with wStri. The conditions for the stability of wild- $ N.\,lugens $ N.lugens-eradication periodic solution of two systems are obtained by applying the Floquet theorem and comparison theorem. And the sufficient conditions for the persistence in the mean of wild $ N.\,lugens $ N.lugens are also given. In addition, the sufficient conditions for the extinction and persistence of the wild $ N.\,lugens $ N.lugens in the subsystem without wLug are also obtained. Finally, we give numerical analysis which shows that increasing the release amount or decreasing the release period are beneficial for controlling the wild $ N.\,lugens $ N.lugens, and the efficiency of population replacement strategy in controlling wild populations is higher than that of population suppression strategy under the same release conditions.


Introduction
Nilaparvata lugens (N.lugens) is a monophagous pest, which can only feed and reproduce on rice and common wild rice.It is the most destructive pest on rice in many Asian countries by sucking rice phloem sap and transmitting rice ragged stunt virus (RRSV) [1,2].Few feasible control strategies are obtainable because of the evolution of high levels of insecticide resistance, and new environmentally friendly methods are urgently needed [3].In recent studies, disease control methods based on artificial Wolbachia infection to inhibit mosquito vector-borne pathogens have been applied [4,5].Scientists are currently working on similar methods for controlling agricultural pests [6].
Nilaparvata lugens can be naturally infected by Wolbachia strain wLug, which lacks the ability to induce cytoplasmic incompatibility (CI) [7].Many experimental results show that the N. lugens infected with wLug has stronger reproductive ability than the uninfected N. lugens , and show imperfect maternal transmission characteristics [6,8,9].Fortunately, Gong et al. [6] successfully developed a stable artificial Wolbachia infection of N. lugens by introducing the Wolbachia strain wStri from L. striatellus host into N. lugens .The results [6] showed that N. lugens infected with wStri maintained perfect maternal transmission and induced moderately high levels of CI.When the wStri-infected males mated with either uninfected or the wLug-infected females, the mean hatch rates per female were 32.5% and 27.7%, respectively.The results of Gong et al. [6] lay a foundation for future experiments on paddy fields.
In order to theoretically study the interaction mechanism between N. lugens infected with wStri and wild N. lugens population, Liu and Zhou [10] proposed and studied a Wolbachia spreading dynamics model in N. lugens with two strains, and obtained sufficient conditions for the N. lugens infected with wStri to invade wild N. lugens successfully.But the authors did not address when and how many N. lugens infected with wStri is released to suppress or replace wild N. lugens , so studying the release of wStri-infected N. lugens to control wild N. lugens for future field trials with wStri is biologically significant.Currently, there are few models to study how to release the N. lugens infected Wolbachia, but many mosquito population dynamics models have been proposed and studied [11][12][13][14][15][16][17][18][19][20][21][22][23][24].Cai et al. [11] considered three strategies for continuous release of sterile mosquitoes: constant release rate, release rate proportional to the wild mosquitoes and proportional release rate with saturation, and developed corresponding continuous mathematical models to study the influences of the three release strategies on the interactive dynamics of mosquitoes.In practice, continuous release of sterile mosquitoes is difficult to realize in practical applications.Huang et al. [16] formulated and investigated two mathematical models with impulsive releases of sterile mosquitoes.The first model considered the periodic pulse release of sterile mosquitoes strategy and obtained a sufficient condition for the stability of the wild mosquito-eradication periodic solution.The second model considered the state-feedback pulse release strategy and proved the existence of the first-order periodic solution.In studies [17][18][19][20], the authors adopted a new modelling idea that only those sexually active sterile mosquitoes were considered in the modelling process.Yu et al. [19] developed and analyzed a population suppression model considering that the release period was longer than the sexual life of sterile mosquitoes.They obtained sufficient conditions for the global asymptotic stability of positive periodic solutions.Later, Zheng et al. [17] considered the following situation that the release period was shorter than the sexual lifespan of sterile male mosquitoes.Li and Ai [18] incorporated the maturation process of mosquito larvae to adults into their model and employed time delay to describe the maturation period of the larvae, the results showed that the delay affects the control of wild mosquitoes.In addition, some scholars considered the interaction between mosquitoes and disease transmission (such as Dengue fever, Zika, etc.), and established some mathematical models to control mosquito-borne diseases [23,24].For example, Taghikahani et al. [23] formulated a new two-sex mathematical model for the population ecology of dengue fever and Wolbachia-infected mosquitoes, and used it to evaluate the impact of periodic release the Wolbachian-infected mosquitoes on the population-level.However, we notice from studies [11][12][13][14][15][16][17][18][19][20][21][22][23][24] that most models were established around the release of male mosquitoes infected with Wolbachia (i.e.population suppression strategy), and the research on the impulsive release strategy of female mosquitoes infected with Wolbachia mostly adopts numerical simulation method.
All the models mentioned above about the release strategies of mosquitoes infected with Wolbachia also provide good help for the release of N. lugens infected with wStri.
Compared with the transmission characteristics of mosquitoes infected with Wolbachia, the N. lugens infected with Wolbachia has many differences.For example, the cytoplasmic incompatibility induced by the mating of male N. lugens infected with wStri with wild uninfected female N. lugens or wild female N. lugens infected with wlug is incomplete [6].To determine when is the best time to release N. lugens infected with wStri and the number of N. lugens infected with wStri per release.In 2023, Liu et al. [25] established and discussed two semi-continuous models with state-feedback impulsive releases of N. lugens infected with wStri.But the authors only considered the scenario where all wild N. lugens populations were uninfected N. lugens , and there are few studies of wild N. lugens including both the wild N. lugens uninfected and infected with wLug.In this paper, we adapt the modelling idea in [10] and consider the periodic impulsive release of male or female N. lugens infected with wStri into the field.And then according to the transmission characteristics of two Wolbachia strains in N. lugens , we establish two impulsive mathematical models with periodic impulsive release of N. lugens infected with wStri: population suppression model and population replacement model.We theoretically discuss the stability of the wild-N.lugens -eradication periodic solution for both models and the corresponding numerical analyses are also carried out.Furthermore, the sufficient conditions of persistence in the mean of the wild N. lugens are also established.Finally, we compare the degree of control of two periodic impulsive release strategies on the wild N. lugens within a short time.
This paper is organized as follows: In Section 2, population suppression and population replacement models with periodic impulsive release are proposed.In Section 3, we study the stability of wild-N.lugens -extinction periodic solution of two models, respectively, and discuss the persistence of wild N. lugens .Then we numerically analyze the influences of release period and release amount on the control of wild N. lugens in Section 4. Finally, we present our conclusions in Section 5.

Model formulation
Let U f (t) and U m (t) be the densities of the wild female N. lugens and male N. lugens uninfected by wLug and wStri at time t, respectively.L f (t) and L m (t) denote the densities of the wild female N. lugens and male N. lugens infected with wLug at time t, respectively.S f (t) and S m (t) represent the densities of the female N. lugens and male N. lugens infected with wStri at time t.
In order to establish the mathematical model more conveniently, we give a chart to show all possible mating patterns of N. lugens according to the research conclusions of studies [10,25].The fourth line of Figure 1 shows that the males N. lugens infected with wStri S m can induce higher CI level when they mate with U f and L f , 0 < c 1 < 1 and 0 < c 2 < 1 represent the CI intensity of S m against U f and L f , respectively.The third and fourth columns of Figure 1 indicate that the N. lugens infected with wStri has perfect maternal transmission, while the wild N. lugens infected with wLug has imperfect maternal transmission.0 < θ < 1 is the percentage of uninfected progeny produced by a wLug-infected mother.Ju et al. [8] shows that Wolbachia of the wlug strain has a reproductive promoting effect on its natural host N. lugens , but wStri does not.Therefore, we assume that the uninfected N. lugens female U f and the wStri-infected N. lugens female S f have the same birth rate, while the birth rate of wLug-infected N. lugens female L f is greater than U f and S f .Let a > 0 be the birth rate of the uninfected N. lugens , and ρa is the birth rate of the wLug-infected N. lugens , where ρ > 1.Moreover, we adhere to the conventional method by substituting the natural death of N. lugens population with a Logistic-like density dependent term [10,25].Let d 1 > 0, d 2 > 0 and d 3 > 0 denote the decay rate constants of the uninfected N. lugens , wLug-infected N. lugens and wStri-infected N. lugens , respectively.Similar to Refs.[10,26,27], we always assume that the proportion of individuals born male is equal to the proportion of individuals born female in this paper, that is, U f = U m , L f = L m and S f = S m .
From Figure 1, we give all the mating patterns of N. lugens , such as U f come from the mating patterns U f × U m , U f × L m and U f × S m .Then the expression denotes the probability of a female N. lugens mates with an uninfected male N. lugens, (t) represents the probability of a female N. lugens mates with a wLuginfected male N. lugens , and (t) gives the probability of incompatible crossing that a wild female N. lugens mates with a wStri-infected male N. lugens .According to the above assumptions and mating patterns of N. lugens , we propose the following two mathematical models.

Population suppression model with periodic impulsive release
To begin with, we consider the case that the wStri-infected males N. lugens are released into the paddy field.It is easy to see from Figure 1 that the mating pattern of the third column will not appear in this ecosystem.According to the mating patterns of N. lugens, we first propose the following population suppression model with periodic release: where b ≥ 0 represents the release amount of male N. lugens infected with wStri each time.τ is the release period, and Denote U = U m + U f and L = L m + L f .Then model (1) can be simplified to the following model: Using a linear scaling for system (2), and rewriting d./ds as d./dt, system (2) can be transmuted into

Population replacement model with periodic impulsive release
We also establish a population replacement model with periodic release.The females N. lugens infected with wStri and males N. lugens infected with wStri are released into the farmland ecosystem, then the farmland ecosystem has uninfected N. lugens , wLuginfected N. lugens and wStri-infected N. lugens population.According to the mating patterns of N. lugens , a population replacement model with periodic impulsive release is given as follows: where Then model ( 4) can be simplified to the following model: Using a linear scaling for system (5), and rewriting d./ds as d./dt, then we transmute system (5

Main results
We will study the dynamics of models ( 3) and ( 6) in this section.For simplicity, denote The population χ is said to be strongly persistent in the mean if lim inf t→+∞ t −1 t 0 χ(s)ds > 0.
Proof: Integrating the first equation of ( 7) between [nT, (n + 1)T], we have By the second equation of ( 7), we can obtain the stroboscopic map: It is easy to calculate that the above discrete system has a unique positive fixed point Because 1, the unique positive fixed point z * is locally asymptotically stable.In addition, So fixed point z * is globally asymptotically stable.Further, the positive periodic solution of system ( 7) is also locally asymptotically stable.Furthermore, The proof is completed.
Consider the following system: Then we give the following lemma according to Lemma 3.2 in Huang et al. [16].
Lemma 3.2 (See [16]): System (8) has a unique periodic solution g(t) with period T, and for any solution g(t) of system (8) with g(0 , t ∈ (nT, (n + 1)T], and g * is the positive root of the following equation In the following, we will discuss the stability of wild-N.lugens -eradication periodic solution (0, 0, z(t)).We first show the periodic solution (0, 0, z(t)) is locally asymptotically stable and then prove it is also a global attractor.

Proof:
The local stability of periodic solution (0, 0, z(t)) can be determined by considering the small-amplitude perturbation of the solution.
Define u(t) = x(t), v(t) = y(t), w(t) = z(t) − z(t).The linearized system of system (3) in (nT, (n + 1)T] is obtained as follows: By simply calculating, the fundamental solution matrix in interval (nT, (n + 1)T] can be given by where the exact expression of function i (i = 1,2,3) is not presented because it is not used in the calculation of the eigenvalues of matrix .It follows from the fourth equations of (3) that Further from the Floquet theory, we obtain that the periodic solution (0, 0, z(t)) is locally asymptotically stable, which can be determined by the absolute values of all eigenvalues of matrix = φ(T) are less than 1.The eigenvalues of matrix are According to the conditions given in Theorem 3.1, we have λ 1 < 1 and λ 2 < 1, thus the periodic solution (0, 0, z(t)) of system (3) is locally asymptotically stable.
We next study the global asymptotical stability of the wild-N.lugens -eradication periodic solution (0, 0, z(t)).Theorem 3.2: If R m3 < 0 and R m4 < 0, the wild-N.lugens -eradication periodic solution (0, 0, z(t)) of system (3) is globally asymptotically stable.Proof: From system (3), we have According to the comparison theorem and Lemma 3.1, we obtain that for any > 0, there always exists a n 1 ∈ Z + such that for t ≥ n 1 T, where W(t) is the solution of the following system: From the third equation of system (3), for any t > n 1 T, we have Then it follows from Lemma 3.2 and the comparison theorem of impulsive differential equation [28] that, if > 0 small enough, there exists a positive integer n 2 > n 1 such that Substituting g(t) − into the second equation of system (3), we have If we can select a small enough such that It follows from (12) that which implies that y(t) → 0 as t → +∞ (n → +∞).Thus, for any > 0 small enough, there must be a positive integer n 3 > n 2 such that y(t) < for t > n 3 T.By the first equation of (3), we derive then it follows from (13) that for sufficiently small 1 > 0, there is a n 4 > n 3 such that T, and then we discuss the first equation of (3) again, we have If conditions R m3 < 0 and R m4 < 0 hold, we can deduce from ( 14) that x(t) → 0 as t → +∞.
Substituting into the third equation of system (3) for x(t) and y(t), we have From Lemma 3.2 and the comparison theorem of impulsive differential equation [28], we obtain that for sufficiently small > 0, there is a positive integer n 5 (n 5 > n 4 ) such that z(t) > z(t) − for t > n 5 T. According to the above discussion, if the conditions of Theorem 3.2 are satisfied, for > 0 small enough, we have 0 < x(t) < , 0 < y(t) < , and z(t) − < z(t) < z(t) + , t ≥ n 5 T.

is strongly persistent in the mean and y(t) goes to extinction.
Proof: From the proof of Theorem 3.2, we obtain that lim t→+∞ y(t) = 0 when R m4 < 0. Further, for any > 0, there exists N 1 > 0 such that By Lemmas 3.1 and 3.2, we can obtain that for > 0 sufficiently small, there exists a N 2 > N 1 such that Substituting inequalities ( 15) and ( 16) into the first equation of (3), we have Dividing ( 17) by x(t) and then integrating both sides of the resulted equation on the interval [N 2 T, t], we can obtain By ( 9), we have lim sup t→+∞ ln(x(t)/x(N 2 T)) t ≤ 0, and Then from (18) and R m1 > 0, we have lim inf t→+∞ t −1 t 0 x(p)dp ≥ that is, the wild N. lugens is strongly persistent in the mean.
Proof: From the first and second equations of (3) and ( 18), we have for t > N 2 T. Similar to the proof of Theorem 3.3, we obtain that lim inf The proof is completed.
The uninfected N. lugens and the wLug-infected N. lugens are two types of natural populations.Since the wild N. lugens infected with wLug is imperfectly maternally transmitted, it is unlikely that only the wild N. lugens infected with wLug occur in the agricultural ecosystem (see Figure 1), but it is possible that only uninfected wild N. lugens populations exist in the agricultural ecosystem.Next, we will show the dynamics of the following subsystem which absents the N. lugens infected with wLug.
From Theorems 3.1-3.4,we can give the following results for system (19).

Corollary 3.1: (a)
The wild-N.lugens -eliminate periodic solution (0, z(t)) of system (19) is locally asymptotically stable if R m1 < 0. (b) The wild-N.lugens -eliminate periodic solution (0, z(t)) of system (19) is globally asymptotically stable if where g * 1 is the positive root of the following equation 19) is strongly persistent in the mean if R m1 > 0.

The dynamics of model (6)
Denote where Z * , h * and h * 1 are given in the following Lemmas 3.3 and 3.4.
To begin with, we consider the following subsystem of system ( 6) when x = 0 and y = 0: Lemma 3.3: System (20) has a unique positive periodic solution Z(t) with period T, and for any solution Z(t) of system (20) with Z(0 + ) ≥ 0, we have |Z(t) − Z(t)| → 0 as t → +∞, where , t ∈ (nT, (n + 1)T], and Z * is the positive root of the following equation: Proof: Integrating the first equation in (20) between pulses, we have for t ∈ (nT, (n + 1)T].
By the second equation of ( 20), we can obtain the stroboscopic map: Consider the equation Z = h(Z), namely, which is equivalent to the standardized quadratic equation: Obviously, system (21) has unique positive fixed point Z * .Hence, system (20) has unique positive periodic solution Similar to Lemma 3.1, we can also get that Z * is globally asymptotically stable for system (21), then the corresponding period solution Z(t) of system ( 20) is also globally asymptotically stable.This completes the proof.

Remark 3.1:
From the Lemma 3.3, we can easily calculate that Z * ≥ 1, thus there is a special case that Z * = 1 when β = 0 and Z(0 + ) > 0, and for any solution Z(t) of system (20), we have Z(t) → 1 as t → +∞.
For the following system: Similar to Lemma 3.
, t ∈ (nT, (n + 1)T], where h * is the positive root of equation h

, the periodic solution h(t) is given by
and h(0 Similar to the discussion of Theorems 3.1-3.4,we have the following results: Theorem 3.5: If R f 1 < 0 and R f 2 < 0, the wild-N.lugens -eradication periodic solution (0, 0, Z(t)) of system (6) is locally asymptotically stable.
. Then the linearized system of system ( 6) in (nT, (n + 1)T] is obtained as follows: Clearly, the fundamental solution matrix in interval (nT, (n + 1)T] is and the expression of function † i (i = 1,2,3) does not need to be given.
From the fourth equations of ( 6), we have The local stability of the periodic solution (0, 0, Z(t)) is determined by the eigenvalues of Φ(T), where they are and According to the Floquet theorem and conditions given in Theorem 3.5, we have λ 1 < 1 and λ 2 < 1.Hence, the periodic solution (0, 0, Z(t)) of system ( 6) is locally asymptotically stable.
Theorem 3.6: If any of the following conditions is true, Then the wild-N.lugens -eradication periodic solution (0, 0, Z(t)) of system (6) is globally asymptotically stable.

Theorem 3.7: If any of the following conditions is true,
Then X(t) is strongly persistent in the mean and Y(t) goes to extinction.
The proofs of Theorem 3.6-3.8are basically similar to those of Theorems 3.2-3.4,therefore, we omit them here.
In the following, we consider the subsystem without wLug of system (6).
From Theorem 3.5, we can obtain the following results for system (23).

Numerical simulation and discussions
In this section, we will verify our results by numerical simulation.From Table 1, we obtain that θ = 0.85, c 1 = 0.675, c 2 = 0.732, and ρ = 2 by taking a = 45 and ρa = 90 at room temperature 25 • C. We choose the parameters δ 1 = 0.85 and δ 2 = 0.2.If we do not release the N. lugens infected with wStri, the ecosystem will be transformed into the following model  According to Theorem 10 in [10], system (24) has unique a positive equilibrium point (1.2911, 0.2089), and it is globally asymptotically stable.The time series of the wild uninfected N. lugens and the wild N. lugens infected with wLug are shown in Figure 2. In the following, we will discuss the influences of the release rate β and release period T of the N. lugens infected with wStri on the dynamics of systems ( 3) and ( 6).

Long-time behaviours of population suppression model
In order to evaluate the influences of the release period T and the release amount β of male N. lugens infected with wStri on the dynamics of system (3).We first take T = 3 and β = 1, by calculating, then R m1 = −0.1399< 0 and R m2 = −0.1203< 0, which satisfy the conditions of Theorem 3.1, the wild-N.lugens -eradication periodic solution (0, 0, z(t)) of system (3) is locally asymptotically stable.System (3) may have two steady states coexisting, the wild-N.lugens -eradication period solution where the wild N. lugens population will be extinct, and a positive periodic solution where the N. lugens population oscillates positively and periodically as shown in Figure 3.These results also indicate that when the number of the wild N. lugens population is small, the wild N. lugens population will be controlled by regularly releasing fewer male N. lugens populations infected with wStri into the field.
However, when we take T = 0.5 and β = 11, then R m3 = −0.2858< 0 and R m4 = −0.0036< 0, from Theorem 3.2, we can get that the periodic solution (0, 0, z(t)) is globally asymptotically stable.It means that whatever the initial value is, the wild N. lugens population goes to extinction for T = 0.5 and β = 11, see Figure 4.If we take T = 6 and β = 1, according to Theorem 3.4, we have R m5 = 0.1961 > 0, and the wild N. lugens population is strongly persistent in the mean as shown in Figure 5. From Figures 2-5, we can also see that increasing the release amount β of male N. lugens infected with wStri or decreasing the release period T are beneficial to suppress the density of wild N. lugens population.

Long-time behaviours of population replacement model
We first take β = 0.1 and T = 6, by calculating, we obtain R f 1 = −0.6335< 0 and R f 2 = −0.6139< 0, there exists a locally asymptotically stable wild N. lugens eradication periodic solution for system (6) where the wild N. lugens population goes to extinction.Furthermore, by selecting appropriate initial values, the numerical simulations show that system (6) also exists a locally asymptotically stable positive periodic solution (see Figure 6).
In addition, we consider a special case of system (6) that β = 0 and Z(0 + ) > 0. We obtain R f 1 = −0.6176< 0 and R f 2 = −0.5980< 0 by Theorem 3.5, and the equilibrium infected with wStri (0,0,1) is locally asymptotically stable as shown in Figure 7.It can be seen that we only need to release the N. lugens infected with wStri once to the field, and it is possible to complete the replacement of wild N. lugens populations.

Control of the wild N. lugens within a short time
In the first two sub-sections, we fixed parameters ρ, θ, c 1 , c 2 , δ 1 and δ 2 , and then showed the long-time behaviours of population suppression model and population replacement model by changing parameters β and T.However, it is very important to control the wild N. lugens population to a low level in a short time in the actual paddy field management.In this subsection, we will study the effects of release amount β and release period T on the control efficiency of wild N. lugens population within a short time under different release strategies.Therefore, we define a concept of control degree as the ratio of the amount of wild N. lugens  6) with T = 6 and β = 0.1 for different initial values.Here we take initial values (0.1,0.02,0.9),(0.2, 0.05, 0.7), (0.3, 0.08, 1), (0.6, 0.1, 0.1), (0.9, 0.15, 0.2) and (1.2, 0.2, 0.4).
Figure 7.The solutions of system (6) with β = 0 for different initial values.Here we take initial values (0.1,0.02,0.9),(0.2, 0.05, 0.7), (0.3, 0.08, 1), (0.6, 0.1, 0.1), (0.9, 0.15, 0.2) and (1.2, 0.2, 0.4).population suppressed or replaced with a finite time to the initial wild N. lugens amount [29], denoted by e, and the calculation formula is shown by where w(t) is the amount of wild N. lugens population at time t, and w(0) denotes the initial amount of wild N. lugens population.We assume that x(0) = 1, y(0) = 0.3, z(0 + ) = β and T 0 = 20.We first consider the effect of the release period T on the control efficiency of wild N. lugens in a finite time T 0 .Fix parameters c 1 = 0.675, c 2 = 0.732, δ 1 = 0.85, δ 2 = 0.2, β = 1.5 and vary T. We take T = 1, 1.25, 2 and 2.5, respectively, Both system (3) and system ( 6) have stable wild-N.lugens -eradication boundary periodic solution.From Figure 9, we can see that the control efficiency of wild N. lugens decreases as T increases.For system (3), Figure 9 shows that when T ≤ 1.25, the control efficiency of the wild N. lugens reaches more than 90% within time T 0 , and when T = 2.5, the control efficiency of the uninfected wild N. lugens and the wild N. lugens infected with wLug within time T 0 is only 29.29% and 55.10%, respectively.However, the control efficiency of the wild N. lugens within time T 0 is more than 95% for system (6).
And then, we fix parameters c 1 = 0.675, c 2 = 0.732, δ 1 = 0.85, δ 2 = 0.2, T = 2 and vary β.Let β = 1.8, 2, 2.5 and 3, there is also a stable wild-N.lugens -eradication boundary periodic solution for both system.Figure 10 shows that the control efficiency of wild N. lugens increases as β increases.If we adopt a population suppression strategy, the control efficiency of uninfected wild N. lugens within time T 0 is more than 93% when β ≥ 2, and the  control efficiency of wild N. lugens infected with wLug within time T 0 is more than 90% when β ≥ 3.However, when we adopt the population replacement strategy, the control efficiency of wild N. lugens within time T 0 is more than 98% when β ≥ 1.8.
In addition, we consider an ecosystem in which the population of N. lugens infected with wLug is absent, fix c 1 = 0.675, δ 1 = 0.85, x(0) = 1, T 0 = 20, and the values of other parameters are the same as those in Figures 9 and 10, then the time series of the control degree of this wild N. lugens population under different release strategies are shown in Figure 11.It can be seen from Figure 11 that no matter which strategy is adopted, the wild N. lugens population will be controlled to more than 99% in T 0 time.Comparing the results in Figures 9-11, the wild N. lugens infected with wLug can reduce the control efficiency of uninfected wild N. lugens.
From Figures 9-11, it is easy to see that under the same release amount and release cycle, the efficiency of population replacement strategy in controlling wild N. lugens within a finite time is significantly higher than that of population suppression strategy.Furthermore, these results also suggest that increasing the release amount β or decreasing the release period T are beneficial for controlling the wild N. lugens.
Finally, we will simulate the change of control degree of wild N. lugens population under a finite release number of times.Fixed parameters c 1 = 0.675, c 2 = 0.732, δ 1 = 0.85, δ 2 = 0.2, β = 0.5, T = 4, x(0) = 1, y(0) = 0.3 and z(0 + ) = β.Based on these parameters, we discuss the situation that the number of release times of N. lugens infected with wStri is 1, 2, 3 and 4 respectively, as shown in Figure 12.It can be seen from Figure 12 that for the population replacement strategy, when the release number of times of N. lugens infected with wStri exceeds 3, the wild N. lugens population will be controlled.However, for population suppression strategy, it is impossible to achieve complete control of wild N. lugens through limited release times.This indicates that the cost of population replacement strategy in controlling wild population will be less than that of population suppression strategy.
In Section 3, we obtained the conditions for the stability of the wild-N.lugenseradication periodic solution and verified them numerically in Subsections 4.1-4.2(see .This also shows that both release strategies are able to control wild N. lugens populations.In this paper, both release strategies assume that the N. lugens infected with wStri is released in periodic pulses, so the important parameters for control measures are the pulse release amount β and pulse release period T of the N. lugens infected with wStri.For population suppression strategy, if the values of the release period T and the release amount β satisfy the conditions of Theorem 3.2, the wild N. lugens will become extinct.If the values of β and T satisfy the conditions of Theorem 3.1 but not the conditions of Theorem 3.2, the eradication of wild N. lugens depends on the initial values.For population replacement strategy, if the values of T and β satisfy the conditions of Theorem 3.6, the wild N. lugens will be replaced.Since the complexity of the expressions for R mi and R fi , we cannot directly compare the two strategies theoretically, but we have employed numerical simulations to compare the two strategies in Subsection 4.3, and the population replacement strategy can control wild N. lugens faster than the population suppression strategy for the same release period T and the same amount of release β (see .Moreover, the population replacement strategy can achieve the replacement of the wild N. lugens population with a finite number of releases (see Figure 12).Of course, the above scenario occurs mainly because there is perfect maternal transmission of female N. lugens infected with wStri.However, if the rice population density is added to models ( 1) and ( 4), both models will become more complex, which will be one of our next major works.Furthermore, we provided the conditions for the persistence in the mean of wild N. lugens in Section 3, but Figure 5 shows the existence of a positive periodic solution for model (3).Unfortunately, we have not found a good mathematical method to solve this problem due to the nonlinearity of the equations, which is also the direction of our future research.

Conclusions
Gong et al. [6] successfully developed a stable artificial Wolbachia infection of N. lugens by introducing the Wolbachia strain wStri from L. striatellus host into N. lugens .The use of N. lugens infected with wStri to control wild N. lugens will be one of the important ways in the future.It is therefore of great interest to study how the release of N. lugens infected with wStri.In this paper, we established two models with periodic impulsive release: the population suppression model (3) and the population replacement model (6).Applying Floquet theory and the comparison theorem of impulsive differential equations, we obtained the conditions for the stability of the wild-N.lugens -eradication periodic solution of both models (see .Meanwhile, sufficient conditions for the persistence in the mean of wild uninfected N. lugens and the extinction of N. lugens infected with wLug were obtained, and the conditions for the persistence in the mean of wild uninfected N. lugens and N. lugens infected with wLug were also given.Numerical simulations were performed to verify our theoretical results.Finally, we compared the control effect of two periodic pulse release strategies on wild N. lugens.The population replacement strategy is obviously much better than the population suppression strategy.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Figure 1 .
Figure 1.All possible mating patterns of N. lugens.The wild N. lugens infected with wLug has imperfect maternal transmission.

Figure 9 .
Figure 9.The control degree of the wild N. lugens within a finite time for different release periods.The first four lines of the legend represent the control efficiency of wild N. lugens in system (3), and the last four lines of the legend represent the control efficiency of wild N. lugens in system (6).(a) The control degree of the uninfected wild N. lugens.(b) The control degree of the wild N. lugens infected with wLug.

Figure 10 .
Figure 10.The control degree of the wild N. lugens within a finite time for different release amounts.The first four lines of the legend represent the control efficiency of wild N. lugens in system (3), and the last four lines of the legend represent the control efficiency of wild N. lugens in system (6).(a) The control degree of the uninfected wild N. lugens.(b) The control degree of the wild N. lugens infected with wLug.

Figure 11 .
Figure11.The control degree of the wild N. lugens within a finite time.The first four lines of the legend represent the control efficiency of wild N. lugens in system(19), and the last four lines of the legend represent the control efficiency of wild N. lugens in system(23).(a) The effect of β on control degree of the wild N. lugens.(b) The effect of T on control degree of the wild N. lugens.

Figure 12 .
Figure 12.Control of the wild N. lugens under a finite release number of times.The second, fourth, sixth, and eighth lines of the legend represent the control degree curve of wild N. lugens by using population replacement strategy, and the first, third, fifth, and seventh lines of the legend represent the control degree curve of wild N. lugens by using population suppression strategy.(a) The control degree of the uninfected wild N. lugens.(b) The control degree of the wild N. lugens infected with wLug.

Table 1 .
Parameter values of the N. lugens in the laboratory.
a Number of eggs laid by a female wild N. lugens uninfected with wLug (day −1 ) (37,55) [8] ρa Number of eggs laid by a female wild N. lugens infected with wLug (day −1 )