Abstract
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--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 are also given. In addition, the sufficient conditions for the extinction and persistence of the wild 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 , and the efficiency of population replacement strategy in controlling wild populations is higher than that of population suppression strategy under the same release conditions.
1. 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 () [Citation1,Citation2]. Few feasible control strategies are obtainable because of the evolution of high levels of insecticide resistance, and new environmentally friendly methods are urgently needed [Citation3]. In recent studies, disease control methods based on artificial Wolbachia infection to inhibit mosquito vector-borne pathogens have been applied [Citation4,Citation5]. Scientists are currently working on similar methods for controlling agricultural pests [Citation6].
Nilaparvata lugens can be naturally infected by Wolbachia strain wLug, which lacks the ability to induce cytoplasmic incompatibility () [Citation7]. Many experimental results show that the infected with wLug has stronger reproductive ability than the uninfected , and show imperfect maternal transmission characteristics [Citation6,Citation8,Citation9]. Fortunately, Gong et al. [Citation6] successfully developed a stable artificial Wolbachia infection of by introducing the Wolbachia strain wStri from host into . The results [Citation6] showed that 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 and , respectively. The results of Gong et al. [Citation6] lay a foundation for future experiments on paddy fields.
In order to theoretically study the interaction mechanism between infected with wStri and wild population, Liu and Zhou [Citation10] proposed and studied a Wolbachia spreading dynamics model in with two strains, and obtained sufficient conditions for the infected with wStri to invade wild successfully. But the authors did not address when and how many infected with wStri is released to suppress or replace wild , so studying the release of wStri-infected to control wild for future field trials with wStri is biologically significant. Currently, there are few models to study how to release the infected Wolbachia, but many mosquito population dynamics models have been proposed and studied [Citation11–24]. Cai et al. [Citation11] 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. [Citation16] 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 [Citation17–20], the authors adopted a new modelling idea that only those sexually active sterile mosquitoes were considered in the modelling process. Yu et al. [Citation19] 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. [Citation17] considered the following situation that the release period was shorter than the sexual lifespan of sterile male mosquitoes. Li and Ai [Citation18] 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 , Zika, etc.), and established some mathematical models to control mosquito-borne diseases [Citation23,Citation24]. For example, Taghikahani et al. [Citation23] 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 [Citation11–24] 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 infected with wStri. Compared with the transmission characteristics of mosquitoes infected with Wolbachia, the infected with Wolbachia has many differences. For example, the cytoplasmic incompatibility induced by the mating of male infected with wStri with wild uninfected female or wild female infected with wlug is incomplete [Citation6]. To determine when is the best time to release infected with wStri and the number of infected with wStri per release. In 2023, Liu et al. [Citation25] established and discussed two semi-continuous models with state-feedback impulsive releases of infected with wStri. But the authors only considered the scenario where all wild populations were uninfected , and there are few studies of wild including both the wild uninfected and infected with wLug. In this paper, we adapt the modelling idea in [Citation10] and consider the periodic impulsive release of male or female infected with wStri into the field. And then according to the transmission characteristics of two Wolbachia strains in , we establish two impulsive mathematical models with periodic impulsive release of infected with wStri: population suppression model and population replacement model. We theoretically discuss the stability of the wild- -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 are also established. Finally, we compare the degree of control of two periodic impulsive release strategies on the wild 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- -extinction periodic solution of two models, respectively, and discuss the persistence of wild . Then we numerically analyze the influences of release period and release amount on the control of wild in Section 4. Finally, we present our conclusions in Section 5.
2. Model formulation
Let and be the densities of the wild female and male uninfected by wLug and wStri at time t, respectively. and denote the densities of the wild female and male infected with wLug at time t, respectively. and represent the densities of the female and male 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 according to the research conclusions of studies [Citation10,Citation25]. The fourth line of Figure shows that the males infected with wStri can induce higher CI level when they mate with and , and represent the CI intensity of against and , respectively. The third and fourth columns of Figure indicate that the infected with wStri has perfect maternal transmission, while the wild infected with wLug has imperfect maternal transmission. is the percentage of uninfected progeny produced by a wLug-infected mother. Ju et al. [Citation8] shows that Wolbachia of the wlug strain has a reproductive promoting effect on its natural host , but wStri does not. Therefore, we assume that the uninfected female and the wStri-infected female have the same birth rate, while the birth rate of wLug-infected female is greater than and . Let a>0 be the birth rate of the uninfected , and is the birth rate of the wLug-infected , where . Moreover, we adhere to the conventional method by substituting the natural death of population with a Logistic-like density dependent term [Citation10,Citation25]. Let , and denote the decay rate constants of the uninfected , wLug-infected and wStri-infected , respectively. Similar to Refs. [Citation10,Citation26,Citation27], we always assume that the proportion of individuals born male is equal to the proportion of individuals born female in this paper, that is, , and .
From Figure , we give all the mating patterns of , such as come from the mating patterns , and . Then the expression denotes the probability of a female mates with an uninfected male , represents the probability of a female mates with a wLug-infected male , and gives the probability of incompatible crossing that a wild female mates with a wStri-infected male . According to the above assumptions and mating patterns of , we propose the following two mathematical models.
2.1. Population suppression model with periodic impulsive release
To begin with, we consider the case that the wStri-infected males are released into the paddy field. It is easy to see from Figure that the mating pattern of the third column will not appear in this ecosystem. According to the mating patterns of , we first propose the following population suppression model with periodic release: (1) (1) where represents the release amount of male infected with wStri each time. τ is the release period, and , , , . .
Denote and . Then model (Equation1(1) (1) ) can be simplified to the following model: (2) (2) Using a linear scaling for system (Equation2(2) (2) ), and rewriting as , system (Equation2(2) (2) ) can be transmuted into (3) (3)
2.2. Population replacement model with periodic impulsive release
We also establish a population replacement model with periodic release. The females infected with wStri and males infected with wStri are released into the farmland ecosystem, then the farmland ecosystem has uninfected , wLug-infected and wStri-infected population. According to the mating patterns of , a population replacement model with periodic impulsive release is given as follows: (4) (4) where .
Denote , and . Then model (Equation4(4) (4) ) can be simplified to the following model: (5) (5) Using a linear scaling for system (Equation5(5) (5) ), and rewriting as , then we transmute system (Equation5(5) (5) ) into (6) (6)
3. Main results
We will study the dynamics of models (Equation3(3) (3) ) and (Equation6(6) (6) ) in this section. For simplicity, denote
Definition 3.1
The population χ is said to be extinct if .
The population χ is said to be strongly persistent in the mean if .
3.1. The dynamics of model (3)
Let be the solution of system (Equation3(3) (3) ) with initial value , and . Clearly, is a piecewise continuous function, where . From [Citation28], the global existence and uniqueness of solutions of system (Equation3(3) (3) ) is guaranteed by the smoothness properties of , which denotes the mapping defined by the right-hand side of system (Equation3(3) (3) ). Denote where and will be given in the following Lemmas 3.1 and 3.2.
If and , the subsystem of system (Equation3(3) (3) ) is presented by impulsive differential equations (7) (7)
Lemma 3.1
System (Equation7(7) (7) ) has a unique periodic solution with period T, and for any solution of system (Equation7(7) (7) ) with , we have as , where and .
Proof.
Integrating the first equation of (Equation7(7) (7) ) between , we have By the second equation of (Equation7(7) (7) ), we can obtain the stroboscopic map: It is easy to calculate that the above discrete system has a unique positive fixed point Because , the unique positive fixed point is locally asymptotically stable. In addition, So fixed point is globally asymptotically stable. Further, the positive periodic solution of system (Equation7(7) (7) ) is also locally asymptotically stable.
Furthermore, The proof is completed.
Consider the following system: (8) (8) Then we give the following lemma according to Lemma 3.2 in Huang et al. [Citation16].
Lemma 3.2
See [Citation16]
System (Equation8(8) (8) ) has a unique periodic solution with period T, and for any solution of system (Equation8(8) (8) ) with , as , where and is the positive root of the following equation
In the following, we will discuss the stability of wild- -eradication periodic solution . We first show the periodic solution is locally asymptotically stable and then prove it is also a global attractor.
Theorem 3.1
The wild- -eradication periodic solution of system (Equation3(3) (3) ) is locally asymptotically stable if and .
Proof.
The local stability of periodic solution can be determined by considering the small-amplitude perturbation of the solution.
Define The linearized system of system (Equation3(3) (3) ) in is obtained as follows: By simply calculating, the fundamental solution matrix in interval can be given by where the exact expression of function (i=1,2,3) is not presented because it is not used in the calculation of the eigenvalues of matrix Φ.
It follows from the fourth equations of (Equation3(3) (3) ) that Further from the Floquet theory, we obtain that the periodic solution is locally asymptotically stable, which can be determined by the absolute values of all eigenvalues of matrix are less than 1. The eigenvalues of matrix Φ are According to the conditions given in Theorem 3.1, we have and , thus the periodic solution of system (Equation3(3) (3) ) is locally asymptotically stable.
We next study the global asymptotical stability of the wild- -eradication periodic solution .
Theorem 3.2
If and , the wild- -eradication periodic solution of system (Equation3(3) (3) ) is globally asymptotically stable.
Proof.
From system (Equation3(3) (3) ), we have and According to the comparison theorem and Lemma 3.1, we obtain that for any , there always exists a such that (9) (9) for , where is the solution of the following system: From the third equation of system (Equation3(3) (3) ), for any , we have (10) (10) Then it follows from Lemma 3.2 and the comparison theorem of impulsive differential equation [Citation28] that, if small enough, there exists a positive integer such that (11) (11) Substituting into the second equation of system (Equation3(3) (3) ), we have (12) (12) If we can select a ϵ small enough such that It follows from (Equation12(12) (12) ) that which implies that as (). Thus, for any small enough, there must be a positive integer such that for . By the first equation of (Equation3(3) (3) ), we derive (13) (13) then it follows from (Equation13(13) (13) ) that for sufficiently small , there is a such that for , and then we discuss the first equation of (Equation3(3) (3) ) again, we have (14) (14) If conditions and hold, we can deduce from (Equation14(14) (14) ) that as .
Substituting ϵ into the third equation of system (Equation3(3) (3) ) for and , we have From Lemma 3.2 and the comparison theorem of impulsive differential equation [Citation28], we obtain that for sufficiently small , there is a positive integer such that for .
According to the above discussion, if the conditions of Theorem 3.2 are satisfied, for small enough, we have Letting , we obtain , and as , which means that the periodic solution of system (Equation3(3) (3) ) is a global attractor. Combining Theorem 3.1, we obtain that the periodic solution is globally asymptotically stable when conditions and hold. The proof is completed.
Theorem 3.3
If and , is strongly persistent in the mean and goes to extinction.
Proof.
From the proof of Theorem 3.2, we obtain that when . Further, for any , there exists such that (15) (15) By Lemmas 3.1 and 3.2, we can obtain that for sufficiently small, there exists a such that (16) (16) Substituting inequalities (Equation15(15) (15) ) and (Equation16(16) (16) ) into the first equation of (Equation3(3) (3) ), we have (17) (17) Dividing (Equation17(17) (17) ) by and then integrating both sides of the resulted equation on the interval , we can obtain (18) (18) By (Equation9(9) (9) ), we have , and Then from (Equation18(18) (18) ) and , we have The proof is completed.
Theorem 3.4
If , we have that is, the wild is strongly persistent in the mean.
Proof.
From the first and second equations of (Equation3(3) (3) ) and (Equation18(18) (18) ), we have for . Similar to the proof of Theorem 3.3, we obtain that The proof is completed.
The uninfected and the wLug-infected are two types of natural populations. Since the wild infected with wLug is imperfectly maternally transmitted, it is unlikely that only the wild infected with wLug occur in the agricultural ecosystem (see Figure ), but it is possible that only uninfected wild populations exist in the agricultural ecosystem. Next, we will show the dynamics of the following subsystem which absents the infected with wLug. (19) (19) From Theorems 3.1–3.4, we can give the following results for system (Equation19(19) (19) ).
Corollary 3.1
(a) | The wild- -eliminate periodic solution of system (Equation19(19) (19) ) is locally asymptotically stable if . | ||||
(b) | The wild- -eliminate periodic solution of system (Equation19(19) (19) ) is globally asymptotically stable if where is the positive root of the following equation | ||||
(c) | System (Equation19(19) (19) ) is strongly persistent in the mean if . |
3.2. The dynamics of model (6)
Denote where , and are given in the following Lemmas 3.3 and 3.4.
To begin with, we consider the following subsystem of system (Equation6(6) (6) ) when x=0 and y=0: (20) (20)
Lemma 3.3
System (Equation20(20) (20) ) has a unique positive periodic solution with period T, and for any solution of system (Equation20(20) (20) ) with , we have as , where and is the positive root of the following equation:
Proof.
Integrating the first equation in (Equation20(20) (20) ) between pulses, we have for .
By the second equation of (Equation20(20) (20) ), we can obtain the stroboscopic map: (21) (21) Consider the equation , namely, which is equivalent to the standardized quadratic equation: Obviously, system (Equation21(21) (21) ) has unique positive fixed point . Hence, system (Equation20(20) (20) ) has unique positive periodic solution with .
Similar to Lemma 3.1, we can also get that is globally asymptotically stable for system (Equation21(21) (21) ), then the corresponding period solution of system (Equation20(20) (20) ) is also globally asymptotically stable. This completes the proof.
Remark 3.1
From the Lemma 3.3, we can easily calculate that , thus there is a special case that when and , and for any solution of system (Equation20(20) (20) ), we have as .
For the following system: (22) (22) Similar to Lemma 3.3, we obtain the following results:
Lemma 3.4
System (Equation22(22) (22) ) has a unique periodic solution with period T, and for any solution of system (Equation22(22) (22) ) with , we have as .
When , the periodic solution is given by where is the positive root of equation
When d=1, the periodic solution is given by and
Similar to the discussion of Theorems 3.1–3.4, we have the following results:
Theorem 3.5
If and , the wild- -eradication periodic solution of system (Equation6(6) (6) ) is locally asymptotically stable.
Proof.
Define , , . Then the linearized system of system (Equation6(6) (6) ) in is obtained as follows: Clearly, the fundamental solution matrix in interval is and the expression of function (i=1,2,3) does not need to be given.
From the fourth equations of (Equation6(6) (6) ), we have The local stability of the periodic solution is determined by the eigenvalues of , where they are and According to the Floquet theorem and conditions given in Theorem 3.5, we have and . Hence, the periodic solution of system (Equation6(6) (6) ) is locally asymptotically stable.
Theorem 3.6
If any of the following conditions is true,
(i) | and . | ||||
(ii) | and . | ||||
(iii) | 1−d=0, and . |
Then the wild- -eradication periodic solution of system (Equation6(6) (6) ) is globally asymptotically stable.
Theorem 3.7
If any of the following conditions is true,
(i) | , , and . | ||||
(ii) | 1−d=0, , and . |
Then is strongly persistent in the mean and goes to extinction.
Theorem 3.8
If , we have
The proofs of Theorem 3.6–3.8 are 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 (Equation6(6) (6) ). (23) (23)
From Theorem 3.5, we can obtain the following results for system (Equation23(23) (23) ).
Corollary 3.2
(a) | The wild- -eliminate periodic solution of system (Equation23(23) (23) ) is locally asymptotically stable if . | ||||||||||||||||
(b) | If any of the following conditions is true,
The wild- -eliminate periodic solution of system (Equation23(23) (23) ) is globally asymptotically stable. | ||||||||||||||||
(c) | System (Equation23(23) (23) ) is strongly persistent in the mean if . |
4. Numerical simulation and discussions
In this section, we will verify our results by numerical simulation. From Table , we obtain that , , , and by taking a=45 and at room temperature C. We choose the parameters and . If we do not release the infected with wStri, the ecosystem will be transformed into the following model (24) (24) According to Theorem 10 in [Citation10], system (Equation24(24) (24) ) has unique a positive equilibrium point , and it is globally asymptotically stable. The time series of the wild uninfected and the wild infected with wLug are shown in Figure . In the following, we will discuss the influences of the release rate β and release period T of the infected with wStri on the dynamics of systems (Equation3(3) (3) ) and (Equation6(6) (6) ).
4.1. 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 infected with wStri on the dynamics of system (Equation3(3) (3) ). We first take T=3 and , by calculating, then and , which satisfy the conditions of Theorem 3.1, the wild- -eradication periodic solution of system (Equation3(3) (3) ) is locally asymptotically stable. System (Equation3(3) (3) ) may have two steady states coexisting, the wild- -eradication period solution where the wild population will be extinct, and a positive periodic solution where the population oscillates positively and periodically as shown in Figure . These results also indicate that when the number of the wild population is small, the wild population will be controlled by regularly releasing fewer male populations infected with wStri into the field.
However, when we take T=0.5 and , then and , from Theorem 3.2, we can get that the periodic solution is globally asymptotically stable. It means that whatever the initial value is, the wild population goes to extinction for T=0.5 and , see Figure . If we take T=6 and , according to Theorem 3.4, we have , and the wild population is strongly persistent in the mean as shown in Figure . From Figures –, we can also see that increasing the release amount β of male infected with wStri or decreasing the release period T are beneficial to suppress the density of wild population.
4.2. Long-time behaviours of population replacement model
We first take and T=6, by calculating, we obtain and , there exists a locally asymptotically stable wild eradication periodic solution for system (Equation6(6) (6) ) where the wild population goes to extinction. Furthermore, by selecting appropriate initial values, the numerical simulations show that system (Equation6(6) (6) ) also exists a locally asymptotically stable positive periodic solution (see Figure ).
In addition, we consider a special case of system (Equation6(6) (6) ) that and . We obtain and by Theorem 3.5, and the equilibrium infected with wStri (0,0,1) is locally asymptotically stable as shown in Figure . It can be seen that we only need to release the infected with wStri once to the field, and it is possible to complete the replacement of wild populations.
And take and T=0.5, we can get 1−d=−9, , and . It follows from Theorem 3.6 that there exists a globally asymptotically stable wild- -eradication periodic solution as shown in Figure . Figure also shows that whatever the initial value is, the wild population will be replaced by the infected with wStri when and T=0.5.
4.3. Control of the wild within a short time
In the first two sub-sections, we fixed parameters ρ, θ, , , and , 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 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 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 population suppressed or replaced with a finite time to the initial wild amount [Citation29], denoted by e, and the calculation formula is shown by where is the amount of wild population at time t, and denotes the initial amount of wild population. We assume that , , and .
We first consider the effect of the release period T on the control efficiency of wild in a finite time . Fix parameters , , , , and vary T. We take T=1, 1.25, 2 and 2.5, respectively, Both system (Equation3(3) (3) ) and system (Equation6(6) (6) ) have stable wild- -eradication boundary periodic solution. From Figure , we can see that the control efficiency of wild decreases as T increases. For system (Equation3(3) (3) ), Figure shows that when , the control efficiency of the wild reaches more than within time , and when T=2.5, the control efficiency of the uninfected wild and the wild infected with wLug within time is only and , respectively. However, the control efficiency of the wild within time is more than for system (Equation6(6) (6) ).
And then, we fix parameters , , , , T=2 and vary β. Let and 3, there is also a stable wild- -eradication boundary periodic solution for both system. Figure shows that the control efficiency of wild increases as β increases. If we adopt a population suppression strategy, the control efficiency of uninfected wild within time is more than when , and the control efficiency of wild infected with wLug within time is more than when . However, when we adopt the population replacement strategy, the control efficiency of wild within time is more than when .
In addition, we consider an ecosystem in which the population of infected with wLug is absent, fix , , , , and the values of other parameters are the same as those in Figures and , then the time series of the control degree of this wild population under different release strategies are shown in Figure . It can be seen from Figure that no matter which strategy is adopted, the wild population will be controlled to more than in time. Comparing the results in Figures –, the wild infected with wLug can reduce the control efficiency of uninfected wild .
From Figures –, it is easy to see that under the same release amount and release cycle, the efficiency of population replacement strategy in controlling wild 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 .
Finally, we will simulate the change of control degree of wild population under a finite release number of times. Fixed parameters , , , , , T=4, , and . Based on these parameters, we discuss the situation that the number of release times of infected with wStri is 1, 2, 3 and 4 respectively, as shown in Figure Equation12(12) (12) . It can be seen from Figure Equation12(12) (12) that for the population replacement strategy, when the release number of times of infected with wStri exceeds 3, the wild population will be controlled. However, for population suppression strategy, it is impossible to achieve complete control of wild 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- -eradication periodic solution and verified them numerically in Subsections 4.1–4.2 (see Figures – and –). This also shows that both release strategies are able to control wild populations. In this paper, both release strategies assume that the 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 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 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 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 will be replaced. Since the complexity of the expressions for and , 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 faster than the population suppression strategy for the same release period T and the same amount of release β (see Figures –). Moreover, the population replacement strategy can achieve the replacement of the wild population with a finite number of releases (see Figure ). Of course, the above scenario occurs mainly because there is perfect maternal transmission of female infected with wStri. However, if the rice population density is added to models (Equation1(1) (1) ) and (Equation4(4) (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 in Section 3, but Figure shows the existence of a positive periodic solution for model (Equation3(3) (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.
5. Conclusions
Gong et al. [Citation6] successfully developed a stable artificial Wolbachia infection of by introducing the Wolbachia strain wStri from host into . The use of infected with wStri to control wild will be one of the important ways in the future. It is therefore of great interest to study how the release of infected with wStri. In this paper, we established two models with periodic impulsive release: the population suppression model (Equation3(3) (3) ) and the population replacement model (Equation6(6) (6) ). Applying Floquet theory and the comparison theorem of impulsive differential equations, we obtained the conditions for the stability of the wild- -eradication periodic solution of both models (see Figures – and –). Meanwhile, sufficient conditions for the persistence in the mean of wild uninfected and the extinction of infected with wLug were obtained, and the conditions for the persistence in the mean of wild uninfected and 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 . The population replacement strategy is obviously much better than the population suppression strategy.
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References
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