Dynamic analysis of a delayed di ff erential equation for Tropidothorax elegans pests

: In this paper, we establish an infectious disease model of Tropidothorax elegans to study the impact of them on plants. Our model involves the time delay for Tropidothorax elegans to hatch eggs, which is influenced by temperature. Second, we theoretically analyze the existence and the stability of the equilibrium and the normal form near the Hopf bifurcating critical point. Next, we choose three groups of parameters for numerical simulations to verify theoretical analysis of our model. Then, based on numerical simulations, we give bioanalysis which are consistent with the patterns of Tropidothorax elegans pests, such as dying o ff in large numbers of adults during the winter and one or two generations a year.


Introduction
In the 21st century, the trend of global warming is becoming more and more pronounced, and industrial gases such as carbon dioxide are a major culprit in global warming.Forests play a key role in carbon sequestration to mitigate the process of global warming, and the capacity of forests to sequester carbon is assessed in [1,2].To achieve the double carbon goal, one aspect is to protect forests from forest diseases and pests.Therefore, many scholars are studying ways to protect the forest to avoid the damage of forest diseases and pests.The key is to master the patterns of forest pest and disease outbreaks.
To find the patterns of forest pest and disease, studying the effects of factors such as meteorological factors and human intervention on the outbreaks of forest diseases and pests is significant.Various models of forest diseases and pests were studied [3][4][5][6].Many scholars provided theoretical knowledge for modeling analysis [7][8][9].In the work of Yao et al. [4], the relation of environment meteorological conditions with the area of forest diseases and pest was clearly studied by using the method of abnormal index analysis.Kiewra et al. [5] established the weather research and forecasting model to show the influence of different meteorological variables.Likewise, Wang et al. [6] used a MaxEnt model and incorporated annual meteorological and human activity factors to study the outbreaks of pine wilt disease.Obviously, meteorological condition is an important factor in outbreaks of forest pest and disease.
In addition, human intervention is also a key factor affecting forest pests and diseases.In References [10][11][12][13], scholars studied many managements to control forest pest and disease, such as using the GIS-based system which is a geographic information system to project the outbreaks of pest, using the skill of genetically engineered baculoviruses for pest, using chemical control and biological control and so on.These are efforts made by people to control pests.
In this paper, we focus on pests caused by the Tropidothorax elegans.This is a class of Lygaeidae with an extremely wide distribution.The appearance and biological habits of Lygaeidae are studied [14,15].All pests of Lygaeidae are a class of stinging pests and harm plants through their mouthparts.They reproduce one or two generations per year, get through the winter as adults and lay eggs in the following year.During the period of damage, both Tropidothorax elegans nymphs and adults suck the leaves, stem and roots of plants.The sucked parts of the leaves will cause the plant to be infected with sooty blotch under the action of some spores, forming a black spot on the leaves, which will greatly affect the photosynthesis of the plant and prevent the plant from absorbing nutrients.This mode of transmission is similar to infectious diseases, and many scholars studied the transmission patterns of many types of infectious diseases and predicted the development of the infectious disease by establishing the model of infectious diseases [16][17][18].Some other scholars gave different measures based on the model analysis to block the spread of the virus [19][20][21][22].
After gathering preliminary knowledge of Tropidothorax elegans and the current research status, we establish a model about Tropidothorax elegans and arrange the rest of the context as follows.In Section 2, according to the effect of Tropidothorax elegans on plants, we establish an infectious disease model on Tropidothorax elegans.In Section 3, we discuss the stability of the equilibrium point of the model and the existence of the Hopf bifurcation.In Section 4, we calculate the normal form of the Hopf bifurcation.In Section 5, we use numerical simulations to verify the correctness of bioanalysis.Finally, we draw the corresponding conclusions in Section 6.

Mathematical modeling
Based on the stinging characteristics of Tropidothorax elegans, we assume that a susceptible plant becomes an infected plant when it is sucked by Tropidothorax elegans.Therefore, we divide all plants into two groups.One group is susceptible plants (S (t), 10 4 plants), which have not been harmed by Tropidothorax elegans.The other group is infected plants (I(t), 10 4 plants), i.e., from susceptible plants that are sucked by Tropidothorax elegans (S (t) → I(t)).The infected plants have two types of changes: somes become healthy after treatment in time, and the others will die without timely treatment.Besides, the recovered plants can be reinfected again.We denote the number of Tropidothorax elegans as X(t) (10 6 pests).Based on above information, the relationship between the variables can be given in Figure 1.In addition, the hatching time of Tropidothorax elegans is also an important factor.We make the hatching time of Tropidothorax elegans as the time delay in our study.Subject to resource constraints, we view the reproductive capacity of Tropidothorax elegans as a decreasing function related to the number of prehatch Tropidothorax elegans.Thus, we get α(t) = α 1 (1 − X(t−τ) N ), where N is the environmental capacity of Tropidothorax elegans and τ is the time delay of the pest hatching period (year).We establish the model as follows: where α 1 , α 2 , β 1 , β 3 , γ, θ, N are nonnegative parameters.The specific descriptions are given in Table 1.In this table, all variables and parameters are nonnegative.

Dynamical analysis
In this section, the system (2.1) is considered.When α 2 = 0, we calculate a nonisolated boundary solution E 0 = (S 0 , 0, 0), where S 0 is an arbitrary value.It means that there is no local infestation of Tropidothorax elegans.Therefore, there are no infected plants.However, we focus on the presence of Tropidothorax elegans.Thus, We consider the following assumption: When (A1) holds, we can obtain the unique and positive equilibrium of system (2.1): where . Since α 1 represents the reproductive capacity of Tropidothorax elegans and β 1 represents the mortality rate of Tropidothorax elegans, (A1) implies that the reproductive capacity of insects is much greater than the mortality rate of insects in fact.
Proof.First, we prove X(t) ≥ 0 when t ≥ 0 under the initial condition of system (2.1).We assume that X(t) ≥ 0 is not always nonnegative for t ≥ 0 and make t 1 be the time that X(t 1 ) = 0 and X ′ (t 1 ) < 0 hold.
According to the third equation of system (2.1), we can obtain X ′ (t 1 ) = 0.The two conclusions are contradictory.Therefore, X(t) ≥ 0 when t > 0.

Stability analysis of equilibrium and the existence of Hopf bifurcation
In this section, we consider the stability and the existence of Hopf bifurcation of E. We can obtain the characteristic equation of system (2.1) at E as follows: When τ = 0, it becomes: 3) has three roots: let λ 1 , λ 2 be the roots of the part of quadratic equation in Eq (3.3), (A1) holds and other parameters are positive.Thus, Eq (3.3) has three negative roots and E is locally asymptotically stable for τ = 0.

Normal form of Hopf bifurcation
In this section, we derive the normal form of Hopf bifurcation for the system (2.1) by using the multiple time scales method.To reflect the actual situation, we focus on the delay in hatching and study the patterns of Tropidothorax elegans pests.We consider the time-delay τ as a bifurcation parameter.Let τ = τ c + εµ, where τ c = τ ( j) ( j = 0, 1, 2, • • • ) is the Hopf bifurcation critical value, µ is perturbation parameter and ε is dimensionless scale parameter.When τ = τ c , the Eq (3.4) has eigenvalue λ = iω (ω > 0), at which system (2.1) undergoes a Hopf bifurcation at equilibrium E.
To translate the equilibrium point E = (S * , I * , X * ) to the origin and simplify the expression, we write S as S , Ĩ and X are the same.At the same time, to normalize the delay, we let t → t τ rescale the time.Then, we obtain following equations: We write Z(t) = (S (t), I(t), X(t)) T , Z(t − 1) = (S (t − 1), I(t − 1), X(t − 1)) T .The Eq (4.1) can be written as: where Then, we make h the eigenvector corresponding to eigenvalue λ = iωτ of the linear section of Eq (4.2) and h * the eigenvector corresponding to eigenvalue λ = −iωτ of adjoint matrix of the linear section of Eq (4.2).Then, h and h * should satisfy ⟨h * , h⟩ = h * T h = 1.According to the above conditions, we obtain: The solution of Eq (4.2) is assumed as follows: where The derivative, with regard to t, is transformed into where D i is differential operator, and From Eq (4.4), we can obtain Then, using the Taylor expansion to expand where Combing in Eqs (4.4)-(4.6)and separating the coefficients of ε, ε 2 and ε 3 on both sides of the equation, the expressions of the coefficient before ε, ε 2 and ε 3 are shown from (4.7)-(4.9): (4.9) The solution of Eq (4.7) can be assumed as: where h is given by Eq (4.3).We substitute Eq (4.10) into the right side of the Eq (4.7) and write the coefficient of e iωτ c T 0 as m 1 .According to the solvability condition ⟨h * , m 1 ⟩ = 0, the expression of ∂G ∂T 1 is obtained as follows: where Since µ is a perturbation term, its effect on higher order is smaller.Thus, we only consider its effect on the linear part and ignore its effect on higher order.We assume that the solution of Eq (4.8) is as follows: where and h 1 , h 2 , h 3 is given by Eq (4.3).
Then we substitute Eq (4.12) into the right side of Eq (4.9), and the coefficient vector of e iωτ c T 0 is denoted by m 2 .According to the solvability condition ⟨h * , m 2 ⟩ = 0, the expression of ∂G ∂T 2 can be obtained as follows: where and h 3 , g 3 , l 3 are given by Eqs (4.3) and (4.12).Let G → G/ε.Then, the deduced third-order normal form of Hopf bifurcation of system (2.1) is: where M is given in Eq (4.11) and H is given in Eq (4.13).Substituting G = re iθ into Eq (4.14), the following normal form of Hopf bifurcation in polar coordinates is obtained: Based on the normal form of Hopf bifurcation in polar coordinates, we only need to consider the first equation in system (4.15).Thus, the following theorem holds: Re(H)α 1 τ c < 0, the system (4.15)exits a semitrivial fixed point r = − Re(M)µ Re(H)α 1 τ c , and the system (2.1) has a periodic solution.1) If Re(M)µ < 0, the periodic solution reduced on the center manifold is unstable.
2) If Re(M)µ > 0, the periodic solution reduced on the center manifold is stable.

Numerical simulations and bioanalysis
In this section, we choose three groups of parameters to simulate the effects of temperature changes and artificial interventions on the model.Then, we give the corresponding biological explanations based on the simulations.Finally, we discuss what humans should do to reduce the harm caused by Tropidothorax elegans.

Determination of parameter values
First, we choose the first group of parameters to describe the situation that Tropidothorax elegans grow freely.Second, considering the context of global warming, we choose the second group of parameters to simulate the effects of global warming on Tropidothorax elegans pests.At higher latitudes, the increase in temperature will lead to a decrease in the number of adults that freeze to death in winter, and the best temperature for the survival of Tropidothorax elegans can be reached more easily, so the incubation time can be shortened [23].At lower latitudes, increased temperatures will lead to a decrease in the mortality rate of adults, so there will be more Tropidothorax elegans laying eggs in next year.Thus, globally higher temperature leads to lower mortality and higher reproduction of Tropidothorax elegans around the world.Finally, we take the third group of parameters to simulate the effect of timely artificial control on Tropidothorax elegans pests.Timely artificial control can increase the mortality rate of Tropidothorax elegans [24].Thus, increasing the recovery rate of infected plants and reducing the mortality rate of infected plants.
In addition, α 2 is the replenishment rate of plants, β 2 is the mortality rate of infective plants and θ is the transition rate from I(t) to S (t).In fact, because people find plants dead and promptly replenish them with the same number of plants, we consider the replenishment rate to be the same as the mortality rate of plants.The infected plants have two types of changes: some become healthy after treatment in time and the others will die without timely treatment.Thus, we consider α 2 + θ = 1.
In Figure 2, S , I, X become stable after a short period of variations.In the early stage, the number of Tropidothorax elegans increases rapidly, and peak after a period of time.Since then, the number of Tropidothorax elegans begins to decline.We suspect this is because limited resources act as a constraint on their growth.After a short period of intraspecific competition, the number of Tropidothorax elegans becomes relatively stable.This also means that when the incubation time of the Tropidothorax elegans is small enough, the Tropidothorax elegans will remain locally forever.This may happen at low latitude regions where temperature is high enough.
For τ = 0.18 > τ c , we calculate the normal form of Hopf bifurcation and obtain Re(M) > 0, Re(H) < 0 from Eq (4.15).According to the Theorem 4.1, the periodic solution is stable when µ > 0. The solution is shown in Figure 3.
Clearly, S , I, X all form the stable periodic solutions in Figure 3.This means that this pest has become an endemic disease and is difficult to eradicate with τ = 0.18.In Figure 3(c), the infestation period of Tropidothorax elegans is about one month and outbreaks once a year which is consistent with the habits of the Tropidothorax elegans (http : //museum.ioz.ac.cn/topic d etail.aspx?id = 7358).
In addition, the number of Tropidothorax elegans tends to zero for a period of time.Then, it gradually increases again, repeating the trend of the first cycle.The biological meaning is that Tropidothorax elegans spend the winter as adults, the number of Tropidothorax elegans decreases considerably due to the low temperature.However, since Tropidothorax elegans go through the winter everywhere, such as near plants, on the backs of plant leaves, under rocks or in soil burrows, it is difficult to kill.Therefore, the number of the Tropidothorax elegans will increase again in next year because of their strong reproductive ability, repeating the experience of the previous year.Besides, the external environment is stable, so the environmental capacity rarely changes and the peak value remains constant.For the group of parameters (2): We calculate the equilibrium E = (S * , I * , X * ) = (0.789983015, 1, 0.97) and τ c = 0.08093.We know that E is locally asymptotically stable for τ ∈ (0, τ c ) and unstable for τ > τ c .Similar to the solution of the first group of parameters when τ = 0.18, the periodic solution of the second group of parameters with τ = 0.12 > τ c is also stable when µ > 0. We choose the initial functions [S (θ), I(θ), X(θ)] = [0.78,0.99, 0.96] (θ ∈ [−τ c , 0]) for simulation and the result of simulation is shown in Figure 4.
The overall trend is the same as Figure 3, the actual situation of these parameters is: under the background of global warming, temperature has increased.This leads to a lower natural mortality rate and a higher reproduction capacity of Tropidothorax elegans, which are more suitable for their survival.The time of the number of Tropidothorax elegans tends to zero is significantly shorter in Figure 4(c).That means that the frequency of infestation is significantly higher and the damage to the forest is greater.Thus, the increase of temperature will lead to the increase of damage frequency and damage degree of Tropidothorax elegans.

Electronic Research Archive
Volume 31, Issue 11, 6947-6963.The Figure 5 corresponds to the actual situation: after people found this pest, people will kill pests in time.Obviously, the more timely the manual control, the longer it takes the number of Tropidothorax elegans to tend to zero, which shows that the manual control has suppressed the harm of the Tropidothorax elegans to the plants.When we choose the prevention methods, we should choose the optimal control method according to the different characteristics of the pest outbreak respectively.
Based on the research [11,12,27], the advantage of biological control is that it is highly selective and safer to the environment.However, its killing effect is slow and the economic cost is high.Compared with it, the killing effect of chemical control is faster and the economic cost is low, but it will bring more environmental pollution and other hidden problems.For example, when chemical control is carried out, pests and their natural enemies are killed in large quantities, at which time the effectiveness of biological control is greatly reduced, but it does achieve the purpose of eliminating pests.With time going by, the remaining chemical reagents are reduced, and the ability to kill pests is weakened.Then, biological control is relied on to kill pests [28].Therefore, in the case of sudden outbreak of Tropidothorax elegans, we can use chemical control to prevent and control the bug urgently, and in the later stage of pest management, we can use biological control technology to reduce the pollution.
In addition, we compare the curves above three groups together as shown in Figure 6: We can see the similarities of these curves: 1) The trend of the change of the number of infected plants and the number of susceptible plants is opposite, this is because that the total number of plants is a constant.which is consistent with the actual situation.
2) In the stage of increasing number of Tropidothorax elegans, after the number of Tropidothorax elegans peaking, the number of susceptible and the number of infected plants peak.There is a delay between the time which the number of Tropidothorax elegans peaks with the time which the number of susceptible plants and infected plants peak.We analyze that this is because the replenishment rate of the plants is smaller than the reproductive capacity of Tropidothorax elegans and the mortality rate of the plants is also smaller than the mortality rate of Tropidothorax elegans, so the change of the number of plants is smaller and slower, while the change of Tropidothorax elegans will be more flexible.
3) In the first and third groups of parameters, we choose different parameters and the same τ = 0.18.In the Figure 6, we find that the two curves almost coincide.However, the time-delay of the second group of parameters is different from other groups, and the curves are different.This means that the effect of parameters variations on the patterns of Tropidothorax elegans pests is less than the effect of temperature on the patterns of Tropidothorax elegans pests.Therefore, we believe that the speed of reproduction determines the outbreak of Tropidothorax elegans.
Some differences in the Figure 6: 1) When the time-delay is the same, we compare the curves for the first group of parameters with the curves for the third group of parameters.The peak of the curves of the third group is lower than the peak of the curves of the first group in Figure 6(a),(b).In fact, this represents a reduction in the number of mortality plants through manual control during the outbreak.In addition, we compare the curves of the first and third group of parameters in Figure 6(c), we find that the breakout period of the curve of the artificial intervention is later than the breakout period of the first set of parameters.This means that artificial intervention can effectively postpone the outbreak of the pest.Therefore, we should do a good job of patrolling in the forest in time, so that the insect pests can be detected and treated in time to ensure the safety of the plants.
2) When the time-delay is different, we compare the first group of parameters with τ = 0.18 and the second group of parameters with τ = 0.12 in Figure 6, and find that the curves of the second group of parameters change more fastly.As the temperature rises, the incubation time of Tropidothorax elegans will be shortened and the number of infected plants and susceptible plants will also change rapidly.Due to resource constraints and more intense intraspecific competition, the maximum number of Tropidothorax elegans in the region has decreased.

Conclusions
In this paper, considering the characteristics of Tropidothorax elegans to sting plants, we have constructed a S IX model with a time delay for Tropidothorax elegans to hatch eggs, which is influenced by temperature and manual control.We have studied the stability of the equilibrium and the existence of Hopf bifurcation.Then, we have analyzed the stability and bifurcating direction of the Hopf bifurcating periodic solution by calculating the normal form with the multiple time scales method.
Based on three groups of parameters, we have conducted numerical simulations.First, we have simulated the change of temperature through the first and the second groups of parameters to verify that Tropidothorax elegans pests become more frequent with higher temperature and that the laws of outbreaks is once or twice a year, which is consistent with actual situation.Second, we have simulated the effects of human intervention through the first and third groups of parameters and have compared them to conclude that human intervention can delay the outbreak of Tropidothorax elegans pests and reduce the number of Tropidothorax elegans at the peak of the outbreak.Further more, based on the analyzed characteristics of Tropidothorax elegans, we could choose different control methods and apply the right medicine to the right situation.In the future, we will further break down the effects of different control methods on this pest, focusing on two methods: pesticide control and natural enemy control, so that we could remove pests more efficiently and protect the environment at the same time.

Figure 3 .
Figure 3.When τ = 0.18, there exists a stable periodic solution near equilibrium E.

Figure 4 .
Figure 4.When τ = 0.12, there exists a stable periodic solution near equilibrium E.

Figure 5 .
Figure 5.When τ = 0.18, there exists a stable periodic solution near equilibrium E.

Figure 6 .
Figure 6.Comparison of three sets of parameters.

Table 1 .
The descriptions of variables and parameters.