1.Introduction

In our work, we tested the effectiveness of our product using the Plagiodera versicolora .In order to better predict and guide the experiment, we established a mathematical population model to predict changes in the population of the Plagiodera versicolora. Unlike other biological populations, insects undergo several distinct insect states throughout their lifetime. Therefore, we establish a mathematical model of the population dynamics of the Plagiodera versicolora with an age structure, discuss its population changes, and predict its growth trend.

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2.Model

Due to the fact that population growth is not only related to the total population size, but also to the age structure of the population, we have established a discrete model that considers the age structure of the population, the Leslie matrix model. Based on existing experimental data, the Leslie model was used to simulate the population dynamics and quantity prediction of the willow leaf beetle, and compared with measured data.

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3. Research content

1. Assuming no consideration is given to the immigration and emigration of the Plagiodera versicolora population.

2. Assuming that the fertility of each adult during the same period is equal.

3. Assuming that the Liulan leaf beetle population has a stable age structure.

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4. Parameters

Parameter	Representative
$$\begin{array}{rl}& n_i(k)\end{array}$$	Population in the i-th insect state on day k
$$\begin{array}{rl}& n_{i,j}(k)\end{array}$$	Population in the i-th insect state on day k, age j
$$\begin{array}{rl}& a_{i,j}(k)\end{array}$$	The transfer rate of individuals aged j days in the i-th stage to the next age
$$\begin{array}{rl}& a_{i,j}(k)\end{array}$$	The transfer rate of individuals aged j days in the i-th insect state to the next insect state

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5. Model establishment and solution

According to age, the willow orchid leaf beetle is divided into six insect states: egg, first instar, second instar, third instar, pupa, and adult：

$$\mathrm{n}(\mathrm{k})=\left(\begin{array}{c}{n}_1(k)\\ n_2(k)\\ \cdots \\ n_5(k)\\ n_6(k)\end{array}\right)$$

Represents the population vector of the willow orchid beetle on day k, and each insect state consists of individuals of different ages:

$$\mathrm{n}(\mathrm{k})=\left(\begin{array}{c}{n}_1(k)\\ n_2(k)\\ \cdots \\ n_5(k)\\ n_6(k)\end{array}\right)$$

In the age group of the i-th insect state, there are individuals aged 0 to days (i=1,..., 6).

According to the fact that individuals aged 0 days on day k+1 of each insect state (i.e. individuals who have just entered this insect state) are derived from the development of an individual in the insect state on day k, while individuals of other ages are derived from the development of individuals in the insect state on the previous day, we can obtain the following recurrence formula:

$$\{\begin{array}{c}{n}_{i+1,0}(k+1)=\sum _{i=0}^{d_i-1}{b}_{i,j}{n}_{ij}(k),\mathrm{j}=0,\dots ,d_i-1\\ n_{i,j+1}(k+1)=a_{i,j}{n}_{ij}(k),\mathrm{i}=1,\dots ,6\end{array}$$

Population age transfer matrix and worm state transfer matrix were introduced：

$$A_i=\left(\begin{array}{ccccc}0& 0& \cdots & 0& 0\\ a_{i,0}& 0& \cdots & 0& 0\\ 0& a_{i,1}& \cdots & 0& 0\\ 0& 0& \ddots & 0& 0\\ 0& 0& \cdots & a_{i,d_i}-1& 0\end{array}\right)B_i=\left(\begin{array}{ccccc}{b}_{i,0}& b_{i,1}& \cdots & b_{i,d_i-1}& b_{i,d_i}\\ 0& 0& \cdots & 0& 0\\ 0& 0& \cdots & 0& 0\\ 0& 0& \ddots & 0& 0\\ 0& 0& \cdots & 0& 0\end{array}\right)$$

Among them, represents the transfer rate of individuals aged j days in the i-th insect state to individuals aged j+1 days, and the matrix achieves internal transfer within the i-th insect state; The matrix achieves the transition from the i-th wormstate to the i-th+1 wormstate. Below, we will use the age transition matrix and the worm state transition matrix to jointly construct the Leslie matrix vector:

$$\mathrm{L}=\left(\begin{array}{ccccc}{A}_1& 0& \cdots & 0& {\mathrm{B}}_6\\ {\mathrm{B}}_1& A_2& \cdots & 0& 0\\ 0& {\mathrm{B}}_2& \cdots & 0& 0\\ 0& 0& \ddots & A_5& 0\\ 0& 0& \cdots & {\mathrm{B}}_5& A_6\end{array}\right)$$

Transforming the equation into matrix form yields:

$$\mathrm{n}(\mathrm{k}+1)=\mathrm{Ln}(\mathrm{k})$$

Approach：

$$\left(\begin{array}{c}{n}_1(k+1)\\ n_2(k+1)\\ \cdots \\ n_5(k+1)\\ n_6(k+1)\end{array}\right)=\mathrm{L}\left(\begin{array}{c}{n}_1(k)\\ n_2(k)\\ \cdots \\ n_5(k)\\ n_6(k)\end{array}\right)$$

Recursive：

$$n_i(k+1)=l^{t+1}{n}_i(\mathrm{k}),\mathrm{i}=1,\dots ,6$$

After obtaining experimental data on population development duration, survival rate, and fertility, we used software MATLAB 2021a to simulate the Leslie model mentioned above, and compared the calculated results with experimental data to obtain a simulation effect diagram of the Leslie model.

Our model fits well with the data collected in the laboratory, and it can be seen that the survival rate of the willow blue leaf beetle is between 60% -70% under laboratory conditions. In future experiments, we will introduce a new variable based on this, which is to use our product. Comparing the results before and after introducing this factor can reflect the effectiveness of our product to a certain extent.

The observations of Plagiodera versicolora's development in each experimental group aligned with the predictions of our population dynamics model. Analysis of the data revealed that Pseudomonas B3-3G had a certain impact on the pupation and feathering of Plagiodera versicolora, leading to a moderate increase in mortality rate. However, the effect was not significant. In contrast, the dsRNA expression vector B3-3G+KC5, derived from B3-3G, significantly increased Plagiodera versicolora's mortality rate and hindered its growth, as evident from the growth curve. Additionally, knockout of ribozyme III further enhanced the insect resistance of engineered bacteria. The experimental group coated with B3-P+KC5 exhibited a notable increase in insect mortality rate. In the control group, normal circumstances led to peak pupation on the fourth day, whereas in the B3-P+KC5 experimental group, the peak occurred on the sixth day, indicating delayed development. The results of B3-G3+KC6 and B3-P+KC6 in the experimental group indicated that the GFP protein did not affect the insect resistance of the engineered bacteria.

Overall, the results of our biological tests aligned with our expectations. Through our experimental design, we successfully achieved the synergistic result of Pseudomonas chlororaphis insect-resistant coRNAi, demonstrating that 1+1>2.

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1. Introduction

Considering the biosafety issues, besides incorporating the lactose operator in the constructed plasmid, we also inserted the KillerRed gene, which encodes a phototoxic red fluorescent protein, to ensure that the engineered bacterial strain would have difficulty surviving in the natural environment. The fluorescent pigment of KillerRed can generate reactive oxygen species (ROS) when stimulated by visible light. ROS compounds are highly reactive and toxic, causing cell damage and subsequent cell death.

Experimental evidence suggests that among numerous phototoxic proteins, KillerRed exhibits the most significant lethal effect. We have validated the lethal effect of KillerRed through experiments and described its effect using ordinary differential equations.

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2.Mathematical Model

Considering the death process of bacterial populations under light conditions, we can establish the following ordinary differential equation model:

$$\frac{\mathrm{d}[N]}{\mathrm{d}\mathrm{t}}=-k[N]$$

Points earned:

$$\begin{array}{rl}& -\frac{N_t}{N_0}=e^{-kt}\end{array}$$

Formula:

$$\begin{array}{rl}& t=\frac{1}{K}\ln \frac{N_0}{Nt}=\frac{2.303}{K}\lg \frac{N_0}{Nt}\end{array}$$

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3. Parameter Definitions

N: Number of viable bacteria

t: Duration of light exposure

k: Bacterial mortality rate

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4. Equation Solving

We used the "ode45" function in MATLAB to solve the differential equation. The obtained results were compared with the actual data, and the error was calculated. The "fminsearch" function was used to fit the unknown parameter k, minimizing the error. Since there may be multiple local optimal solutions, multiple fittings were performed to find the global optimal solution. Based on the fitting results, the differential equation was solved again to obtain the final model, and the fitting curve was plotted as shown in Figure (). The fitted value of parameter k was found to be 0.07.

Exponential Fit:

We used the curve fitting toolbox in MATLAB to fit the given data. After comparing the R2 and root mean square error (RMSE) of different approximation results, we found that exponential approximation provided the best fit. The adjusted R2 value was as high as 97.03%, and the RMSE was 5.883. The final expression for the fitted curve is:

$$f(x)=98.75\ast {\exp }^{\wedge }((-0.06922)\ast x)$$

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5. Conclusion

The model is similar to the logarithmic residue model of high-temperature sterilization bacteria. From the experimental and predicted results, it can be observed that after introducing KillerRed, a short period of light exposure is sufficient to kill the majority of bacteria, although not as thorough as high-temperature sterilization, which can eliminate almost all bacteria. This suggests that bacterial death caused by the biological termination switch is relatively mild compared to physical extermination methods, but it still meets the requirements of biosafety. Additionally, for used engineered strains, high-temperature sterilization is employed to further ensure biosafety.

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