**2. Mathematical Modeling**

In this paper, we consider carbon emission and carbon absorption together, and analyze the problem of carbon neutrality under China's industrial adjustment. With the rapid development of economy and technology, we assume that carbon emissions and absorption are in a competitive relationship as a whole; this is because in the early stage of China's economic development, the proportion of traditional industries has increased year by year. In 2007, the added value of China's secondary industry accounted for 47.6% of the total proportion. At the same time, China's clean energy development technology is not mature enough, the coal consumption is large and the utilization rate is low. In order to achieve economic growth, traditional high-carbon emission industries are developed, and natural resources are over-exploited, resulting in a significant increase in carbon emissions, immature carbon storage technology, and a corresponding reduction in carbon absorption. As the global climate is gradually warming, the greenhouse effect is obvious year on year, and mankind is facing serious natural disasters. China has gradually realized this great development idea of lucid waters and lush mountains are invaluable assets. In order to implement this correct development idea, our government is actively committed to reducing the coal proportion, improving the energy utilization rate, developing clean energy, shifting from the traditional high-carbon secondary industry to the green and sustainable tertiary industry, reducing carbon emissions, increasing the vegetation coverage, and striving to build a green city. When we only consider the competitive relationship between carbon emission and absorption, we can obtain the following model,

$$\begin{cases} \frac{dx(t)}{dt} = x(t)(a\_1 - \frac{a\_1 S\_1}{N\_2} y(t)),\\ \frac{dy(t)}{dt} = y(t)(a\_2 - \frac{a\_2 S\_2}{N\_1} x(t)), \end{cases} \tag{1}$$

where *a*<sup>1</sup> represents the annual growth rate of carbon emission, *a*<sup>2</sup> represents the annual growth rate of carbon absorption, *x*(*t*) represents China's carbon emission amount at time *t*, *y*(*t*) represents China's carbon absorption amount at time *t*, *N*<sup>1</sup> means the maximum capacity of carbon emissions and *N*<sup>2</sup> means the maximum capacity of carbon absorption. *S*<sup>1</sup> means the competition coefficient coefficient of carbon emissions relative to carbon absorption and *S*<sup>2</sup> represents the competition coefficient of carbon absorption relative to carbon emission.

We think that adding carbon adsorption saturation term to the model will make the model more realistic. This is because China has a vast territory, diverse climates, wide latitudes, and a large distance from the sea. In addition, the terrain is different, and the terrain types and mountain ranges are diverse, which leads to various combinations of temperature and precipitation and different combinations of water temperatures form different types of forest vegetation. This is because the net carbon absorbed by each vegetation is the same under certain conditions every year. Furthermore, from the technical point of view, we know that the progress of carbon storage technology promotes the increase of carbon absorption, but with the relative backwardness of technology, the carbon storage technology will improve relatively slowly, resulting in the decrease of the change rate of carbon storage.

Considering that the dual wheels of optimizing industrial structure and energy structure proposed by Guo [10] could make great contributions to national emission reduction, we can use the quadratic function simulated by previous articles to express the relationship between time and carbon emissions. We can assume that the distance between annual carbon emissions and peak carbon emissions represents the speed of carbon emissions, which is reasonable, because the smaller the distance between them, the larger the carbon emissions, and the smaller the slope of the curve. In practice, it shows that as the industrial structure is gradually transferred from the secondary industry to the tertiary industry, the energy structure is also changed from coal-based primary energy to natural gas-based clean energy, and the carbon emission changes slowly.

We know that there will be a series of processes from the transformation of industrial structure and technology research and development to the application of technology in time production, which will take a certain amount of time. If the relationship between carbon emission and carbon absorption in 2022 is simulated, the carbon emission reduction technology in 2022 will increase compared with the carbon emission when the technology is mature, because the carbon emission reduction technology is just successful but immature. Therefore, we should choose the distance from the peak to the carbon emissions before 2022 as the factor that will affect the carbon emissions in 2022. Increased investment from the government in carbon emission reduction technologies and the rapid development of carbon emission reduction technologies will accelerate the transformation of industrial structure and energy structure, as well as increasing the efficiency during the period of putting into use. For this reason, we establish the following model, the descriptions of these parameters are given in Table 1, and we note that these parameters are all positive,

$$\begin{cases} \frac{d\mathbf{x}}{dt} = \mathbf{x}(a\_1 - \frac{a\_1 S\_1}{N\_2}\mathbf{y}) + k(m - \mathbf{x}(t - \tau)),\\ \frac{d\mathbf{y}}{dt} = \mathbf{y}(a\_2 - \frac{a\_2 S\_2}{N\_1}\mathbf{x} - \frac{a\_2}{N\_2}\mathbf{y}). \end{cases} \tag{2}$$

For convenience, we denote that

$$c\_1 = \frac{a\_1 S\_1}{N\_2},\ b\_2 = \frac{a\_2 S\_2}{N\_1},\ c\_2 = \frac{a\_2}{N\_2}.$$

then, model (2) becomes

$$\begin{cases} \frac{d\mathbf{x}}{dt} = \mathbf{x}(a\_1 - c\_1\mathbf{y}) + k(m - \mathbf{x}(t - \tau)),\\ \frac{d\mathbf{y}}{dt} = \mathbf{y}(a\_2 - b\_2\mathbf{x} - c\_2\mathbf{y}). \end{cases} \tag{3}$$

,

According to the initial condition of the system (3), we present a theorem about the nonnegtivity of solution of the system (3).

**Theorem 1.** *If x*(0) > 0, *y*(0) > 0*, the solution x*(*t*), *y*(*t*) *of the system* (3) *with τ* = 0 *is positive.*


**Table 1.** Descriptions of variables and parameters in the model (2).

**Proof.** First, we prove *y*(*t*) > 0 when *t* > 0 under the positive initial condition of the system (3) with *τ* = 0.

We assume that *y*(*t*) is not always positive for *t* > 0 and make *t*<sup>1</sup> be the first time that *y*(*t*1) = 0, *y* (*t*1) < 0. According to the second equation of the system (3), we can obtain *y* (*t*1) = 0. The two conclusions we obtain are contradictory. Therefore, under the positive initial condition, the solution *y*(*t*) of the system (3) is positive for *t* > 0. Then, we prove *x*(*t*) > 0 when *t* > 0 under the positive initial condition of the system (3) with *τ* = 0. We assume that *x*(*t*) is not always positive for *t* > 0 and make *t*<sup>2</sup> be the first time that *x*(*t*2) = 0, *x* (*t*2) < 0. According to the first equation of the system (3), we can obtain *x* (*t*2) = *km* > 0. The two conclusions we reach are contradictory. Therefore, under the positive initial condition, the solution *x*(*t*) of the system (3) with *τ* = 0 is also positive for *t* > 0.

**Remark 1.** *We prove if x*(0) > 0, *y*(0) > 0*, the solution x*(*t*), *y*(*t*) *of the system* (3) *with τ* = 0 *is positive. It is also not easy for us to prove the solution of the system* (3) *is positive when τ* > 0*. However, according to the numerical simulation of a group of real parameters, we find that the solution of the system* (3) *is always positive, which is not contradictory to the positivity of the solution of the system* (3)*.*

Next, we will consider the dynamics phenomena of the system (3).
