**1. Introduction**

Unreasonable development has caused environmental destruction, therefore, it is the general trend to protect the environment and save energy. Global collaboration promotes the rapid development of the world; immoderate use of natural resources, however, has given rise to land degradation, deforestation and biodiversity loss, and so on. Three wastes in industrial production cause soil pollution, water pollution and air pollution, the rapid development of the global secondary industry promotes the burning of fossil fuels in large quantities, and the greenhouse gas released intensifies the greenhouse effect, causing global warming. The melting of the polar ice cap causes the sea level to rise, and some river deltas with low altitude and fertile land are submerged. At the same time, it also causes seawater to pour into the harbor, which pollutes underground water sources and aggravates the salinization of land. We can know from the notice issued by "The State of Global Climate 2020" that the global average temperature in 2020 was about 1.2 degrees Celsius higher than the pre-industrial level. In the face of natural disasters, human beings are extremely helpless. In order to slow down the trend of climate warming, the United Nations adopted the United Nations Framework Convention on Climate Change in New York on 9 May 1992. In 1997, the Kyoto Protocol of the United Nations Framework Convention on Climate Change was

**Citation:** Han, L.; Sui, H.; Ding, Y. Mathematical Modeling and Stability Analysis of a Delayed Carbon Absorption-Emission Model Associated with China's Adjustment of Industrial Structure. *Mathematics* **2022**, *10*, 3089. https://doi.org/ 10.3390/math10173089

Academic Editors: Mariano Torrisi and Mustafa R.S. Kulenovic

Received: 6 July 2022 Accepted: 23 August 2022 Published: 27 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

successfully formulated, and it provided legally binding quantitative emission reduction and emission limitation targets for developed countries. In December 2017, twenty-nine countries around the world had signed the Joint Statement on Carbon Neutrality. By September 2019, 66 countries had agreed at the United Nations Climate Action Summit that lucid waters and lush mountains are invaluable assets, and had formed the Climate Ambition Alliance. All these measures have accelerated global carbon neutrality. Britain, Sweden, France, Denmark, New Zealand, Hungary and other countries have written the goal of carbon neutrality into their laws. The EU announced that it will become the first "carbon neutral" land in the world in 2050.

China is a big carbon emitter, so it is imperative to promote peak carbon dioxide emissions and achieve carbon neutrality. According to the data of the seventh national census, China's population has exceeded 1.4 billion, accounting for 21.5% of the world's total population. Abundant human resources have promoted the rapid development of the secondary industry, which is dominated by manufacturing. China's economy is developing steadily, among which the traditional manufacturing industry with high energy consumption and high carbon emission is still the main industry in China. In 2019, China's total carbon emissions reached 10.17 billion tons, accounting for 28% of the global carbon emissions, and China's industrial carbon emissions accounted for more than 50% of China's total carbon emissions. Therefore, Zhang et al. [1] suggested that adjusting the industrial structure and the energy structure have become two obstacles on the road of carbon neutrality in China. Cai et al. [2] used standard methods to calculate urban carbon dioxide emissions, and established a data set of urban carbon dioxide emissions in China. As the largest developing country in the world and a responsible big country, China passed the Energy Conservation Law of the People's Republic of China on 1 November 1997 in order to slow down the global warming trend and play a leading role among developing countries. China released the white paper "China's Policies and Actions to Address Climate Change" in October 2008. In the meantime, China also actively participates in global climate change negotiations, strengthens communication, coordination and cooperation with other countries in the world, and makes contributions to jointly addressing the challenges of climate change and promoting global sustainable development. In September 2020, China proposed at the United Nations General Assembly that carbon dioxide emissions would peak before 2030, and that it strives to achieve carbon neutrality before 2060; In 2021, at the National People's Congress, peak carbon dioxide emissions and carbon neutrality were written into the government work report for the first time. In the same year, the basic ideas and important measures to realize peak carbon dioxide emissions and carbon neutrality were put forward at the ninth meeting of the Central Committee of Finance and Economics. At the National People's Congress in 2021, peak carbon dioxide emissions and carbon neutrality were written into the government work report for the first time, and China put forward the basic ideas and important measures to realize peak carbon dioxide emissions and carbon neutrality at the ninth meeting of the Central Committee of Finance and Economics in the same year.

China is a big carbon emitting country and a big energy consumption country. In 2010, the proportion of carbon emissions from coal in primary energy accounted for about 70%. For this reason, Zou et al. [3] found that the research and development of new energy will greatly promote the realization of carbon neutralization in China. New energy has become the protagonist of the third energy transformation, and will lead the future of carbon neutrality. In [4–7], the authors suggest that developing low-carbon cities, optimizing industrial structure, reducing carbon emissions from steel industry, improving carbon emission reduction technology and reducing carbon sequestration cost are important measures for China to realize peak carbon dioxide emissions and carbon neutrality ahead of schedule.

In the field of applied mathematics, there are a few researches on China's carbon neutrality. Wang et al. [8] innovatively constructed traditional Markov probability transfer matrix and spatial Markov probability transfer matrix to explore the temporal and spatial evolution of China's urban carbon emission performance and predict the long-term trend of carbon emission performance. Chen [9] put forward the energy supply and demand model under two related carbon emission scenarios, namely, China's planned peak and advanced peak scenarios, and suggested that low carbon would be a basic feature of the change of energy supply and demand structure, and non-fossil energy would replace oil as the second largest energy source. Industrial structure and energy consumption structure all have significant influence on carbon dioxide emissions, especially industrial energy intensity. In [10], Guo used economic accounting methods to estimate the potential of China's industrial carbon emission reduction from the perspective of structural emission reduction and intensity emission reduction, and further discussed the influence of industrial internal structure adjustment and energy structure optimization on industrial carbon emission peak and emission reduction potential. According to the data, during the 20 years from 2000 to 2019, the proportion of coal decreased from the original peak of 72.5% to 57.7%, and the natural gas resources increased from 2.2% to 8.4%. Li et al. [11] used the generalized Weng model to predict the regional natural gas production in China, and the prediction results show that the peak natural gas production will reach 323 billion cubic meters per year in 2036. In [12], the scholars investigate the relationship between energy consumption, economic growth and carbon dioxide emissions in Pakistan by using the annual time series data from 1965 to 2015. The estimation results of ARDL show that energy consumption and economic growth have both increased CO2 emissions in Pakistan in the short and long term. In [13], based on China's provincial panel data from 2004 to 2016, Liu et al. empirically analyzed the impact of ecological civilization construction on carbon emission intensity by using spatial Durbin model based on STIRPAT model. The above-mentioned scholars only consider one of carbon emission and carbon absorption, but not both. We know that only by considering both of them can we accurately and reasonably forecast the time when China will achieve peak carbon dioxide emissions and carbon neutrality. In this paper, we selected industrial structure and energy consumption structure as the influencing factors of carbon emissions.

As the processes of carbon emission and carbon absorption are time-varying processes, we can describe them by continuous differential equations. Furthermore, considering that carbon emission and carbon absorption are not only related to the current time, but also to the past time, we can use the delayed differential equation model to describe the phenomenon of the dynamic system of carbon emission and carbon absorption more truly and accurately. There is a lot of research work on delayed differential equations in many fields, such as biology, medicine, physics, and so on [14–18]. At present, there are few research achievements in describing carbon emission and carbon absorption model by using delayed differential equations, so the purpose of this paper is to use delayed differential equations to describe carbon emission and carbon absorption model.

The motivation of this paper is as follows. Firstly, according to the Chinese government's goal of achieving peak carbon dioxide emissions by 2030 and carbon neutrality by 2060, we want to make some predictions and analyze whether China can achieve it under the current policy based on the carbon absorption and emission model. If there is some gap between the simulated results of the model and the ideal goal, we can put forward some policy suggestions to achieve China's peak carbon dioxide emissions carbon neutrality goal by combining the model with the actual situation. Secondly, considering that many scholars have studied the peak carbon dioxide emissions and carbon neutrality in China from different fields, but there are few related studies on the use of delayed differential equations to describe this problem, and we try to establish a carbon absorption and emission model from the perspective of delayed differential equations to analyze the problem, and analyze this problem from different angles to see if we can get new results. Thirdly, this paper establishes a two-dimensional delayed differential equation model of carbon emission and carbon absorption, which is different from models models cited in the literature [3–13]. We focus on analyzing the existence and stability of equilibrium point, and the existence of system bifurcation, and studying the long-term change process of

carbon emission and carbon absorption. The models cited in the literature [3–13] include traditional Markov probability transfer matrix and spatial Markov probability transfer matrix, energy supply and demand model, generalized Weng's model, spatial Durbin model based on STIRPAT model, etc. The above models do not analyze the amount of carbon absorption, but the two-dimensional delayed differential equation proposed by us is not only related to carbon emission, but also to carbon absorption. We also use the knowledge of differential equations to analyze the stability of the model, focusing on the long-term stability of carbon emission and absorption, and so on.

The rest of the content is arranged as follows. In Section 2, we establish a delayed carbon absorption-emission model based on the carbon emission and carbon absorption. In Section 3, we analyze the existence and stability of equilibria and the existence of Hopf bifurcation for the model with time delay. In Section 4, we derive the normal form of the Hopf bifurcation of the above model and analyze the stability of the bifurcating periodic solutions. In Section 5, we present numerical simulations to verify the correctness of our analysis. Finally, the conclusion is drawn in Section 6.
