Accurate reporting and prediction of PM2.5 concentration is very important for improving public health. In this article, we use spectral clustering algorithm to cluster 15 cities in the Pearl River Delta. On this basis, we propose a special difference equation model, especially the use of nonlinear diffusion equations to characterize the temporal and spatial dynamic characteristics of PM2.5 propagation between and within clusters for real‐time prediction. For example, through the analysis of PM2.5 concentration data for 91 consecutive days in the Pearl River Delta, and according to different accuracy definitions, the average prediction accuracy of the difference equation model in all city clusters is 97% or 88%. The mean absolute error (MAE) of the forecast data for each urban agglomeration is within 7 units ( ). Experimental results show that the difference equation model can effectively reduce the prediction time and improve the prediction accuracy. Therefore, based on the spectral clustering algorithm and the difference equation model, the fastest prediction speed and the best prediction result can be obtained, and the problem of PM2.5 concentration prediction can be effectively solved. The research can provide decision support for local air pollution early warning and urban integrated management.

<abstract><p>In this paper, we investigate the complex dynamics of a classical discrete-time prey-predator system with the cost of anti-predator behaviors. We first give the existence and stability of fixed points of this system. And by using the central manifold theorem and bifurcation theory, we prove that the system will experience flip bifurcation and Neimark-Sacker bifurcation at the equilibrium points. Furthermore, we illustrate the bifurcation phenomenon and chaos characteristics via numerical simulations. The results may enrich the dynamics of the prey-predator systems.</p></abstract>

Accurate reporting and prediction of PM 2.5 concentration are very important for improving public health. In this article, we use a spectral clustering algorithm to cluster 44 cities in the Bohai Rim Region. On this basis, we propose a special difference equation model, especially the use of nonlinear diffusion equations to characterize the temporal and spatial dynamic characteristics of PM 2.5 propagation between and within clusters for real-time prediction. For example, through the analysis of PM 2.5 concentration data for 92 consecutive days in the Bohai Rim Region, and according to different accuracy definitions, the average prediction accuracy of the difference equation model in all city clusters is 97% or 90%. The mean absolute error (MAE) of the forecast data for each urban agglomeration is within 7 units μg / m 3 . The experimental results show that the difference equation model can effectively reduce the prediction time, improve the prediction accuracy, and provide decision support for local air pollution early warning and urban comprehensive management.

In this paper, we study the stability and bifurcation analysis of a class of discrete-time dynamical system with capture rate. The local stability of the system at equilibrium points are discussed. By using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Hopf bifurcation in the interior of R+2 are proved. The numerical simulations show that the capture rate not only affects the size of the equilibrium points, but also changes the bifurcation phenomenon. It was found that the discrete system not only has flip bifurcation and Hopf bifurcation, but also has chaotic orbital sets. The complexity of dynamic behavior is verified by numerical analysis of bifurcation, phase and maximum Lyapunov exponent diagram.

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This study elucidates the sufficient conditions for the first-order nonlinear differential equations with periodic coefficients and time-varying delays to have positive periodic solutions. Our results are proved using the Krasnosel’skii fixed point theorem. In this article, we have identified two sets Δ \Delta and ∇ \nabla and proved that at least one positive periodic solution exists in the interval between the point belonging to Δ \Delta and the point belonging to ∇ \nabla . We propose simple conditions that guarantee the existence of sets Δ \Delta and ∇ \nabla . In addition, we obtain the necessary conditions for the existence of positive periodic solutions of the first-order nonlinear differential equations when the periodic coefficients satisfy certain conditions. Finally, examples and numerical simulations are used to illustrate the validity of our results.

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