This paper investigates the global dynamics of a broad class of nonlinear rational difference equations given by $x_{n+1}={\displaystyle \frac{a{x}_{n+1-2k}}{b+c{x}_{n+1-k}{x}_{n+1-2k}}},n=0,1,\dots ,$ which generalizes several known models in the literature. We establish the existence of exactly three equilibrium points and show that the trivial equilibrium is globally asymptotically stable when the parameter ratio $\alpha =(b/a)$ lies in $(-1,1)$. The nontrivial equilibria are shown to be always unstable. An explicit general solution is derived, enabling a detailed analysis of solution behavior in terms of initial conditions and parameters. Furthermore, we identify and classify minimal period $2k$ and $4k$ solutions, providing necessary and sufficient conditions for the occurrence of constant and periodic behaviors. These analytical results are supported by numerical simulations, confirming the theoretical predictions. The findings generalize and refine existing results by offering a unified framework for analyzing a wide class of rational difference equations.

In this study, the effects of meteorological factors, i.e., atmospheric temperature and humidity, the pollutant oxides i.e., sulfur dioxide (SO2) and nitrogen dioxide (NO2) are examined in connection to particulate matter 2.5 (PM2.5) concentrations, which is an important air pollutant that is harmful to both human health and the environment. It is essential to comprehend the variables that cause PM2.5 levels to increase in urban areas and that contribute to efficient air quality management plans. The dataset is taken from multiple monitoring sites throughout the city of Islamabad on meteorological factors along with pollutant oxides from year 2019 to 2023 containing particulate matter, nitrogen dioxide, sulfur dioxide, atmospheric temperature, and humidity from Pakistan Environmental Protection Agency, comprising the observations of PM2.5 and associated measures, i.e., NO2, SO2, atmospheric temperature, and humidity, examined using a statistical regression model. The main goal is to assess how well NO2, SO2, atmospheric temperature, and humidity can predict changes in PM2.5 concentrations. The mean value of PM2.5 is 35.26 µg/m³ (SD = 21.16 µg/m³), and the model explains 44% of the variation in particulate matter concentration. The overall model is statistically significant (F Statistic = 274.52, p < 0.01), and the study showed a statistically significant link between the response variable (PM2.5) and the predictors i.e., NO2, SO2, atmospheric temperature, and humidity (p-values < 0.01). These findings enable that air quality management strategies can offer a better comprehension of the dynamics of air pollution by offering insights into the variables impacting PM2.5 concentrations. Policymakers and environmental organizations can build focused measures to reduce and regulate PM2.5 pollution with the help of the implications of this research. It may be possible to lower PM2.5 levels and improve overall air quality by addressing the contributing elements mentioned in this study, such as NO2, SO2, atmospheric temperature, and humidity, thereby protecting public health and the environment.