Abstract
This study introduces a novel nonlinear mathematical model by considering three dynamic variables: densities of human population and industries, and concentration of atmospheric pollutants. The formulated model assumes that the industries are established proportional to the density of human population for their livelihood, posing serious health risks due to the emission of pollutants. Also, due to atmospheric pollutants, the government mandates the relocation of industries. The formulated model exhibits a unique, two or three-interior equilibria depending on parameter values, which further elicits the robust behavior and rich dynamics. Bifurcation analysis shows the emergence of transcritical, saddle-node, and supercritical Hopf bifurcations when the establishment rate of industries crosses a critical threshold. The model is further extended using a sustainable pollution mitigation strategy ‘water spraying’, revealing the results that when water is sprayed proportional to the atmospheric pollutants, the equilibrium value of concentration of pollutants decreases under a specific condition; this minimizes the negative effect of pollution on human population and terminates the limit cycle oscillations, which leads to gain stability along with the industrialization. Also, at the fixed establishment rate of industries, the stability region enhances on increasing the spraying rate of water. It is also obtained that to reduce the atmospheric pollutants, the natural depletion rate of pollutants as well as scavenging rate of pollutants due to sprayed water should be large; and if the relocation of industries is unattainable then the minimization of pollutant’s emission due to anthropogenic or industrial activities or both together also work effectively.
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Acknowledgements
The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2026).
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The APC was supported by the Deanship of Graduate Studies and Scientific Research at Qassim University (QU-APC-2026).
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G.A.: Writing—review & editing, Writing—original draft, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. A.K.M.: Writing– review & editing, Writing—original draft, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization. A.K.A.: Writing—review & editing, Visualization, Supervision, Methodology, Investigation, Formal analysis. M.S.: Writing—review & editing, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization. All authors reviewed the manuscript.
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Agrawal, G., Misra, A.K., Agrawal, A.K. et al. Modeling the impact of aerial water spray on the dynamics of anthropogenic pollutants to sustain industrialization. Sci Rep (2026). https://doi.org/10.1038/s41598-026-42300-w
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DOI: https://doi.org/10.1038/s41598-026-42300-w
