A dynamical study to combat atmospheric pollutants using aerial water spray and sustain industrialization

In pursuit of global development and civilization, the accelerated pace of industrialization has brought significant atmospheric challenges, posing serious risks to public health. This study examines the implications of industrial establishment and relocation/closure, focusing on the emission of atmospheric pollution, all associated with the human population. To capture the interwoven dynamics, a novel three-dimensional nonlinear mathematical model is formulated. Bifurcation analysis of the model reveals complex dynamical behaviors, including the emergence of transcritical, saddle-node, subcritical, and supercritical Hopf bifurcations. Critical thresholds are identified for the establishment rate (and the relocation/closure rate) of industries, beyond which the system loses (and gains) its stability. To overcome with the industrial and anthropogenic atmospheric pollution, the model system is extended by introducing a new dynamic variable representing aerial water spraying as a pollution removal technique. The extended system significantly suppresses bifurcations, terminates limit cycle oscillations, and promotes asymptotic stability. Further findings reveal that increasing the water spraying rate reduces the concentration of atmospheric pollutants and simultaneously increases the densities of human population and industries. The system's stability region expands as the spraying rate increases, along with the increasing industrial establishment rate. Also, the concentration of atmospheric pollutants attains a minimum when both the aerial water spraying rate and the scavenging rate of atmospheric pollution through aerial water spraying are maximized. The study further identifies an optimal control strategy for aerial water spraying that minimizes the atmospheric pollutants at the lowest possible cost.