# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021169
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## Modeling the influence of human population and human population augmented pollution on rainfall

 1 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi - 221 005, India 2 Department of Mathematics, Amity University, Uttar Pradesh, Lucknow, India 3 Department of Mathematical & Statistical Sciences,, Shri Ramswaroop Memorial University, Barabanki 225 003, India

* Corresponding author: kusum41@gmail.com

Received  March 2021 Revised  May 2021 Early access June 2021

Worldwide, human population is increasing continuously and this has magnified the level of pollutants in the environment. Pollutants affect the human population as well as the environmental ecology including rainfall. Here, we formulate a mathematical model comprising ordinary differential equations to see the effect of human population and pollution caused by human population on the dynamics of rainfall. In the modeling process, it is assumed that the augmentation in the density of human population increases the concentration of pollutants; however, decreases the rate of formation of cloud droplets. It is also assumed that pollutants have negative impact on human population and affect the precipitation. The feasibility of all equilibrium and their stability properties are discussed. Further, to capture the effect of environmental randomness, the proposed model is also analyzed by incorporating white noise terms. For the proposed stochastic model, we have established the existence and uniqueness of global positive solution. It is also shown that system possesses a unique stationary distribution with some restrictions. The model analysis reveals that rainfall may decrease or increase due to the anthropogenic emission of pollutants in the atmospheric environment. Finally, for the validation of analytical findings, numerical simulations are presented.

Citation: A. K. Misra, Gauri Agrawal, Kusum Lata. Modeling the influence of human population and human population augmented pollution on rainfall. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021169
##### References:
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Friedman, Stochastic Differential Equations and their Applications, Academic Press, New York, 1976.   Google Scholar [7] R. Z. Hasminskii, Stochastic stability in differential equations, in Mechanics and Analysis, {Monogr. Textb. Mech. Solids Fluids}, 7, Sijthoff $&$ Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.  Google Scholar [8] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar [9] Z. Huang, Q. Yang and J. Cao, A stochastic model for interactions of hot gases with cloud droplets and raindrops, Nonlinear Anal. Real World Appl., 12 (2011), 203-214.  doi: 10.1016/j.nonrwa.2010.06.008.  Google Scholar [10] V. Jha, W. R. Cotton, G. G. Carrió and R. 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Wang, Analysis of astochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response, Advances in Difference Equations, 224 (2015). doi: 10.1186/s13662-015-0448-0.  Google Scholar [15] X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997.  Google Scholar [16] X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar [17] A. K. Misra and K. Lata, Modeling the effect of time delay on the conservation of forestry biomass, Chaos Soltions Fract., 46 (2013), 1-11.   Google Scholar [18] A. K. Misra and A. Tripathi, A stochastic model for making artificial rain using aerosols, Physica A., 505 (2018), 1113-1126.  doi: 10.1016/j.physa.2018.04.054.  Google Scholar [19] A. K. Misra and A. Tripathi., Stochastic stability of aerosols-stimulated rainfall model, Physica A., 527 (2019), 121337. doi: 10.1016/j.physa.2019.121337.  Google Scholar [20] B. G. Pachpatte, A Note on Gronwall-Bellman Inequality, J. Math Anal. Appl., 44 (1973), 758-762.  doi: 10.1016/0022-247X(73)90014-0.  Google Scholar [21] Pollution and Health Metrics: Global, Regional, and Country, Analysis December 2019. Available from: https://gahp.net/wp-content/uploads/2019/12/PollutionandHealthMetrics-final-12_18_2019.pdf Google Scholar [22] Y. Qian, D. Gong, J. Fan, L. R. Leung, R. Bennartz, D. Chen and W. Wang, Heavy pollution suppresses light rain in China: Observations and Modeling, Journal of Geophysical Research, 114 (2009), D00K02. doi: 10.1029/2008JD011575.  Google Scholar [23] D. Rosenfeld, Suppression of rain and snow by urban and Industrial air pollution, Science, 287 (2000), 1793-1796.   Google Scholar [24] D. Rosenfeld and W. Woodley, Pollution and Clouds, 14 (2001), 33., doi: 10.1088/2058-7058/14/2/30.  Google Scholar [25] D. Rosenfeld, U. Lohmann, G. B. Raga, C. D. O. Dowd, M. Kulmala, S. Fuzzi, A. Reissell and M. O. Andreae, Flood or drought: How do aerosols affect precipitation?, Science, 321 (2008), 1309-1313.   Google Scholar [26] J. B. Shukla, M. Verma and A. K. Misra, Effect of global warming on sea level rise: A modeling study, Ecol. Complex., 32 (2017), 99-110.   Google Scholar [27] J. B. Shukla, A. K. Misra, R. Naresh and P. Chandra, How artificial rain can be produced? A mathematical model, Nonlinear Anal. Real World Appl., 11 (2010), 2659-2668.   Google Scholar [28] J. B. Shukla, S. Sundar, A. K. Misra and R. Naresh, Modeling the effects of aerosols to increase rainfall in regions with shortage, Meteorol .Atmos. Phys., 120 (2013), 157-163.   Google Scholar [29] S. Sundar and R. K. Sharma, The role of aerosols to increase rainfall in the regions with less intensity rain: A modeling study, Comput. Ecol. Softw., 3 (2013), 1-8.   Google Scholar [30] S. Sundar, R. Naresh, A. K. Misra and J. B. Shukla, A nonlinear mathematical model to study the interactions of hot gases with cloud droplets and raindrops, Appl Math Model., 33 (2009), 3015-3024.  doi: 10.1016/j.apm.2008.10.032.  Google Scholar [31] R. K. Upadhyay, R. D. Parshad, K. Antwi-Fordjour, E. Quansah and S. Kumari, Global dynamics of stochastic predator-prey model with mutual interference and prey defense, J. Appl. Math. Comput., 60 (2019), 169-190.  doi: 10.1007/s12190-018-1207-7.  Google Scholar [32] Y. Wang, M. Wang, R. Zhang, S. J. Ghan, Y. Lin, J. Hu, B. Pan, M. Levy, J. H. Jiang and M. J. Molina, Assessing the effects of anthropogenic aerosols on Pacific storm track using a multiscale global climate model, Proc. Natl. Acad. Sci. USA., 111 (2014), 6894-6899.   Google Scholar [33] B. P. Yadav, A. K. Das, K. V. Singh and S. K. Manik, Rainfall Statistics of India - $(2017)$. http://www.imd.gov.in., Google Scholar [34] C. Zhao, X. Tie and Y. Lin, A possible positive feedback of reduction of precipitation and increase in aerosols over eastern central China, Geophysical Research Letters, 33 (2006), L11814. doi: 10.1029/2006GL025959.  Google Scholar

show all references

##### References:
 [1] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York (1972).  Google Scholar [2] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101.  doi: 10.1016/j.jmaa.2007.11.005.  Google Scholar [3] B. Dubey and A. S. Narayanan, Modelling effects of Industrialization, population and pollution on a renewable resource, Nonlinear Anal. Real World Appl., 11 (2010), 2833-2848.  doi: 10.1016/j.nonrwa.2009.10.007.  Google Scholar [4] J. Fan, L. R. Leung, Z. Li, H. Morrison, H. Chen, Y. Zhou, Y. Qian and Y. Wang, Aerosol impacts on clouds and precipitation in eastern China: Results from bin and bulk microphysics, Journal of Geophysical Research, 117 (2012), D00K36. doi: 10.1029/2011JD016537.  Google Scholar [5] J. A. Foley, R. DeFries, G. P. Asner, C. Barford, G. Bonan, S. T. Carpenter and R. K. Snyder, Global consequences of land use, Science, 309 (2005), 570-574.   Google Scholar [6] A. Friedman, Stochastic Differential Equations and their Applications, Academic Press, New York, 1976.   Google Scholar [7] R. Z. Hasminskii, Stochastic stability in differential equations, in Mechanics and Analysis, {Monogr. Textb. Mech. Solids Fluids}, 7, Sijthoff $&$ Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.  Google Scholar [8] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar [9] Z. Huang, Q. Yang and J. Cao, A stochastic model for interactions of hot gases with cloud droplets and raindrops, Nonlinear Anal. Real World Appl., 12 (2011), 203-214.  doi: 10.1016/j.nonrwa.2010.06.008.  Google Scholar [10] V. Jha, W. R. Cotton, G. G. Carrió and R. Walko, Sensitivity studies on the impact of dust and aerosol pollution acting as cloud nucleating aerosol on Orographic precipitation in the Colorado river basin, Advances in Meteorology, (2018), 3041893. doi: 10.1155/2018/3041893.  Google Scholar [11] C. Ji and D. Jiang., Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.  Google Scholar [12] J. H. Jiang, H. Su, L. Huang, Y. Wang, S. Massie, B. Zhao, A. Omar and Z. Wang, Contrasting effects on deep convective clouds by different types of aerosols, Nature Communications, 9 (2018). doi: 10.1038/s41467-018-06280-4.  Google Scholar [13] K. Lata and A. K. Misra, The influence of forestry resources on rainfall: A deterministic and stochastic model, Appl Math Model., 81 (2020), 673-689.  doi: 10.1016/j.apm.2020.01.009.  Google Scholar [14] S. Li and S. Wang, Analysis of astochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response, Advances in Difference Equations, 224 (2015). doi: 10.1186/s13662-015-0448-0.  Google Scholar [15] X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997.  Google Scholar [16] X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar [17] A. K. Misra and K. Lata, Modeling the effect of time delay on the conservation of forestry biomass, Chaos Soltions Fract., 46 (2013), 1-11.   Google Scholar [18] A. K. Misra and A. Tripathi, A stochastic model for making artificial rain using aerosols, Physica A., 505 (2018), 1113-1126.  doi: 10.1016/j.physa.2018.04.054.  Google Scholar [19] A. K. Misra and A. Tripathi., Stochastic stability of aerosols-stimulated rainfall model, Physica A., 527 (2019), 121337. doi: 10.1016/j.physa.2019.121337.  Google Scholar [20] B. G. Pachpatte, A Note on Gronwall-Bellman Inequality, J. Math Anal. Appl., 44 (1973), 758-762.  doi: 10.1016/0022-247X(73)90014-0.  Google Scholar [21] Pollution and Health Metrics: Global, Regional, and Country, Analysis December 2019. Available from: https://gahp.net/wp-content/uploads/2019/12/PollutionandHealthMetrics-final-12_18_2019.pdf Google Scholar [22] Y. Qian, D. Gong, J. Fan, L. R. Leung, R. Bennartz, D. Chen and W. Wang, Heavy pollution suppresses light rain in China: Observations and Modeling, Journal of Geophysical Research, 114 (2009), D00K02. doi: 10.1029/2008JD011575.  Google Scholar [23] D. Rosenfeld, Suppression of rain and snow by urban and Industrial air pollution, Science, 287 (2000), 1793-1796.   Google Scholar [24] D. Rosenfeld and W. Woodley, Pollution and Clouds, 14 (2001), 33., doi: 10.1088/2058-7058/14/2/30.  Google Scholar [25] D. Rosenfeld, U. Lohmann, G. B. Raga, C. D. O. Dowd, M. Kulmala, S. Fuzzi, A. Reissell and M. O. Andreae, Flood or drought: How do aerosols affect precipitation?, Science, 321 (2008), 1309-1313.   Google Scholar [26] J. B. Shukla, M. Verma and A. K. Misra, Effect of global warming on sea level rise: A modeling study, Ecol. Complex., 32 (2017), 99-110.   Google Scholar [27] J. B. Shukla, A. K. Misra, R. Naresh and P. Chandra, How artificial rain can be produced? A mathematical model, Nonlinear Anal. Real World Appl., 11 (2010), 2659-2668.   Google Scholar [28] J. B. Shukla, S. Sundar, A. K. Misra and R. Naresh, Modeling the effects of aerosols to increase rainfall in regions with shortage, Meteorol .Atmos. Phys., 120 (2013), 157-163.   Google Scholar [29] S. Sundar and R. K. Sharma, The role of aerosols to increase rainfall in the regions with less intensity rain: A modeling study, Comput. Ecol. Softw., 3 (2013), 1-8.   Google Scholar [30] S. Sundar, R. Naresh, A. K. Misra and J. B. Shukla, A nonlinear mathematical model to study the interactions of hot gases with cloud droplets and raindrops, Appl Math Model., 33 (2009), 3015-3024.  doi: 10.1016/j.apm.2008.10.032.  Google Scholar [31] R. K. Upadhyay, R. D. Parshad, K. Antwi-Fordjour, E. Quansah and S. Kumari, Global dynamics of stochastic predator-prey model with mutual interference and prey defense, J. Appl. Math. Comput., 60 (2019), 169-190.  doi: 10.1007/s12190-018-1207-7.  Google Scholar [32] Y. Wang, M. Wang, R. Zhang, S. J. Ghan, Y. Lin, J. Hu, B. Pan, M. Levy, J. H. Jiang and M. J. Molina, Assessing the effects of anthropogenic aerosols on Pacific storm track using a multiscale global climate model, Proc. Natl. Acad. Sci. USA., 111 (2014), 6894-6899.   Google Scholar [33] B. P. Yadav, A. K. Das, K. V. Singh and S. K. Manik, Rainfall Statistics of India - $(2017)$. http://www.imd.gov.in., Google Scholar [34] C. Zhao, X. Tie and Y. Lin, A possible positive feedback of reduction of precipitation and increase in aerosols over eastern central China, Geophysical Research Letters, 33 (2006), L11814. doi: 10.1029/2006GL025959.  Google Scholar
Plot of isoclines (12) in blue color and (13) in red color
Global stability of $(N^*, C_r^*, P^*)$ and $(N^*, C_d^*, C_r^*)$ in $N-C_r-P$ and $N-C_d-C_r$ spaces, respectively
Variation of $C_r(t)$ with time for different values of $\eta$ (a) for $\pi_1 = 0.09$ and (b) for $\pi_1 = 0.9.$
Path of $N(t), C_d(t), C_r(t)$ and $P(t)$ for stochastic model (17) with $\sigma_1 = 0.0001, \sigma_2 = 0.006, \sigma_3 = 0.004, \sigma_4 = 0.008$ as well as deterministic model (1)
Path of $N(t), C_d(t), C_r(t)$ and $P(t)$ for stochastic model (17) with $\sigma_1 = 0.08, \sigma_2 = 0.2, \sigma_3 = 0.09, \sigma_4 = 0.1$ as well as deterministic model (1)
The stationary distribution of $N(t), C_d(t), C_r(t)$ and $P(t)$ obtained at $t = 100$ from $10,000$ simulation run for the stochastic model (17), for $\sigma_1 = 0.0001, \sigma_2 = 0.006, \sigma_3 = 0.004, \sigma_4 = 0.008.$
The stationary distribution of N(t), Cd(t), Cr(t) and P(t) obtained at t = 100 from 10, 000 simulation run for the stochastic model (17), for σ1 = 0.08, σ2 = 0.2, σ3 = 0.09, σ4 = 0.1.
The stationary distribution of N(t), Cd(t), Cr(t) and P(t) obtained at t = 100 from 10, 000 simulation run for the stochastic model (17), for σ1 = 0.09, σ2 = 0.3, σ3 = 0.1, σ4 = 0.2.
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