Effect of automobiles on global warming: A modeling study

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1 ticle ffect of automobiles on global waming: modeling study Shyam Sunda shish uma Misha Ram aesh Depatment of Mathematics P. S. Institute of Technology anpu-895 India Depatment of Mathematics School of Basic and pplied Sciences H. B. Technical Univesity anpu-8 India -mail: Received 4 pil 7; ccepted May 7; Published Septembe 7 bstact Global waming theatens ou envionment as well as basic human needs. In the pesent scenaio inceasing demand and excessive use of automobiles have inceased the level of cabon dioxide emission in the envionment poviding a significant contibution to incease the global waming. This pape deals with the modeling of the effect of automobiles on global waming. Fo this thee nonlinealy inteacting vaiables namely; density of human population density of automobiles and the concentation of cabon dioxide have been taken into account. In the modeling pocess it is assumed that the density of automobiles inceases in popotion to human population following a logistic gowth. The model is analyzed using stability theoy of odinay diffeential equations. Local and global stability conditions ae established to study the feasibility of the model system. It is shown that with incease in human population the demand fo automobiles inceases which has significant effect on global waming incease. eywods mathematical model; automobile; cabon dioxide; stability. omputational cology and Softwae ISS 7X URL: vesion.asp RSS: mail: ces@iaees.og dito in hief: WenJun Zhang Publishe: Intenational cademy of cology and nvionmental Sciences Intoduction The majo factos esponsible fo inceasing the level of cabon dioxide in the atmosphee ae mainly due to man made activities. The human activities mainly buning of fossil fuels excessive use of automobiles land use changes etc. ae esponsible fo the obseved ise in the concentation of atmospheico. It is estimated that appoximately 9% contibution is due to the emissions fom fossil fuel combustion and the est is due to agicultue defoestation land use changes P 4; IP 4). utomobiles ae seious global theat fo incease in global waming due to emission of cabon dioxide gas. Ove the past centuy thee ae many factos which poduce an effect of automobile emissions on global waming such as fuel used oad gadient inceasing vehicle age payload cold stats and maintenance condition of the vehicle. utomobiles ae one of the significant souces of vaious gases and paticulate mattes. The black smoke eleased into the ai by vehicles is the common fom of pollution that we outinely

2 face. abon dioxide emitted fom automobiles is one of the lagest contibutos to the global waming ng and Thomas 5; Shaw 5; Sunda and aesh 5; Opae 6). Due to apid gowth of human population density of automobiles is inceasing atociously. Due to incease in ath s aveage tempeatue vaious new diseases ae occuing fequently since the bacteia can suvive bette in such conducive situations and multiply faste. Futhe many undesiable consequences such as economical loss loss of biodivesity dought flood etc. may aise due to global waming Douglas and Selden 995; Robinson et al. 7; Gao et al. ; Wu et al. ; Feaini ; Wu and Zhang ; Zhang and Liu ; Zhang and Zhang ; Rai and Rai a b). The concentation of geenhouse gases continues to incease due to inceasing demand and excessive use of automobiles which can affect the envionment in seveal ways. These geenhouse gases contibute to global waming and affect the espiatoy epoductive immune and neuological systems of animals. These gases ae also main contibutos to acid ain which changes the ph of wateways and soils. abon dioxide O ) is natually pesent in the atmosphee which is the pimay geenhouse gas emitted though human activities. In O accounted fo about 8% of all U.S. geenhouse gas emissions fom human activities. Human activities ae changing the cabon cycle-both by adding moe O to the atmosphee and by influencing the ability of natual sinks like foests to emove O fom the atmosphee I-). Fom the above it may be pointed out that the continuous incease in population and the associated use of automobiles have significant impact on inceasing the global waming buden. Theefoe in this study ou main aim is to exploe the ole of automobiles on global waming. Fo this we popose a model consisting of thee nonlinealy inteacting vaiables namely; density of human population density of automobiles and the concentation of cabon dioxide. Model To develop a mathematical model we conside the density of human population as t) t) be the density of automobiles and t) be the concentation of O in the atmosphee. In the modeling pocess the human population density is assumed to follow logistic gowth with intinsic gowth ate and caying capacity. It is assumed that the emission of cabon dioxide affects the gowth of human population and influences the caying capacity with depletion ate constants and due to O. It is natual to assume that as the population inceases the demand of automobiles also inceases. Thus we assume that the density of automobiles is diectly popotional to the density of human population with a ate with its natual depletion ate coefficient. Let be the constant ate of emission of cabon dioxide fom vaious souces into the atmosphee which is futhe enhanced by the emission fom automobiles with a ate. The constant epesents natual depletion ate coefficient of cabon dioxide. In view of the above assumptions the dynamics of the system is assumed to be govened by the following system of nonlinea diffeential equations d ) dt d dt d dt ) )

3 with ) ) ). Hee all the paametes ae assumed to be positive. Lemma The egion of attaction in the positive octant is given by the set : R 4) Remak Fom equation ) it is noted that in pesence of cabon dioxide the intinsic gowth ate of human population at any time t is. quilibium and Stability nalysis The model system ) ) has only one non-tivial equilibium ). The solution of ) is given by the following set of algebaic equations 5) 6) 7) Using equations 5) and 6) in equation 7) we get 8) Fom which we get 4 let) Using this value of ) we can find and fom equations 5) and 6) espectively. Diffeentiating 8) with espect to and noting emak ) we get d d 9)

4 This implies that inceases as inceases. Similaly we can show easily that d d d d and d d. This analysis eveals that the numbe density of automobiles inceases as human population inceases and consequently the concentation of cabon dioxide in the atmosphee also inceases which in tun may decease the gowth of population density. 4 Stability nalysis 4. Local stability of the equilibium ) To pesent local stability behaviou of the model system ) ) coesponding to the equilibium ) we compute the following Jacobian matix ) J igenvalues of the Jacobian matix ) J ae given by the following chaacteistic equation x x x whee ) It may be easily noted that povided the following condition is satisfied ) ) Theefoe by Routh-Huwitz citeia eigenvalues of the Jacobian matix coesponding to the equilibium ) ae all negative o have negative eal pat. Hence equilibium is locally asymptotically stable. Theoem The equilibium is locally asymptotically stable povided the condition ) is satisfied. 4. onlinea stability of the equilibium )

5 4 To establish nonlinea stability we conside the following positive definite function m V m ln m ) whee i m i ae positive constants to be chosen appopiately. Diffeentiating ) with espect to t and using the model system ) ) we get dv dt m m ) m ) m ) ) m ) ) m ) ) hoosing m dv will be negative definite if the following condition is satisfied. dt ) ) Theoem The equilibium is nonlinealy asymptotically stable within the egion of attaction povided the condition ) is satisfied. 5 umeical Simulation In this pape we popose a thee-dimensional nonlinea model to study the effect of automobiles on global waming which is analyzed using stability theoy of diffeential equations. The model incopoates thee nonlinealy inteacting vaiables viz the density of human population the density of automobiles and the concentation of cabon dioxide. The equilibium analysis of the model eveals that the density of automobiles inceases as human population inceases leading to incease the concentation of cabon dioxide in the atmosphee. The model has only one non-tivial unique equilibium point which is locally asymptotically stable unde condition ). Moeove this equilibium is also found to be nonlinealy asymptotically stable inside the egion of attaction unde condition ). To validate the analytical findings we pefom numeical simulation of the model system ) ) with espect to fo diffeent values of paametes. Fo that the system ) ) is integated numeically with the help of MPL 7 by consideing the following set of paametes values The equilibium so obtained is igenvalues of the Jacobian matix coesponding to ae and Since all the eigenvalues ae negative the inteio equilibium is locally asymptotically stable. To see the nonlinea stability behaviou fo the above set of paametes compute geneated gaph in is shown in Fig. whee the tajectoies with diffeent initial stats appoaches the equilibium. It depicts that is nonlinealy stable.

6 5 Fig. onlinea stability in plane. Fig. Vaiation of with time t fo diffeent values of. In Fig. the vaiation of density of human population with time t is shown fo diffeent values of λ the gowth ate of automobiles. Fom the figue it is clea that the density of human population deceases as the gowth ate of automobiles inceases. In Fig. the vaiation of concentation of cabon dioxide with time t is shown fo diffeent values of gowth ate of automobiles. Fom this figue it is seen that the concentation of cabon dioxide inceases as the gowth ate of automobiles inceases. Thus the incease in the gowth ate of automobiles inceases the level of cabon dioxide in the atmosphee which in tun advesely affects the human population. s discussed in intoduction demand of automobiles inceases as the population inceases which in tun incease the concentation of cabon dioxide in the atmosphee. The effect of automobiles on human population and cabon dioxide concentation is plotted in Figs. 4 and 5 espectively. Thus in Figs. 4 and 5 the vaiation of density of human population ) and the concentation of cabon dioxide ) with time t is

7 6 shown fo diffeent values of δ the emission ate of cabon dioxide due to automobiles espectively. Fom these figues it is obseved that the density of human population deceases and the concentation of cabon dioxide inceases with incease in the emission ate of cabon dioxide fom automobiles. This elevated level of cabon dioxide has significant effect on global waming. Fig. Vaiation of with time t fo diffeent values of. Fig. 4 Vaiation of with time t fo diffeent values of. Fig. 5 Vaiation of with time t fo diffeent values of.

8 7 6 onclusion In this pape an attempt has been made to compehend the effect of automobiles on global waming. It is assumed that the model system consists of thee nonlinealy dependent vaiables namely density of human population density of automobiles and the concentation of cabon dioxide. The poposed model is analysed using stability theoy of odinay diffeential equations and numeical simulations. It is shown analytically and numeically that the concentation of cabon dioxide inceases as the density of automobiles inceases. This elevated level of cabon dioxide in the atmosphee due to emissions fom automobiles may significantly contibute to incease the global waming. The buden of global waming could be educed significantly if the emission fom automobiles is esticted by way of enfocing stingent pollution noms and by educing the unnecessay load of automobiles. Refeences Douglas H Selden TM Stoking the fies? O emissions and economic gowth. Jounal of Public conomics 57): 85- ng JOM Thomas G. 5. utomotive exhaust emissions and its implications fo envionment sustainability. Intenational Jounal of dvanced cademic Reseach ): - P. 4. Geenhouse gas emissions fom a typical passenge vehicle. poduction/files/6-/documents/4f44a.pdf. ccessed 5 Mach 7 Feaini.. I think diffeent: Models of climate waming impact on plant species ae unealistic. nvionmental Skeptics and itics ): - Gao Xu J Gao G Li Y Hutchins D et al.. Rising O and inceased light exposue synegistically educe maine pimay poductivity. atue limate hange : 59-5 IP. 4. limate hange 4: Mitigation of limate hange. a5/wg/ipcc_wg_a5_full.pdf. ccessed Mach 7 Opae. 6. Vehicle emissions o exhaust gas on ou oads effects ecommendations. modenghana.com/news/686/vehicle-emissions-o-exhaust-gas-on-ou-oads- effects-eco.html. ccessed Mach 7 Rai PR Rai P. a. nvionmental and socio-economic impacts of global climate change: n oveview on mitigation appoaches. nvionmental Skeptics and itics 4): 6-48 Rai PR Rai P. b. Paadigms of global climate change and sustainable development: Issues and elated policies. nvionmental Skeptics and itics ): -45 Robinson B Robinson Willie S. 7. nvionmental effects of inceased atmospheic cabon dioxide. Jounal of meican Physicians and Sugeons : 79-9 Shaw J. 5. as and Global Waming: 5 Ways diesel engines ae wose fo the envionment. newsmax.com/fastfeatues/as-and-global-waming- Diesel-ngines-nvionment-limate-hange/ 5//8/id/6959/. ccessed Mach 7 Sunda S aesh R. 5. Modeling and analysis of the suvival of a biological species in a polluted envionment: ffect of envionmental tax. omputational cology and Softwae 5): - U.S. negy Infomation dministation.. ccessed Mach 7

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