Deforestation, Carbon Dioxide Increase in The Atmosphere and Global Warming: A Modelling Study
Deforestation, Carbon Dioxide Increase in The Atmosphere and Global Warming: A Modelling Study
Deforestation, Carbon Dioxide Increase in The Atmosphere and Global Warming: A Modelling Study
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Prabir Panja
Haldia Institute of Technology, Haldia, India
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To cite this article: Prabir Panja (2019): Deforestation, Carbon dioxide increase in the atmosphere
and global warming: A modelling study, International Journal of Modelling and Simulation, DOI:
10.1080/02286203.2019.1707501
ARTICLE
CONTACT Prabir Panja prabirpanja@gmail.com Department of Applied Science, Haldia Institute of Technology, Purba Medinipur, Haldia, West Bengal
721657, India
© 2019 Informa UK Limited, trading as Taylor & Francis Group
2 P. PANJA
There are also some research articles [13–17] which crops also draw in carbon dioxide and release oxygen
discuss the causes and effects of global warming on in the atmosphere, but forests store up to 100 times
human beings as well as earth temperature. Global more carbon than agricultural fields of the same area.
warming is harming the environment in several ways, To formulate the mathematical model, the following
including desertification, increased melting of snow and assumptions have been made:
ice, sea level rise, stronger hurricanes and cyclones, and
forests, farms and cities will face troublesome new pests, ● Here CðtÞ; H ðtÞ; F ðt Þ and Gðt Þ denote the concentra-
heat waves, heavy downpour and increased flooding. All tion of carbon dioxide in the atmosphere, density of
these factors will damage or destroy agriculture and human population, density of forest biomass and the
fisheries as well as human populations. In 2009, van quantity of global warming at time t, respectively.
der Werf et al. [18] published a research article on global ● Carbon dioxide increases in the atmosphere with
warming. He claimed that carbon dioxide increases in a constant rate due to different natural calamities
the atmosphere due to forest loss may increase global such as respiration and volcano eruptions and dif-
warming. There are some studies [19–22] which suggest ferent activities of human such as deforestation,
that carbon dioxide increase in the atmosphere is one of land use changes and burning fossil fuels.
the main reasons for global warming. ● Carbon dioxide decreases in the atmosphere due to
From the above literature survey, it is seen that the the use of it during photosynthesis by plants as well
global warming has a significant effect on different preda- as due to several natural reasons.
tor–prey dynamical system on animals as well as human ● It is assumed that the human population grows
populations. In 2015, Panja and Mondal [23] studied logistically. It is also assumed that the human popu-
a predator–prey interaction model among Phytoplankton, lation will be decreased due to the death of human
Zooplankton and Fish population. After that, Panja et al. for the increase of carbon dioxide in the atmosphere.
[24] investigated the impacts of anti-predator behaviour of ● It is assumed that the human population increase
the predator–prey system by using mathematical models. by consuming different foods or resources which
Then, Panja et al. [25] studied the effects of toxicants on are coming directly or indirectly from forests.
phytoplankton, zooplankton and fish dynamics and also on ● Again, it is assumed that forest’s population grows
the fish harvesting. The impacts of global temperature logistically with the help of atmospheric carbon diox-
change in the predator–prey system have been studied by ide during photosynthesis in the presence of sunlight.
some researchers [26–31]. From the literature survey, it is ● It is also assumed that forests decrease in the earth
observed that there are very few amount of research paper due to several activities by human such as rapid
on mathematical modelling where global warming is con- growth of human population, industrialization,
sidered. So in this paper, our objective is to study mathe- modern lifestyle etc.
matically the impacts of deforestation and global warming ● It is assumed that global warming increases with
on human population. a constant rate due to several natural reasons,
increase of carbon dioxide in the atmosphere and
increase of human population and deforestation.
2. Model formulation
Forests are very much essential for life, home for mil- Keeping the above assumptions in mind, a mathematical
lions of species throughout the world. Also, forests model has been developed as follows:
produce oxygen, store carbon dioxide and help to 9
dC
¼ A þ αH βC γCF >
make the balance of oxygen, carbon dioxide in the dt >
>
dH
¼ rH 1 Hk α1 CH þ β2 β1 HF =
environment. Forests are also vital for human beings dt
(1)
to live as they provide us with food, shelter and medi-
dF
¼ r1 F 1 kF1 β1 HF þ γ1 γCF > >
dt >
;
cines as well as many other useful things. During photo- dG
dt ¼ B þ λC þ λ1 H dG
synthesis, they produce oxygen which is very much
essential to survive for human beings. Due to rapid with initial conditions Cð0Þ 0; H ð0Þ 0; F ð0Þ 0;
increase of human population, industrialization etc., Gð0Þ 0. The biological meaning of all the parameters
deforestation increases day by day. Deforestation have been given in Table 1.
damages the atmosphere and also plays a huge role in
the carbon cycle on our world. When the forests are cut
3. Boundedness of solutions
down and the woods burned, then the carbon stored in
the trees is released into the atmosphere. It is experi- In this section, we have studied the boundedness of
mentally proved that smaller crops and agricultural solutions of our proposed system.
INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION 3
Table 1. Biological meaning of parameters. Theorem 2. The human and forest’s free equilibrium
Symbol Biological meaning point E0 is always unstable.
A Constant increase of carbon dioxide
α Rate of increase of carbon dioxide due to different human
activities Proof. See Appendix A.
β Natural depletion rate coefficient of atmospheric carbon dioxide
γ Depletion rate coefficient of atmospheric carbon dioxide due to
forest biomass Theorem 3. The forest’s free equilibrium point E1 is locally
r Intrinsic growth rate of human population asymptotically stable if r1 < β1 H1 γ1 γC1 ; r < 2rH k þ
1
● χ ðγ Þ ¼ 0:
5. Stability analysis ● σ 21 σ 1 σ 04 σ 02 σ 3 ðσ 1 σ 2 2σ 3 Þ σ 1 σ 03 σ 01 σ 3 Þ0
In this section, the local stability of the system ð1Þ has and all other eigenvalues are of negative real parts,
been investigated in the neighbourhood of the equili- where xðγÞ is purely imaginary at γ ¼ γ .
brium points. The Jacobian matrix of the system ð1Þ is
given by Proof. See Appendix A.
4 P. PANJA
concluded that if the depletion rate coefficient of if we change carrying capacity k of human population
atmospheric carbon dioxide increases, then the from 1500 to 3000, then our proposed system remains
solution of our proposed system becomes stable. stable within 1500 k 1560 and but solution becomes
That is, the concentration of carbon dioxide in unstable when k > 1560. Hence, it can be concluded that,
the atmosphere has a big role for the stability of if the density of human population gradually increases,
our proposed model. then our proposed system becomes unstable due to lim-
Again, bifurcation diagram of system ð1Þ with respect ited resource of food, shelter etc. Again, it can be con-
to carrying capacity ðkÞ of the human population has cluded that atmospheric carbon dioxide as well as global
been drawn in Figure 3. From this figure, it is seen that warming gradually increases due to cut down of forests by
human for food, shelter etc. and our system may become k1 7800 but becomes unstable if k1 > 7800. Hence, it
unstable. can be concluded that if the forest biomass increases,
Using the same set of parametric values used in then the carbon dioxide in the atmosphere gradually
Figure 2, the bifurcation diagram of system ð1Þ with increases due to photosynthesis which makes our sys-
respect to carrying capacity of forest biomass ðk1 Þ has tem unstable.
been presented in Figure 4. From this figure, it is Again, the bifurcation diagram of system ð1Þ with
observed that if we change k1 from 7000 to 10,000, respect to growth rate of forest biomass ðr1 Þ has been
then our proposed system remains stable in 7000 shown in Figure 5. If we change the value of r1 changes
from 0:0 to 0:2, then our proposed system shows can control the birth rate of human in under control,
unstable solution in 0 r1 0:1046 but becomes stable then our proposed system becomes stable.
when r1 > 0:1046. From this figure, it can be concluded Finally, the change of global warming with the varia-
that if the growth rate of forest biomass r1 increases, tion of α (rate of increase of increase of carbon dioxide
then our proposed system tends to stability. But our in the atmosphere) and λ (rate of increase of global
proposed system becomes unstable if the value of r1 warming due to increase of carbon dioxide in the atmo-
exceeds a critical value. That is if we can increase the sphere) has been plotted in Figure 8. From this figure, it
plantation program, then the increase level of carbon is seen that as the value of α and λ increases, then the
dioxide will be under control. It will be helpful to con- global warming gradually increases. Hence, it can be
trol global warming. concluded that if the carbon dioxide in the atmosphere
Also, the bifurcation diagram of system ð1Þ with increases, then the global warming gradually increases.
respect to depletion rate of forest biomass due to
human β1 has been shown in Figure 6. From this
figure, it is seen that if we change the value of β1 from 8. Conclusion
0:0005 to 0:0007, then our system shows stable solution In this paper, a mathematical model of carbon dioxide in
in 0:0005 β1 0:0006 but becomes unstable when the atmosphere, human population, forest biomass and
β1 > 0:0006. So, it can be concluded that if the rate of global warming has been developed. It is assumed that
cut down forest biomass increases, then the carbon carbon dioxide in the atmosphere increases due to different
dioxide in the atmosphere as well as global warming natural resources such as earthquake, different human
gradually increases which may make our ecological sys- activities such as rapid industrialization, environmental
tem unstable. pollution etc. Also, carbon dioxide decreases in the atmo-
Again, the bifurcation diagram of system ð1Þ with sphere due to natural depletion rate. It is assumed that
respect to depletion rate of human population due to human population grows logistically. Again, the human
carbon dioxide ðα1 Þ has been presented in Figure 7. population decreases due to carbon dioxide increase in
From this figure, it is observed that if we change α1 the atmosphere because it makes some death on human
from 0:0 to 0:00001, then our proposed system shows population. It is also assumed that forest biomass increases
unstable solution in 0 α1 0:0000039 but the system logistically. Also, it is assumed that the growth rate of forest
becomes stable when α1 > 0:0000039. So, it can be con- biomass depends on atmospheric carbon dioxide during
cluded that if the density of human population decreases, photosynthesis. Again, it is considered that the temperature
then our proposed system becomes stable. That is if we of the earth’s surface that is global warming increases due to
different natural resources and also carbon dioxide if we implement some laws for the conservation of forest
increases in the atmosphere due to different human activ- biomass, then the increase level of environmental atmo-
ities. Then, mathematical model is formulated and different spheric carbon dioxide will be under control. Also, if the
possible equilibrium points are evaluated. After that, the human population gradually increases, then the forest bio-
local stability analysis of the system ð1Þ is investigated mass decreases due to food, shelter, rapid industrialization
around these equilibrium points. From the theoretical ana- etc. which can make our proposed model unstable. Again,
lysis of stability theory, it is seen that the stability of the it is also found that if the forest biomass highly
interior equilibrium point depends on the parameters increases and human population decrease, then the carbon
γ; k; k1 ; r1 ; β1 ; α1 etc. Also, Hopf bifurcation analysis of dioxide in the atmosphere gradually increases due to
our proposed system ð1Þ has been done with respect to photosynthesis of plants which may make our proposed
some important parameters. system unstable. Also, it can be concluded that if the rate of
From the numerical simulations of our proposed math- cut down of forest biomass by human beings increases,
ematical model, it is found that the interior equilibrium then the carbon dioxide in the atmosphere as well as global
point is locally asymptotically stable. Also, it is seen that if warming gradually will be increased that can make our
the value of γ changes from 0 to 0.005, then system solution proposed ecological system unstable. It is also concluded
becomes unstable within 0:00015 γ 0:0033 and stable that if we can increase the growth rate of forest biomass ðr1 Þ
when γ > 0:0033. Again, it is observed that if the carrying by taking different strategies, then we can able to make our
capacity ðkÞ of human population changes from 1500 to proposed system stable. So from our theoretical and
3000, then our proposed system remains stable within numerical study, it can be concluded that we can restrict
1500 k 1560 and the system becomes unstable when the rate of increase of carbon dioxide in the atmosphere by
k > 1560. Also, it is seen that if the value of environmental taking plantation program.
carrying capacity of forest biomass k1 changes from 7000 to
10,000, then our system remains stable within the range Disclosure statement
7000 k1 7800 but system becomes unstable when
k1 > 7800. If the value of r1 varies from 0.0 to 0.2, then No potential conflict of interest was reported by the author.
our proposed system remains unstable within 0 r1
0:1046 and the system becomes stable when r1 > 0:1046. Notes on contributor
It is observed that if we change the value of β1 from 0:0005
to 0:0007, then our proposed system becomes stable up to Prabir Panja is working as an Assistant Professor at the
Department of Applied Science, Haldia Institute of
the range 0:0005 β1 0:0006 but the system becomes Technology, Haldia, West Bengal, India. He teaches mathe-
unstable when β1 > 0:0006. Also, it is observed that if the matics at the undergraduate level. He has completed his Ph.D
value of α1 changes from 0:0 to 0:00001, then our proposed degree in Mathematical Biology from Vidyasgar University,
system shows unstable solution in 0 α1 0:0000039 West Bengal, India in 2017. Dr. Panja has published more
but the system becomes stable when α1 > 0:0000039. than 17 papers in different reputed international journals,
such as Nonlinear Dynamics, Chaos Solitons & Fractals,
From these bifurcation diagrams, it can be concluded that Computational and Applied Mathematics, International
if we increase forest biomass ðγÞ by plantation program or Journal of Biomathematics, Theory in Biosciences,
INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION 9
International Journal of Nonlinear Sciences and Numerical an options assessment report. Meridian Institute for the
Simulation etc. Dr. Panja’s research areas are Mathematical Government of Norway; 2009. p.75–77.
Ecology and Epidemiology. [17] Defries R, Achard F, Brown S, et al. Earth observations
for estimating greenhouse gas emissions from defores-
tation in developing countries. Environ Sci Policy.
References 2007;10(4):385–394.
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and the Global Environment, Harvard T.H. Chan School [19] Lonngren EK, Bai EW. On the global warming pro-
of Public Health, Harvard University. [cited 2017 May 15]. blem due to carbon dioxide. Energy Policy. 2008;36
[2] Wilcox BA, Ellis B. Forests and emerging infectious dis- (4):1567–1568.
eases of humans, United Nations Food and Agriculture [20] Ghommem M, Hajj RM, Puri IK. Influence of natural
Organization Corporate Document Repository. and anthropogenic carbon dioxide sequestration on
[3] McMichael AJ, Woodruff RE, Hales S. Climate change global warming. Ecol Model. 2012;235–236:1–7.
and human health: present and future risks. Lancet. [21] Florides GA, Christodoulides P. Global warming and
2006;367:859–869. carbon dioxide through sciences. Environ Int.
[4] Khasnis AA, Nettleman MD. Global warming and 2009;35:390–401.
infectious disease. Arch Med Res. 2005;36:689–696. [22] Newell ND, Marcus L. Carbon dioxide and people.
[5] Kahn LH. Deforestation and emerging diseases, Palaios. 1987;2:101–103.
Bulletin of the Atomic Scientists. 2011 15 Feb [cited [23] Panja P, Mondal SK. Stability analysis of coexistence of
2017 Oct 11]. Available from: Thebulletin.org three species prey–predator model. Nonlinear Dyn.
[6] Moslemi JM, Snider SB, MacNeill K, et al. Impacts of an 2015;81:373–382.
Invasive Snail. PLoS ONE. 2012;7(6):e38806. [24] Panja P, Mondal SK, Chattopadyay J. Dynamical effects
[7] Julia B, Byker TS. Does forest loss increase human of anti-predator behaviour of adult prey in a
disease? Evidence from Nigeria. Am Econ Rev. predator-prey model with ratio-dependent functional
2017;107(5):516–521. response. Asian J Math Phys. 2017;1:19–32.
[8] Hansen J, Ruedy R, Sato M, et al. Global surface tem- [25] Panja P, Mondal SK, Jana DK. Effects of toxicants on
perature change. Rev Geophys. 2010;48:RG4004. Phytoplankton-Zooplankton-Fish dynamics and
[9] U.K. Met Office. Warming: A guide to climate change. harvesting. Chaos Solit Fract. 2017;104:389–399.
Exeter, U.K.: Met Office Hadley Centre; 2011. [26] Misra AK, Verma M. A mathematical model to study
[10] IPCC. Summary for Policymakers. In: Climate Change the dynamics of carbon dioxide gas in the atmosphere.
2007: The Physical Science Basis. Contribution of Appl Math Comput. 2013;219:8595–8609.
Working Group I to the Fourth Assessment Report of [27] Uszko W, Diehl S, Englund G, et al. Effects of warm-
the Intergovernmental Panel on Climate Change. ing on predator-prey interactions-a resource-based
Cambridge: Cambridge University Press. New York, NY; approach and a theoretical synthesis. Ecol Lett.
2007. 2017;20:513–523.
[11] http://www.climateandweather.net/global-warming/cli [28] Panja P. Fuzzy parameter based mathematical model on
mate-change-and-animals.html [cited 2017 Jul 15]. forest biomass. Biophys Rev Lett. 2018;13:179–193.
[12] Beaudry F Climate change in the Arctic. ThoughtCo. [29] Panja P. Optimal control analysis of a cholera epidemic
2016 May 20. Available from: thoughtco.com/climate- model. Biophys Rev Lett. 2019;14:1–22.
change-in-the-arctic-3994022. [30] Amarasekare P. Effects of temparature on consumer-
[13] http://www.climateandweather.net/global-warming resource interaction. J Anim Ecol. 2015;84:665–679.
/deforestation.html [cited 2017 Aug 28] [31] Binzer A, Guill C, Brose U, et al. The dynamics of food
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from Deforestation and Forest Degradation (REDD): Ginn Boston; 1982.
10 P. PANJA
Also, from the fourth equation of system ð1Þ, we have Proof of Theorem 5
dG The characteristic equation of the Jacobian matrix J ðE Þ
þ dG B þ λCn þ λ1 Hn
dt is given by
Again, applying differential inequality [33], we have x4 þ σ 1 x3 þ σ 2 x 2 þ σ 3 x þ σ 4 ¼ 0 (2)
0 < G BþλCndþλ1 Hn :
where σ 1 ¼ a11 þ a44 a22 a33 ; σ 2 ¼ a11 a22
Proof of Theorem 2 a12 a21 a22 a33 þ a22 a44 þa11 a33 a11 a44 þ a33 a44
a23 a32 a13 a31 ; σ 3 ¼ a11 a22 a33 a11 a22 a44 a12 a21 a33 þ
The eigenvalues of the Jacobian matrix J ðE0 Þ are a12 a21 a44 þ a22 a33 a44 a11 a33 a44 þa23 a32 a44 þ a11 a23 a32
β; d; r α1 C0 and r1 þ γγ1 C0 . Since all the biologically þa13 a21 a32 þ a13 a31 a44 þ a31 a12 a23 þ a13 a31 a22 ; σ 4 ¼
feasible parameters associated in this proposed model a11 a22 a33 a44 a12 a21 a33 a44 þ a11 a23 a32 a44 þ a13 a21
are positive, so one of the eigenvalue r1 þ γγ1 C0 > 0 is a32 a44 a31 a12 a23 a44 a13 a31 a22 a44 ; a ¼ β þ γF ; a12 ¼
INTERNATIONAL JOURNAL OF MODELLING AND SIMULATION 11
ω20 x3 x4 ¼ σ 4 (6)
Solving for ζ 0 ðγÞ at γ ¼ γ , we have
qffiffiffiffi d
where ω0 ¼ Imx1 ðγ Þ. By the above relations, ω0 ¼ σσ 31 . Rexj ðγÞ γ¼γ ¼ ζ 0 ðγ Þ
Now, if x3 ; x4 are complex conjugate, then from equa- dγ
M2 ðγ ÞM4 ðγ Þ þ M1 ðγ ÞM3 ðγ Þ
tion ð4Þ, it follows that 2Rex3 ¼ σ 1 ; if they are real ¼
roots, then by ð4Þ and ð5Þ, x3 < 0 and x4 < 0. M12 ðγ Þ þ M22 ðγ Þ
To complete the proof, now we verify the transvers-
σ 21 σ 1 σ 04 σ 02 σ 3 ðσ 1 σ 2 2σ 3 Þ σ 1 σ 03 σ 01 σ 3
ality condition. As χ ðγ Þ is a continuous function of all its ¼ Þ0
2σ 31 σ 3 þ 2ðσ 1 σ 2 2σ 3 Þ2
roots, so there exists an open interval γðγ ; γ þ Þ,
where x1 and x2 are complex conjugate for γ. Suppose Thus, the transversality conditions hold and hence Hopf
there general forms in this neighbourhood are bifurcation occurs at γ ¼ γ .