a mathematical model to study the dynamics of carbon dioxide gas in the atmosphere

Applied Mathematics and Computation 219 (2013) 8595–8609

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Applied Math ematics and Computati on

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A mathematical model to study the dynamics of carbon dioxide gas in the atmosphere

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.02.058

⇑ Corresponding author. E-mail address: [email protected] (A.K. Misra).

A.K. Misra ⇑, Maitri Verma Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221 005, India

a r t i c l e i n f o a b s t r a c t

Keywords:Mathematical model CO2 gasHuman population Forest biomass StabilityHopf-bifurcation

A nonlinear mathematical model to explore the effects of human population and forest biomass on the dynamics of atmospheric carbon dioxide (CO2) gas has been proposed and analyzed. In the modeling proce ss, it is assumed that the concentration of CO 2 in the atmosphere increases due to natural as well as anthropogenic factors. Further, it is assumed that the atmospheric CO 2 is absorbed by forest biomass and other natural sinks. Equilibria of the model have been obtained and their stability discussed. The model analysis reveals that human population declines with an increase in anthropogenic CO 2 emissions into the atmosphere. Further, it is found that the depletion of forest biomass due to human population (deforestation) leads to increase in the atmospheric concentration of CO 2. It is also found that deforestation rate coefficient has destabili zing effect on the dynamics of the system and if it exceeds a thres hold value, the system loses its stability and periodic solutions may aris e through Hopf-bifurcation. The stability and direction of these bifurcating periodic solutions are analyzed by using center manifold theory. Numerical simulation is performed to support theoretical results.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The elevated level of carbon dioxide (CO2) in the atmosphere is the main driving force behind the threat of global warming. The concentration of atmosph eric CO 2 has increased from its pre-industrial level of 270 ppm (parts per million) [1] to395 ppm as recorded in 2013 [2]. If the rising trend of atmospheric CO 2 concentr ation continues in future, it is very likely that the menace of climate change becomes more forbidding. Mitigation of the climate change is crucial due to its adverse impacts on human population and ecosystem. The climate changes exacerbate many of the health risks in human population. The frequency and intensity of extreme weather events like floods, windstorms , droughts , etc., increase in warmer climate. These events cause many deaths due to direct injuries, malnutrition, and increase the risk of infectious diseases in the population [3–5]. The climate changes also affect the incidence of vector-borne infections due to increase in the abundance and distribut ion of vectors which transmit the disease [6,7]. It is estimate d that climate changes are responsible for 3% of diarrhoea, 3% of malaria, and 3.8% of dengue fever deaths worldwide reported in 2004 [8]. The cardiovascular and respiratory problems also exacerbate due to heat waves. Since the enhanced level of CO 2 is primarily responsible for the ominous climate changes, it is vital to device strategies for the reduction and stabilization of future concentr ations of CO 2. For this, abetter understand ing of the prime factors behind the increased atmospheric concentratio n of CO 2 and their effects on the dynamics of atmosph eric CO 2 is requisite.

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8596 A.K. Misra, M. Verma / Applied Mathematics and Computation 219 (2013) 8595–8609

Human population and forest biomass are the major factors which affect the atmospheric level of CO 2. The human activities, mainly fossil fuel burning and land use changes (deforestation), are responsible for the observed rise in the concentration of atmosph eric CO 2. Estimates show that nearly two-third of the enhancem ent in atmospheric CO 2 is due to fossil fuel burning and the rest is due to land use changes [9]. On the other hand, forest biomass plays a crucial role in the regulation of atmospheric concentr ation of CO 2. Forest sequestrates gigatons of CO 2 each year from the atmosphere during photosyn thesis process and hence helps in reducing the global atmospheric burden of CO 2. In this way, forest biomass influence the dynamics of atmospheric CO 2 significantly. Moreover, destruction of this natural sink of CO 2 because of human activities is one of the prime reason behind the increased level of CO 2 in the atmosphere [10,11]. Thus, understanding the interactions of human population, forest biomass and carbon dioxide provides a better insight for the prediction and control of future concen- trations of atmospheric CO 2.

Some mathemati cal models have been proposed to study the effect of various factors on the atmospheric concentration of CO2. Tennakon e [12] has studied the biomass-carbon dioxide system by using a mathemati cal model. In this study, it is pointed out that excessive deforesta tion destabilizes the biomass-car bon dioxide equilibrium. The connectio n between global warming and human activities has been studied by using a feedback model [13]. In this study, it is shown that the emission of CO 2 due to human activities has destabili zing effect. Caetano et al. [14] have studied a mathematical model which relates the atmospheric concentration of CO 2 with forest area and gross domestic product. In their study, they have considered reforestation and clean technolo gy as a control variable for the atmospheric CO 2 and optimized the total investment on reforestatio n and clean technology to obtain the desired level of CO 2. To the best of our knowledge, in literature, no mathematical model is available which explores the interplay among human population, forest biomass and atmospheric CO 2.Therefore, in the present paper, we have formulated a mathematical model which explores the impact of human population and forest biomass on the dynamics of atmospheric CO 2. The present paper is organized as follows. In the next section, we formulate a mathematical model governing the dynamics of the problem. The stability analysis of the model is presented in Section 3. In Section 4, we obtain the conditions for existence of Hopf-bifurc ation after choosing a suitable bifurcating parameter. The direction and stability of Hopf-bifurc ation is analyzed in Section 5. In Section 6, numerical simulatio n is car- ried out to verify the analytica l findings and finally in Section 7, the paper is concluded .

2. Mathematical model

Let at any time t, in any region under considerati on, XðtÞ; NðtÞ and FðtÞ be the atmospheric concentration of CO 2, human population and forest biomass, respectively. We have considered both natural and anthropoge nic emissions of CO 2. The growth rate of atmospheric CO 2 due to natural sources (e.g., volcanic eruption, respiration process of living organism, changes in oceanic circulation, etc.) is assumed to be a constant. Since the anthropogenic CO 2 emission s depend on the human population, therefore we assume that growth rate of atmospheric CO 2 due to human activities is proportional to human population [15,16]. Since forest uptake CO 2 from the atmosph ere during photosynthesis process, hence it is assumed that the concentratio n of CO 2 depletes due to the forest biomass. The depletion of CO 2 due to natural sinks other than forests (likeoceans etc.) is also considered and it is taken to be proportional to atmospheric concentr ation of CO 2. It is further assumed that the human population and forest biomass follow the logistic growth. As pointed out in the previous section, climate changes due to the elevated level of CO 2 increase the frequency and intensity of extreme weather events, and the incidence of vector borne diseases, which causes declination in human population. Therefore, we assume that the human population declines due to increase in the concentratio n of CO 2. The human population clear forests for agriculture, housing, logging and other purposes. This deforesta tion feedback into the population growth [17–22]. Thus, we have assumed that human population and forest biomass have a predator prey type of relationship , i.e., the forest biomass decreases due to human population whereas the growth rate of human population increases due to forest biomass. Plants tend to grow better in the high level of CO 2 concentratio n due to increased photosyn thesis [23]. This phenomenon is called carbon fertilization. Thus, it is also assumed that the forest biomass increases due to enhancement in the atmospheric concentration of CO 2. Keeping above points in view, the dynamics of the model is governed by the following system of nonlinear ordinary differential equations:

dX dt ¼ Q 0 þ kN � aX � k1XF;

dN dt ¼ sN 1� N

L

� �� hXN þ p/NF;

dF dt ¼ uF 1� F

M

� �� /NF þ p1k1XF;

ð1Þ

where Xð0Þ > 0; Nð0ÞP 0; Fð0ÞP 0 . In the model system (1), Q 0 is the growth rate of atmospheric CO 2 due to natural (non-anthropoge nic) causes which is assumed to be constant and k is its growth rate coefficient due to anthropoge nic factors. The constant a denotes the natural depletion rate coefficient of atmosph eric CO 2 whereas k1 is its depletion rate coefficient due to forest biomass. The constant s s and L represent the intrinsic growth rate and carrying capacity of human population, respectively , while u and M represent the intrinsic growth rate and carrying capacity of forest biomass, respectively . The constant h is the depletion rate coefficient of human population due to CO 2 and / is the deforestatio n rate coefficient.

A.K. Misra, M. Verma / Applied Mathematics and Computation 219 (2013) 8595–8609 8597

The proportional ity constants p and p1 represent growth of human population due to forest biomass and growth of forest biomass due to CO 2, respectivel y. Since the growth of human population is benefited slightly due to deforesta tion, we assume that p is much smaller than 1. All the above constants are assumed to be positive.

The region of attraction [24] for model system (1), which shows that all solutions of model system (1) are bounded is given by the set:

X ¼ ðX;N; FÞ : 0 < X 6 Xm; 0 6 N 6 Nm; 0 6 F 6 Fmf g;

where Xm ¼ ðQ 0 þ kNmÞ=a; Nm ¼ Lþ ðp/L=sÞFm, Fm ¼ sM½uaþ p1k1ðQ 0 þ kLÞ�=ðusa� pp1k1k/MLÞ where usa > pp1k1k/MLand it attracts all the solutions initiating in the interior of the positive octant.

The model system (1) has four non-negativ e equilibria which are listed as follows:

(i) E1ðQ0=a;0;0Þ,(ii) E2ðX2;0; F2Þ,

(iii) E3 sðQ0 þ kLÞ=ðsaþ hkLÞ; Lðsa� hQ0Þ=ðsaþ hkLÞ; 0ð Þ,(iv) E�ðX�;N�; F�Þ.

Here it may be noted that the equilibriu m E3 exists provided s� ðhQ0=aÞ > 0. This condition states that in absence of forest biomass, human population exists only if its intrinsic growth rate is more than its depletion rate due to CO 2. The existence of equilibria E1 and E3 is obvious. The existence of equilibria E2 and E� are given in Appendices A and B,respectively . The equilibrium E3 exists without any condition while the interior equilibriu m E� exists under the following conditions:

s >hQ 0

a; ð2Þ

a/ > p1k1k ð3Þ

and

ðuaþ p1k1Q0Þðsaþ hkLÞ > Lðsa� hQ 0Þða/� p1k1kÞ: ð4Þ

Remark 1. It can be easily found that dN�=dk < 0 and dF�=dk > 0, which imply that the equilibrium level of human population decreases whereas the equilibrium level of forest biomass increases as the growth rate of atmospheric CO 2 due to anthropoge nic factors increases.

Remark 2. It is easy to show that for small values of p; dX�=d/ > 0. This implies that if the growth of human population due to forest biomass is small, an increase in the deforesta tion rate causes an increase in the equilibrium concentr ation of atmospheric CO 2.

3. Stability analysis

The local stability analysis of equilibria E1; E2 and E3 is given in Appendix C. From this analysis, we find that E1 has always stable manifold locally in X-direction and unstable manifold locally in F-direction whereas it has locally unstable manifold in N-directio n whenever E2 exists. Also, we find that equilibria E2 and E3 have locally unstable manifold in N-direction and F- direction, respectively whenever E� exists. The stability behavior of the interior equilibriu m E� is discussed as follows.

The Jacobian matrix evaluated at the equilibriu m E� is given by

P� ¼�ðaþ k1F�Þ k �k1X�

�hN� � sN�

L p/N�

p1k1X� �/F� � uF�

M

0B@

1CA:

The characterist ic equation for P� is given as

w3 þ A1w2 þ A2wþ A3 ¼ 0; ð5Þ

where,

A1 ¼ aþ bþ c;

A2 ¼ aðbþ cÞ þ bc þ p/2N�F� þ khN� þ p1k21X�F�;


A3 ¼ aðbc þ p/2N�F�Þ þ k1h/X�N�F� þ ckhN� þ p1k21bX�F� � p/p1k1kN�F�;

a ¼ aþ k1F�; b ¼ sN�

L; c ¼ uF�

M:

Here, we have A1 > 0; A3 > 0. From Routh–Hurwitz criterion, eigenvalues of the Jacobian matrix P� will be either negative or with negative real part iff the following condition is satisfied:

A1A2 � A3 > 0: ð6Þ

Thus we have the following result:

Theorem 1. The interior equilibrium E�, if exists, is locally asymptotically stable if and only if the condition (6) is satisfied.The above theorem tells that if the initial state of system (1) is near the equilibriu m point E�, the solution trajectories

approach to the equilibrium E� under condition (6). Thus, if the initial values of state variables X; N and F are near to the correspondi ng equilibrium levels, the atmosph eric concentration of CO 2 gets stabilized if and only if condition (6) holds.

The non-linear stability of interior equilibriu m follows from the following theorem:

Theorem 2. The equilibrium E�, if exists, is non-linearly stable provided the following inequality is satisfied:

maxk2

1X2mhM

kpu;kpp2

1k21M

hu

( )< aþ k1F�: ð7Þ

For proof of this theorem see Appendix D.

The above theorem tells that for every initial start within the region of attraction, the solution trajectories approach to the equilibrium state E�, and hence the atmospheric concentratio n of CO 2 get stabilized, if the condition (7) is satisfied.

Remark 3. The condition (6) for the local stability of interior equilibrium E� can be written as

aðbþ cÞðaþ bþ cÞ þ ðbþ cÞðbc þ p/2N�F�Þ þ ðaþ bÞkhN� þ ðaþ cÞp1k21X�F� þ p/p1k1kN�F� � k1h/X�N�F� > 0 ð8Þ

It can be easily noted that for large values of /, the above condition may not be satisfied. Thus there is a possibility that if we increase the value of /, then after some critical value of /, the interior equilibria E� may become unstable and periodic solutions may arise through Hopf-bifurc ation.

4. Existence of Hopf-bifurcat ion

In this section, we investigate for the possibility of Hopf-bifurc ation from the interior equilibrium E� by taking the deforestation rate coefficient ‘/’ as bifurcating paramete r and keeping the other parameters fixed.

Let at a critical value of /, say /c; A1ð/cÞA2ð/cÞ � A3ð/cÞ ¼ 0. Thus, at / ¼ /c , the characterist ic Eq. (5) can be written as

ðwþ A1Þðw2 þ A2Þ ¼ 0:

This equation has three roots w1;2 ¼ �iw and w3 ¼ l, where w ¼ffiffiffiffiffiffiA2p

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðbþ cÞ þ bc þ p/2N�F� þ khN� þ p1k

21X�F�

qand

l ¼ �A1 ¼ �ðaþ bþ cÞ.Thus, at / ¼ /c , the characteri stic Eq. (5) has a pair of purely imaginary roots while the third root is negative. To show the

transversali ty condition, let at any point / in �-neighbor hood of /c; w1;2 ¼ jð/Þ � iqð/Þ. Substituting this in (5) and separat- ing real and imaginary parts, we have

j3 � 3jq2 þ A1ðj2 � q2Þ þ A2jþ A3 ¼ 0; ð9Þ

3j2q� q3 þ 2A1jqþ A2q ¼ 0: ð10Þ

As qð/Þ– 0, from (10) we have

q2 ¼ 3j2 þ 2A1jþ A2:

Substituting this in (9), we have

8j3 þ 8A1j2 þ 2jðA21 þ A2Þ þ A1A2 � A3 ¼ 0: ð11Þ

From the above equation, we get

djd/

� �/¼/c

¼ � 1

2ðA21 þ A2Þ

dd/ðA1A2 � A3Þ

" #/¼/c

– 0; providedd

d/ðA1A2 � A3Þ

� �/¼/c

– 0:

Thus, we have the following result regarding the existence of Hopf-bifurc ation:


Theorem 3. The necessary and sufficient conditions for the occurrence of Hopf-bifurca tion from the interior equilibrium E� is that there exists / ¼ /c such that

ðiÞ Hð/cÞ ¼ A1ð/cÞA2ð/cÞ � A3ð/cÞ ¼ 0;

ðiiÞ dd/ðA1A2 � A3Þ

� �/¼/c

– 0:

5. Stability and direction of Hopf-bif urcation

In previous section, we have obtained the conditions for occurrence of Hopf-bifurc ation. In this section, we analyze the stability and direction of the bifurcating periodic solutions arising through Hopf-bifurc ation, using the center manifold theory and following the idea of Hassard et al. [25].

First of all, we shift the origin of coordina te system to the Hopf-bifurcatio n point E�ðX�;N�; F�Þ by using the following transformat ion:

z1 ¼ X � X�; z2 ¼ N � N�; z3 ¼ F � F�:

Now the model system (1) takes the form,

_Z ¼ P�Z þ GðZÞ; ð12Þ

where,

Z ¼z1

z2

z3

0B@

1CA and G ¼

g1

g2

g3

0B@

1CA ¼

�k1z1z3

� sz22

L � hz1z2 þ p/z2z3

� uz23

M � /z2z3 þ p1k1z1z3

0BB@

1CCA:

The eigenvector s v1 and v2 associated with the eigenvalues w1 ¼ iw and w3 ¼ l of P� at / ¼ /c are found to be

v1 ¼p/2N�F� þ bc þ ðbþ cÞiw�w2

p1k1p/N�F� � ðc þ iwÞhN�

h/N�F� þ p1k1ðbþ iwÞF�

0B@

1CA

and

v2 ¼p/2N�F� þ bc þ ðbþ cÞlþ l2

p1k1p/N�F� � ðc þ lÞhN�

h/N�F� þ p1k1ðbþ lÞF�

0B@

1CA:

Define A ¼ ðImðv1Þ;Reðv1Þ;v2Þ ¼wðbþ cÞ p/2N�F� þ bc �w2 p/2N�F� þ bc þ ðbþ cÞlþ l2

�whN� p1k1p/N�F� � chN� p1k1p/N�F� � ðc þ lÞhN�

p1k1wF� h/N�F� þ p1k1bF� h/N�F� þ p1k1ðbþ lÞF�

0B@

1CA ¼def

p11 p12 p13

p21 p22 p23

p31 p32 p33

0B@

1CA:

The matrix A is a nonsingular matrix such that

A�1P�A ¼0 �w 0w 0 00 0 l

0B@

1CA:

The inverse of matrix A is given by

A�1 ¼ D�1

q11 q12 q13

q21 q22 q23

q31 q32 q33

0B@

1CA

where,

q11 ¼ lh2/N�2F� þ lhp1k1bN�F� þ p1k1lF�ðp1k1p/N�F� � chN�Þ;

q12 ¼ ðh/N�F� þ p1k1bF�Þðw2 þ lðbþ cÞ þ l2Þ � p1k1lF�ðp/2N�F� þ bc �w2Þ;

q13 ¼ hN�ð�lp/2N�F� þ lw2 þ c2lþ cl2 þ cw2Þ � p1k1p/N�F�ðw2 þ ðbþ cÞlþ l2Þ;


q21 ¼ wh2/N�2F� þ p1k1F�fðb� cÞwhN� þwp1k1p/N�F�g;

q22 ¼ wðbþ cÞh/N�F� þwp1k1b2F� � p1k1l2wF� � p1k1wp/2N�F�2;

q23 ¼ whN�ðc2 � p/2N�F� � l2Þ �wðbþ cÞp1k1p/N�F�;

q31 ¼ �wh2/N�2F� � p1k1wF�fðb� cÞhN� þ p1k1p/N�F�g;

q32 ¼ �wðbþ cÞh/N�F� þ p1k1wF�ðp/2N�F� � b2 �w2Þ;

q33 ¼ whN�ð�c2 þ p/2N�F� �w2Þ þ p1k1p/ðbþ cÞwN�F�;

D ¼ �wðw2 þ l2Þðh2/N�2F� þ hp1k1bN�F� þ p21k

21p/N�F�2 � hp1k1cN�F�Þ:

Let Z ¼ AY or Y ¼ A�1Z, where Y= ðy1; y2; y3ÞT . Under this linear transformat ion system (12) becomes

_Y ¼ ðA�1P�AÞY þ FðYÞ; ð13Þ

where, FðYÞ ¼ A�1GðAYÞ, or

FðYÞ ¼f 1

f 2

f 3

0B@

1CA ¼ 1

D

q11g1 þ q12g2 þ q13g3

q21g1 þ q22g2 þ q23g3

q31g1 þ q32g2 þ q33g3

0B@

1CA:

Thus, we have

_y1

_y2

_y3

0B@

1CA ¼

0 �w 0w 0 00 0 l

0B@

1CA

y1

y2

y3

0B@

1CAþ

f 1

f 2

f 3

0B@

1CA: ð14Þ

The system (14) can be written in the form

_U ¼ BU þ f ðU;VÞ;_V ¼ CV þ gðU;VÞ;

ð15Þ

where, U ¼ ðy1; y2ÞT ; V ¼ ðy3Þ; B ¼ 0 �w

w 0

� �; C ¼ ðlÞ; f ¼ ðf 1; f 2Þ and g ¼ ðf 3Þ. Here the matrix B have eigenvalues with

zero real part and C have a negative real eigenvalue. The functions f and g are C2 functions such that f ð0;0Þ ¼ f 0ð0; 0Þ ¼ 0 and gð0;0Þ ¼ g0ð0; 0Þ ¼ 0 (here f 0 means Jacobian matrix of f).

Then, there exists an invariant center manifold WCð0Þ ¼ fðU;VÞ 2 R2 � RjV ¼ hðUÞ; jUj < d; hð0Þ ¼ h0ð0Þ ¼ 0g for small d > 0, where h is C2 function [26,27].

The flow restricted to the center manifold is governed by the two-dimensi onal system

_U ¼ BU þ f ðU; hðUÞÞ: ð16Þ

The following theorem shows that the center manifold can be approximat ed to any desired degree of accuracy.

Theorem 4. Let u be a C1 mapping of a neighborhoo d of the origin in R2 into R with uð0Þ ¼ 0 and u0ð0Þ ¼ 0. Suppose that as U ! 0; ðTuÞðUÞ= OðjUjqÞ where ðTuÞðUÞ ¼ u0ðUÞ½BU þ f ðU;uðUÞÞ� � CuðUÞ � gðU;uðUÞÞ and q > 1. Then as U ! 0,jhðUÞ �uðUÞj= OðjUjqÞ.[26]

We approximat e the central manifold up to a quadratic approximat ion i.e. we take

y3 ¼ hðy1; y2Þ ¼12ðb11y2

1 þ 2b12y1y2 þ b22y22Þ þ h:o:t ð17Þ

here h.o.t stands for higher order terms. Using this in (15), we have

@h@y1

dy1

dtþ @h@y2

dy2

dt¼ ly3 þ f 3; ð18Þ

this gives

wð�b11 þ b22Þy1y2 þwb12y21 �wb12y2

2 �l2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ ¼ Q1y2

1 þ Q2y1y2 þ Q3y22 þ h:o:t; ð19Þ


where,

Q 1 ¼1D�k1q31p11p31 �

sL

q32p221 � hq32p21p11 þ p/q32p21p31 �

uM

p231 q33

h�/q33p21p31 þ p1k1q33p11p31�;

Q 2 ¼1D�k1q31ðp11p32 þ p12p31Þ �

2sL

q32p21p22 � hq32ðp11p22 þ p12p21Þ þ p/q32ðp21p32 þ p22p31Þ�

�2uM

q33p31p32 � /q33ðp21p32 þ p22p31Þ þ p1k1q33ðp11p32 þ p12p31Þ�;

Q 3 ¼1D�k1q31p32p12 �

sL

q32p222 � hq32p22p12 þ p/q32p22p32 �

uM

q33p232 � /q33p22p32 þ p1k1q33p12p32

h i:

Comparing the coefficients of y21; y1y2 and y2

2, we get the following system of linear equation s in b11; b12 and b22 :

�l2

b11 þwb12 ¼ Q 1;

�wb11 � lb12 þwb22 ¼ Q 2;

�wb12 �l2

b22 ¼ Q3:

Solving these equation s, we have

b11 ¼ �w2ðQ 1 þ Q 3Þ þ l

2 ðwQ 2 þ lQ 1Þ� �

l3

4 þw2l ;

b12 ¼ �l2Q2

4 þwl2 ðQ3 � Q1Þ

h il3

4 þw2l ;

b22 ¼ �l2

2 Q 3 � l2 wQ2 þw2ðQ1 þ Q3ÞÞ

h il3

4 þw2l :

We have determined the equation governing the flow on the center manifold. This equation tells about the behavior of the solutions of (15). This follows from the following theorem:

Theorem 5. Suppose that the zero solution of (16) is stable (asymptotically stable) (unstable), then the zero solution of (15) isstable (asymptotically stable) (unstable).[26]

From (16) we have

_y1

_y2

� �¼

0 �w

w 0

� �y1

y2

� �þ f 1

f 2

!;

where,

f 1 ¼ 1Dðq11g1 þ q12g2 þ q13g3Þ;

f 2 ¼ 1Dðq21g1 þ q22g2 þ q23g3Þ;

g1 ¼ �k1 p11y1 þ p12y2 þp13

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h ip31y1 þ p32y2 þ

p33

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h i;

g2 ¼ �sL

p21y1 þ p22y2 þp23

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h i2

� h p11y1 þ p12y2 þp13

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ


p23

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h iþ p/ p21y1 þ p22y2 þ

p23

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ


p33

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h i;


g3 ¼ �uM

p31y1 þ p32y2 þp33

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h i2

� / p21y1 þ p22y2 þp23

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ


p33

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h iþ p1k1 p11y1 þ p12y2 þ

p13

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ


p33

2ðb11y2

1 þ 2b12y1y2 þ b22y22Þ

h i:

Let f kij ¼

@f k

@yi@yj

h ið0;0Þ

and f kijl ¼

@f k

@yi@yj@yl

h ið0;0Þ

.Then

f 111 ¼

1D

q11ð�2k1p11p31Þ þ q12 �2sL

p221 � 2hp21p11 þ 2p/p21p31

� �þ q13 �

2uM

p231 � 2/p21p31 þ 2p1k1p11p31

� �� ; ð20Þ

f 211 ¼

1D

q21ð�2k1p11p31Þ þ q22 �2sL

p221 � 2hp21p11 þ 2p/p21p31

� �þ q23 �

2uM

p231 � 2/p21p31 þ 2p1k1p11p31

� �� ; ð21Þ

f 122 ¼

1D�q11 2k1p32p12ð Þ þ q12 �

2sL

p222 � 2hp22p12 þ 2p/p22p32

� �þ q13 �

2uM

p232 � 2/p22p32 þ 2p1k1p12p32

� �� ; ð22Þ

f 222 ¼

1D�q21 2k1p32p12ð Þ þ q22 �

2sL

p222 � 2hp22p12 þ 2p/p22p32

� �þ q23 �

2uM

p232 � 2/p22p32 þ 2p1k1p12p32

� �� ; ð23Þ

f 112 ¼

1D�q11k1ðp11p32 þ p12p31Þ þ q12 �

2sL

p21p22 � hðp22p11 þ p12p21Þ þ p/ðp21p32 þ p22p31Þ� ��

þ q13 �2uM

p31p32 � /ðp21p32 þ p22p31Þ þ p1k1ðp11p32 þ p12p31Þ� ��

; ð24Þ

f 212 ¼

1D�q21k1ðp11p32 þ p12p31Þ þ q22 �

2sL

p21p22 � hðp22p11 þ p12p21Þ þ p/ðp21p32 þ p22p31Þ� ��

þ q23 �2uM

p31p32 � /ðp21p32 þ p22p31Þ þ p1k1ðp11p32 þ p12p31Þ� ��

; ð25Þ

f 1111 ¼

1D�3q11k1ðp11p33 þ p13p31Þb11 þ q12 �

6sL

p23p21b11 � 3hðp23p11 þ p13p21Þb11 þ 3p/ðp23p31 þ p33p21Þb11

� ��

þ q13 �6uM

p33p31b11 � 3/ðp23p31 þ p33p21Þb11 þ 3p1k1ðp11p33 þ p13p31Þb11

� ��; ð26Þ

f 2222 ¼

1D�3q21k1ðp33p12 þ p13p32Þb22 þ q22 �

6sL

p23p22b22 � 3hðp23p12 þ p13p22Þb22 þ 3p/ðp22p33 þ p23p32Þb22

� ��

þ q23 �6uM

p33p32b22 � 3/ðp23p32 þ p22p33Þb22 þ 3p1k1ðp13p32 þ p33p12Þb22

i; ð27Þ

f 1122 ¼

1D

�q11ð�2k1ðp33p12 þ p13p32Þb12 � k1ðp11p33 þ p13p31Þb22Þ

þ q12 �2sL

p23p21b22 �4sL

p23p22b12 � 2hðp23p12 þ p13p22Þb12 � hðp23p11 þ p13p21Þb22 þ 2p/ðp22p33 þ p23p32Þb12

�

þp/ðp33p21 þ p31p23Þb22

�þ q13 �

2uM

p31p33b22 �4uM

p33p32b12 � 2/ðp22p33 þ p23p32Þb12 � /ðp23p31 þ p33p21Þb22

�

þp1k1ðp11p33 þ p13p31Þb22 þ 2p1k1ðp13p32 þ p33p12Þb12

��; ð28Þ

f 2112 ¼

1D

q21 �k1ðp33p12 þ p13p32Þb11 � 2k1ðp13p31 þ p11p33Þb12

� ��

þ q22 �2sL

p23p22b11 � 4sL

p23p21b12 � hðp23p12 þ p13p22Þb11 � 2hðp23p11 þ p13p21Þb12 þ p/ðp22p33 þ p23p32Þb11

�

þ2p/ðp23p31 þ p33p21Þb12

�þ q23 �

2uM

p32p33b11 �4uM

p31p33b12 � /ðp22p33 þ p23p32Þb11 � 2/ðp23p31 þ p33p21Þb12

�

þp1k1ðp13p32 þ p33p12Þb11 þ 2p1k1ðp13p31 þ p11p33Þb12

��: ð29Þ


The direction and stability of bifurcating periodic solutions of the system (15) can be determined by the following formula [28]

mdjd/

� �/¼/c

¼ 1w½f 1

12 ðf 111 þ f 1

22 Þ � f 212 ðf 2

11 þ f 222 Þ � f 1

11 f211 þ f 1

22 f222 � � ðf 1

111 þ f 2112 þ f 1

122 þ f 2222 Þ:

If m > 0 (m < 0), then the Hopf-bifurc ation is supercritica l (subcritical) and the bifurcating periodic solutions exist for / > /c

(/ < /c). The bifurcating periodic solutions are stable or unstable according as m djd/

h i/¼/c

> 0 or m djd/

h i/¼/c

< 0.

The bifurcating direction of periodic solutions of system (1) is the same as that of (15).

6. Numerical simulation

To check the feasibility of our analysis regarding the existence of interior equilibriu m E� and the correspondi ng stability conditions, we have conducted some numerical computation using MATLAB 7.5.0. For this, we choose the set of parameter values in model system (1) as given in Table 1.

It may be checked that for the above set of paramete rs, the conditions for the existence of interior equilibrium E� (i.e. (2)–(4)) are satisfied. The components of the interior equilibriu m E� are obtained as:

X� ¼ 476:4342 ppm, N� ¼ 660:3704 persons, F� ¼ 684:0234 tons. The eigenvalues of the Jacobian matrix corresponding to the equilibrium E� for the model system (1) are �0.0891,

�0.0286 + 0.0206 i and �0.0286 � 0.0206i, which are either negative or with negative real part. This implies that the equilibrium E� is locally asymptotically stable. For the above data, the solution trajector ies of the model system (1) have been drawn in Fig. 1 with different initial starts. We may see that all the trajectories initiating inside the region of attraction

Table 1Parameter values in model system (1).

Parameter Value Unit

Q0 1 ppm month �1

k 0.05 ppm ðperson month Þ�1

a 0.003 ðmonthÞ�1

k1 0.0001 ðton month Þ�1

s 0.01 month �1

L 1000 person h 0.00001 ðppm month Þ�1

u 0.2 month �1

M 2000 ton p 0.01 person ðtonÞ�1

/ 0.0002 ðperson month Þ�1

p1 0.01 ton ðppmÞ�1

440460

480500

520

640650

660670

680620

640

660

680

700

720

740

760

Carbon dioxide X(t)Human population N(t)

Fore

st b

iom

ass

F(t)

E*

Fig. 1. Nonlinear stability of ðX�;N�; F�Þ in X–N–F space.

0 200 400 600500

550

600

650

700

750

800

Hum

an p

opul

atio

n N(

t)

Time (t)

λ=0.02λ=0.04λ=0.06

0 200 400 600400

500

600

700

800

900

1000

Fore

st b

iom

ass

F(t)

Time (t)

λ=0.02λ=0.04λ=0.06

Fig. 2. Variation of human population and forest biomass with time for different values of k.

0 100 200 300 400 500450

460

470

480

Car

bon

diox

ide

X(t)

Time (t) in months0 100 200 300 400 500

650

655

660

665H

uman

pop

ulat

ion

N(t)

Time (t) in months

0 100 200 300 400 500660

670

680

690

700

Fore

st b

iom

ass

F(t)

Time (t) in months

Fig. 3. Variation of X; N and F with respect to time for / ¼0.0002.


are approaching to the equilibrium point ðX�;N�; F�Þ. This shows the non-linear stability behavior of the interior equilibriu mðX�;N�; F�Þ in X–N–F space. To show the effect of anthropoge nic emissions of CO 2 on human population and forest biomass, the variations in N and F with respect to time for different values of k are shown in Fig. 2. This figure illustrates our analytical results which are stated in Remark 1. It is depicted from this figure that the enhancem ent in atmospheric CO 2 due to human activities ultimately results in declination in human population. The critical value of deforestation parameter ‘/’ (i.e. /c) at which stability loss occurs has been calculated for the above set of parameter values and it is 0.0006. For the above chosen

set of parameter values, the sign of m and m djd/

h i/¼/c

are found to be positive which implies that the Hopf-bifurc ation is super-

critical and the bifurcating periodic solutions are orbitally stable. Figs. 3 and 4 show the variations in atmospheric CO 2, human population and forest biomass with respect to time for

/ ¼ 0:0002 ð< /cÞ and / ¼ 0:0007 ð> /cÞ, respectively. These diagrams show that for / < /c , all the variables attain their equilibrium values but for / > /c , all the variables show oscillatory behavior and do not get stabilized. The bifurcatio n diagrams have been shown in Fig. 5 by taking / as bifurcation parameter. This figure depicts the dynamics of system as the deforestatio n rate increases. From this figure it is evident that for small values of / system is stable but as the value of /crosses a threshold value (/c), the system loses its stability and undergoes Hopf-bifurc ation. Thus the periodic solutions exist for / > /c and hence the Hopf-bifurcatio n is supercrit ical. Fig. 6 has been drawn to show that the bifurcatin g periodic solutions are orbitally stable. It shows that the two solution trajectories one initiating from inside and other from outside the limit cycle, finally approaches to the limit cycle.

0500

10001500

2000

0200

400600

800150

200

250

300

350

400

Carbon dioxide X(t)Forest biomass F(t)

Hum

an p

opul

atio

n N

(t)

Fig. 6. Stable limit cycle for / ¼ 0:0007.

0 1000 2000 30000

500

1000

1500

Car

bon

diox

ide

X(t)

Time (t) in months0 1000 2000 3000

150

200

250

300

350

Hum

an p

opul

atio

n N

(t)

Time (t) in months

0 1000 2000 30000

200

400

600

Fore

st b

iom

ass

F(t)

Time (t) in months

Fig. 4. Variation of X; N and F with respect to time for / ¼0.0007.

2 4 6 8x 10−4

0

500

1000

1500

2000

Car

bon

diox

ide

X(t)

φ2 4 6 8

x 10−4

0

200

400

600

800

Hum

an P

opul

atio

n N

(t)

φ

2 4 6 8x 10−4

0

200

400

600

800

Fore

st b

iom

ass

F(t)

φ

Fig. 5. Bifurcation diagrams of atmospheric CO 2, human population and forest biomass with respect to /. The other parameters are same as given in Table 1.



7. Conclusion

In this paper, a nonlinear mathematical model is proposed and analyzed to explore the dynamics of CO 2 gas in the atmosphere. It is assumed that the concentratio n of CO 2 in the atmosphere increases with a constant rate due to natural processes and with a rate proportional to human population due to anthropogenic factors. The proposed model has four equilibria, and conditions for the existence of these equilibria have been obtained. The local as well as the non-linear stability analysis of the interior equilibriu m have been performed. This analysis provides conditions for the stabilization of atmospheric concentr a- tion of CO 2. Also, it is found that as the growth rate of anthropoge nic CO 2 emissions increases, human population declines whereas forestry biomass increases. The analysis shows that an increase in deforestation rate coefficient ‘/’ leads to increase in the atmospheric concentr ation of CO 2. The Hopf-bifu rcation analysis is performed by taking / as bifurcation parameter. From this analysis, it is found that as the value of / crosses a critical value, the stable equilibrium becomes unstable and sustained oscillations may arise via Hopf-bifurc ation. This critical value of / is obtained for a particular set of parameter values. It is shown that if the deforesta tion rate is below a critical level then the future atmospheric concentr ations of CO 2 getstabilized but if the deforestatio n rate exceeds this critical level, the atmospheric concentratio n of CO 2 shows oscillatory behavior and in this case it is not possible to predict the future concentr ations of CO 2 in the atmosphere. The stability and direction of Hopf-bifurc ation are also analyzed in detail. The expressions are derived for finding the stability and direction of bifurcating periodic solutions using center manifold theory. The model analysis suggests that control on excessive deforestatio n and promoting reforestation are potential strategies for the control and stabilization of the future concentra- tions of CO 2 in the atmosphere. Special attention must be paid to plantatio n of those trees, which consume more CO 2 duringthe photosynthes is process.

Acknowled gements

Authors are grateful to the referees for their useful suggestions . The second author thankfully acknowledges the Univer- sity Grants Commission , New Delhi, India for providing financial assistance in the form of Senior Research Fellowship (20-12/2009(ii) EU-IV).

Appendix A. Existence of E2ðX2;0; F2Þ

The equilibrium E2ðX2;0; F2Þ may be obtained by solving the following set of equation s:

Q0 � aX � k1FX ¼ 0; ðA:1Þ

u 1� FM

� �þ p1k1X ¼ 0: ðA:2Þ

Using the value of X from (A.1) in (A.2), we get the following quadratic polynomial in F:

k1uM

F2 � u k1 �aM

F � ðuaþ Q 0p1k1Þ ¼ 0: ðA:3Þ

The above equation has a unique positive root,

F2 ¼u k1M � að Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 aþ k1Mð Þ2 þ 4up1k

21Q0M

q2k1u

: ðA:4Þ

Using this value of F2 in (A.1), we get the positive value of X ¼ X2.

Appendix B. Existence of E�ðX�;N�; F�Þ

The equilibrium E�ðX�;N�; F�Þ may be obtained by solving the following set of equations:

Q0 þ kN � aX � k1FX ¼ 0; ðB:1Þ

s 1� NL

� �� hX þ p/F ¼ 0; ðB:2Þ

u 1� FM

� �� /N þ p1k1X ¼ 0: ðB:3Þ

From (B.1), we have

X ¼ Q 0 þ kNaþ k1F

: ðB:4Þ


Using (B.4) in (B.2) and (B.3), we get following equations in N and F:

s 1� NL

� �� h

Q0 þ kNaþ k1F

� �þ p/F ¼ 0 ðB:5Þ

and

u 1� FM

� �� /N þ p1k1

Q 0 þ kNaþ k1F

� �¼ 0: ðB:6Þ

Now to show the existence of E�, we plot the isoclines given by Eqs. (B.5) and (B.6).From Eq. (B.5), we may easily note that:

(i) when N ¼ 0, we get following equation in F:

p/k1F2 þ ðp/aþ sk1ÞF þ ðsa� hQ0Þ ¼ 0: ðB:7Þ

The above equation have negative roots. Thus we get a negative value of F, say Fa at N ¼ 0,(ii) when F ¼ 0, we have N ¼ Na ¼ Lðsa�hQ0Þ

saþhkL > 0 if s > hQ0a , and

(iii) dN dF > 0.

From Eq. (B.6), we may easily note that:

(i) when N ¼ 0; F ¼ Fb ¼ F2 > 0.(ii) when F ¼ 0; N ¼ Nb ¼ uaþp1k1Q0

a/�p1k1k > 0 provided a/ > p1k1k, and (iii) dN

dF < 0.

Now the isoclines given by Eqs. (B.5) and (B.6) will intersect at a unique point ðN�; F�Þ in the interior of the positive quad- rant (as shown in Fig. B.1 ) provided:

Nb > Na;

i.e.

ðuaþ p1k1Q0Þðsaþ hkLÞ > Lðsa� hQ 0Þða/� p1k1kÞ: ðB:8Þ

Finally, using these values of N� and F� in (B.4), we get the positive value of X ¼ X�.

Appendix C. Stability analysis of equilibri a E1; E2 and E3

We discuss the stability of equilibria E1; E2 and E3 by finding the sign of the eigenvalues of Jacobian matrix correspondi ng to each equilibrium . The Jacobian matrix for model system (1) is given as follows:

−500 0 500 1000 1500 2000−500

0

500

1000

F

N

(F*, N*)

Fig. B.1. Existence of ðF�;N�Þ.


P ¼�ðaþ k1FÞ k �k1X

�hN s 1� 2NL

� �� hX þ p/F p/N

p1k1F �/F u 1� 2FM

� �� /N þ p1k1X

264

375:

Let Pi be the Jacobian matrix P evaluated at equilibriu m Ei ði ¼ 1;2;3Þ.The eigenvalues of the Jacobian matrix P1 are �a, s� ðhQ0=aÞ and uþ ðp1k1Q 0=aÞ. Thus E1 has always stable manifold lo-

cally in X-direction and unstable manifold locally in F-direction whereas it has locally stable (unstable) manifold in N-direc-tion provided s� ðhQ 0=aÞ < 0 (> 0). Thus if E1 has locally stable manifold in N-direction, then E3 will not exist.

From the Jacobian matrix P2, we find that one eigenvalue of P2 is sþ p/F2 � ðhQ 0=ðaþ k1F2ÞÞ which is positive whenever E3 exists and the other two eigenvalues are negative. This shows that E2 has locally stable manifold in X—F plane and unstable manifold locally in N-direction whenever E3 exists.

Further from the Jacobian matrix P3, we find that one eigenvalue of P3 is uðasþ khLÞ � /Lðas� hQ0Þ þ p1k1sðQ0 þ kLÞ,which is positive whenever E� exists and the other two eigenvalues are either negative or with negative real part. Thus E3

has locally stable manifold in X—N plane and unstable manifold locally in F-direction whenever E� exists.

Appendix D. Proof of Theorem 2

Consider the following positive definite function:

W ¼ 12ðX � X�Þ2 þm1 N � N� � N� ln

NN�

� �þm2 F � F� � F� ln

FF�

� �; ðD:1Þ

where m1 and m2 are positive constants to be chosen appropriately. Now differentiating ‘W’ with respect to ‘t’ along the solution of model system (1), we get

dW dt ¼ �ðaþ k1F�ÞðX � X�Þ2 �m1s

LðN � N�Þ2 �m2u

MðF � F�Þ2 þ ðk�m1hÞðX � X�ÞðN � N�Þ þ ðm1p/�m2/ÞðN

� N�ÞðF � F�Þ � k1XðX � X�ÞðF � F�Þ þm2p1k1ðX � X�ÞðF � F�Þ: ðD:2Þ

Choosing m1 ¼ kh and m2 ¼ pm1 ¼ pk

h , we get

dW dt ¼ �ðaþ k1F�ÞðX � X�Þ2 � ks

hLðN � N�Þ2 � kpu

hMðF � F�Þ2 � k1XðX � X�ÞðF � F�Þ þ kpp1k1

hðX � X�ÞðF � F�Þ: ðD:3Þ

Now we note that dW=dt can be made negative definite inside the region of attraction ‘X’ provided condition (7) is satisfied.

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a mathematical model to study the dynamics of carbon dioxide gas in the atmosphere

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