1. Introduction

The increased concentration of atmospheric has direct effect on the growth of plant biomass such as vegetation, forests, etc. This effect is known as ‘ fertilization’. According to ‘The Intergovernmental Panel on Climate Change’ fertilization is defined as “the enhancement in the net primary productivity of terrestrial vegetation that occurs as a result of elevated atmospheric concentration” ^[12]. It has also been stated in ^[12] that in standard carbon cycle model calculations, fertilization acts as a sink in the carbon cycle. In ecological modeling studies, it has been observed that the terrestrial biomass grows due to uptake of atmospheric carbon dioxide introduced into the atmosphere naturally as well as by human activities. During interaction of plant biomass with, carbon is exchanged naturally through photosynthesis, respiration, decomposition and combustion, etc. ^[12]. The transformation of in to glucose by plant leaves during photosynthesis process is given by the following simple chemical equation ^[11],

Some investigations have been conducted to study the effect of elevated level of atmospheric on growth of plant species ^{[2, 3, 6, 8, 14, 15, 16, 17, 20, 22, 23, 26, 27, 28]}. In particular, Albertine et al. ^[2] have conducted an experiment to study the growth of pollen from flowers at elevated levels of ozone and carbon dioxide concentrations and found considerable increase in pollens produced at elevated carbon dioxide level. Ambavaram et al. ^[3] have investigated the effect of elevated atmospheric carbon dioxide on cereal crops such as rice (Oryza sativa) and found considerable increase in rice yield. Madhu and Hatfield ^[14] have shown that the growth and yield of most of the agricultural crops are significantly enhanced by above and belowground environmental conditions and elevated level of atmospheric carbon dioxide. It has been observed that if the rate of photosynthesis is increased, plants can grow faster attaining their equilibrium with increased biomass ^[15]. Miri et al. ^[17] studied the effect of elevated on vegetative growth of plants such as soybean and lamb’s-quarter and found significant increase in their growth. Poorter and Perez-Soba ^[20] presented an analysis to study the effect of elevated carbon dioxide on growth of plant species. They have shown that increased level of atmospheric carbon dioxide stimulates the rate of photosynthesis enhancing the biomass of plant species. Prior et al. ^[22] presented a review of elevated atmospheric effects on plant growth and water relations. They have shown that plant growth can be enhanced by increasing the concentration of atmospheric . Wolfe-Bellin et al. ^[28] examined the effect of rising concentration of atmospheric on forb phytolacca americana L. (Phytolaccacea) and found considerable increase in its growth.

Further, the depletion of plant biomass is chiefly caused by human activities such as cutting trees for agriculture, food and energy, establishment of industries, household appliances, creating new houses, farmlands, etc. Several investigations have been made to study the effect of population density on degradation of plant biomass ^{[1, 5, 7, 10, 13, 19, 24, 25]}. In this regard, Dubey et al. ^[7] have presented a mathematical model to study the effect of population and population pressure augmented industrialization on forestry resources. They have shown that the equilibrium density of biomass decreases as the equilibrium densities of population and population pressure augmented industrialization increases. Shukla et al. ^[24] have studied the effect of population on forestry biomass using a nonlinear mathematical model. They have shown that the forestry biomass may become extinct if population increases without control.

It is pointed out here that increase in the carbon dioxide concentration in the atmosphere due to human activities is a key factor to stimulate the growth of biomass ^{[4, 5, 18, 21]}. Further, increase in the population density causes depletion of plant biomass but it increases the concentration of in the atmosphere which is used by plant leaves during photosynthesis thus regulating the concentration of in the atmosphere. In view of the above, in this paper, we have proposed and analyzed a nonlinear mathematical model to study the effect of increased carbon dioxide concentration (due to population density) on plant growth.

2. Mathematical Model

To model the dynamics of density dependent emission of carbon dioxide, we have made the following assumptions,

1. The equilibrium level of carbon dioxide is enhanced significantly due to increase in population density and is assumed to be in the direct proportion of population density.

2. The growth rates of plant biomass and population density are assumed to be following logistic equation.

3. The depletion of plant biomass is in the direct proportion of its density as well as the human population density.

4. The growth of plant biomass is in the direct proportion of its density as well as the concentration of carbon dioxide.

5. The growth of population density is in the direct proportion of its density as well as the density of plant biomass.

In the atmosphere, under consideration, let and denote the densities of plant biomass and human population at any time , respectively. Let be the concentration of carbon dioxide , emitted into the atmosphere naturally as well as by human activities, at any time . It is pointed out here that the plant biomass density decreases due to human activities such as cutting of trees for household appliances, paper industries, agriculture, creating new houses and other purposes. Thus, we assume that the plant biomass density decreases due to human population while human population density increases due to plant biomass and therefore the decrease in plant biomass density is assumed to be proportional to the human population as well as plant biomass density. Further, the increase in human population density is assumed to be proportional to the human population as well as plant biomass density. As pointed out in previous section that the growth and yield of plant biomass depend on carbon dioxide concentration and hence it is obvious to assume that the growth rate of plant biomass increases due to carbon dioxide whereas the concentration of carbon dioxide decreases due to plant biomass. Thus, the growth rate of biomass is assumed to be proportional to the density of plant biomass as well as the concentration of carbon dioxide. In the modeling process, the dynamics of the biomass density is assumed to be governed by logistic equation as follows,

The constants and are intrinsic growth rate and carrying capacity of plant biomass respectively. The constant represents the depletion rate coefficient of plant biomass due human population, is the growth rate coefficient of plant biomass due to uptake of . Further, let and be intrinsic growth rate and carrying capacity of human population, respectively. It is assumed that the growth of human population due to biomass is directly proportional to the densities of plant biomass as well as human population (i.e. ). The constant denotes the growth rate coefficient of human population due to plant biomass. In view of these assumptions, the differential equation governing the dynamics of the population is given by,

Increase in carbon dioxide concentration into the atmosphere due to natural sources (e.g. ocean release, the combustion of organic matter, wildfires, the respiration processes of living organisms, etc.) is assumed to be constant (let ). It is noted here that the anthropogenic emissions of carbon dioxide into the atmosphere is due to human population and therefore the growth rate of atmospheric carbon dioxide is assumed to be proportional to the human population density (i.e. ). Further, as discussed above, the depletion of carbon dioxide is assumed to be proportional to the biomass density as well as the concentration of carbon dioxide. The constant denotes the growth rate coefficient of due to human population, be its natural depletion rate coefficient and is its depletion rate coefficient due to plant biomass. Thus, the equation governing the concentration of in the atmosphere is given as follows,

Thus, the dynamics of the system is governed by the following set of nonlinear ordinary differential equations,

(1)

(2)

(3)

All the constants taken here are positive.

Now, we analyze the model (1) – (3) under the following two cases:

(I)

(II)

2.1. Case I.

To analyze the model system (1) – (3), we need the bounds of dependent variables. For this, we establish the region of attraction in the following lemma ^[9].

Lemma 2.1.1. The set

is the region of attraction for all solutions of the model system (1) – (3) initiating in the interior of positive octant, where,

provided,

The model system (1) – (3) has four non-negative equilibria;

(i)

(ii)

(iii)

(iv)

The existence of or is obvious and hence omitted.

2.1.1.1. Existence of .

The positive solution of can be obtained by solving the following algebraic equations,

(4)

(5)

Eliminating from (4) and (5), we get,

Using the value of we can find the value of from equation (4) or (5).

2.1.1.2. Existence of

The positive solution of can be obtained by solving the following algebraic equations,

(6)

(7)

(8)

From (7) and (8), we get respectively,

(9)

(10)

Using (9) and (10) in (6), we get

(11)

Where

From (11), we get, (say).

Thus, we have a unique positive root (say ), provided,

(12)

which makes .

Using this value of , we will get the values of and from equations (9) and (10) respectively.

Remark: From equation (11), it can be checked that and . This implies that the equilibrium density of plant biomass increases due to increase in the rates of introduction of carbon dioxide by natural factors as well as by population density.

Similarly, it can also be proved that, and . This implies that equilibrium density of plant biomass decreases as the growth rate of population density increases.

To establish the local stability behaviour of and , we compute the following general variational matrix for model system (1) – (3),

From the matrix, corresponding to , it can be easily seen that two eigenvalues of the variational matrix will always be positive and hence it is unstable. Also corresponding to , we may easily note that one eigenvalue of the variational matrix is , which is positive for the existence of and hence is unstable whenever exist. Further, corresponding to , we may note that one eigenvalue of the variational matrix is always positive and hence it is also unstable.

The local stability behaviour of interior equilibrium is given in the following theorem.

Theorem 2.1.1. Let the following inequality holds

(13)

Then the equilibrium is locally asymptotically stable.

(See appendix A for proof)

Theorem 2.1.2. Let the following inequality holds in

(14)

Then the interior equilibrium is nonlinearly asymptotically stable.

(See appendix B for proof)

The above theorems imply that under appropriate conditions the dynamical system would remain in equilibrium. The density of plant biomass decreases as the density of population increases and it increases as the rate of introduction of into the atmosphere increases.

Remark: If the rate of introduction of due to population is zero (i.e.), the stability conditions (13) and (14) are satisfied automatically. This implies that has destabilizing effect on the system.

2.2 Case II.

In this case also, we need the bounds of dependent variables, to analyze the model system (1) – (3). For this purpose, we establish the region of attraction in the following lemma ^[9].

Lemma 2.2.1. The set

is region of attraction for all solutions of the model system (1) – (3) initiating in the interior of positive octant, where, .

In this case also, the model system (1) – (3) has four non-negative equilibria;

(i)

(ii)

(iii)

(iv)

The existence of and is obvious hence omitted. The proof of existing in this case is the same as in the case (i) hence it is omitted.

2.2.1.1. Existence of

The positive solution of can be obtained by solving the following algebraic equations,

(15)

(16)

(17)

From (16) and (17), we get respectively,

(18)

(19)

Using (18) and (19) in (15), we get

(20)

where

From (20), we get,

(21)

Thus, we have a unique positive root (say ), provided (12), Using this value of , we will get the values of and from equations (18) and (19) respectively.

Remark: From equation (20), as before, it can be checked that . This implies that the equilibrium density of plant biomass increases as the rate of introduction of carbon dioxide increases due to natural factors. Similarly, it can also be proved that, and . It means that equilibrium density of plant biomass decreases as the growth rate of population increases.

As in case (I), the local stability behaviour of and can be easily checked by computing the variational matrix for model system (1) – (3) and it is found that and are unstable.

The stability behaviour of interior equilibrium is given in the following theorems.

Theorem 2.2.1. The interior equilibrium is locally asymptotically stable without any condition.

(See appendix C for proof)

Theorem 2.2.2. The interior equilibrium is nonlinearly asymptotically stable inside the region of attraction without any condition.

(See appendix D for proof)

The above theorems imply that the dynamical system would remain in equilibrium without any condition.

3. Numerical Simulation and Discussion

In this section, we conduct some numerical simulations using MAPLE 7 to check the feasibility of the dynamical system regarding stability conditions by choosing the following set of parameter values in model system (1) – (3). We have performed the numerical simulation only for case (I).

The equilibrium values of corresponding to above data are given as,

The eigenvalues of variational matrix corresponding to equilibrium for the model system (1) – (3) are and . It is noted here that all eigenvalues of the variational matrix are negative. Hence the interior equilibrium is locally asymptotically stable. For the above set of parameter values, the condition of existence of interior equilibrium i.e. (12), local stability condition i.e. (13) and nonlinear stability condition i.e. (14) are satisfied.

Figure 1. Nonlinear stability in B-N plane

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The nonlinear stability behaviour of in and plane has been shown in figure 1 and figure 2 respectively, with different initial starts. From these figures, it can be seen that all the trajectories initiating inside the region of attraction are approaching towards the equilibrium values and respectively. The variation of concentration of carbon dioxide and density of biomass with time for different values of (i.e. at ) is shown in figure 3 and figure 4, respectively. From these figures, it is shown as the rate of introduction of due to population density increases, the equilibrium concentration of carbon dioxide into the atmosphere increases and hence the density of biomass. The variation of concentration of carbon dioxide and biomass density with time for different values of (i.e. at ) is shown in figure 5 and figure 6, respectively. From these figures it is observed that as the rate of uptake of carbon dioxide with biomass increases, the equilibrium concentration of carbon dioxide in the atmosphere decreases while the density of biomass increases ^{[3, 14, 17, 20, 28]}. The variation of densities of biomass and population with time for different values of is shown in figures 7 and 8 respectively. From figure 7, we note that the equilibrium density of plant biomass decreases as the growth rate of population density increases ^{[1, 10, 13, 19]}. From figure 8, we depict that the density of population increases as increases.

Figure 2. Nonlinear stability in B-C plane

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Figure 3. Variation of concentration of CO₂ with time 't' for different values of δ

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Figure 4. Variation of biomass density with time 't' for different values of δ

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Figure 5. Variation of concentration of CO₂with time 't' for different values of S₂

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Figure 6. Variation of biomass density with time 't' for different values of S₂

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Figure 7. Variation of biomass with time 't' for different values of r₁

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The variation of plant biomass density and concentration of for different values of and with time is shown in figures 9 and 10 respectively. From these figures, it is observed that, when and , the equilibrium density of biomass and concentration of are much less than their values when and ; and . This implies that the population density has significant effect on carbon dioxide concentration in the atmosphere.

Figure 8. Variation of population density with time 't' for different values of r₁

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Figure 9. Variation of biomass (B)with time 't' for different values of Q and δ

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Figure 10. Variation of concentration of CO₂ with time 't' for different values of Q and δ

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4. Conclusion

In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of human population density dependent emission of carbon dioxide on the growth of plant biomass. The dynamics of the model system consists of three nonlinearly interacting dependent variables, namely; plant biomass density, population density and concentration of carbon dioxide. The model analysis is conducted using stability theory of nonlinear ordinary differential equations. It is shown, analytically and numerically, that the density of plant biomass decreases due to increase in population density ^{[1, 5, 19]} and increases due to the presence of carbon dioxide in the atmosphere ^{[4, 18, 21]}. Thus, the plant biomass density increases as the rate of introduction of , due to natural as well as human activity related factors, increases and it decreases as the growth rate of population increases. It is also shown that the concentration of carbon dioxide increases as the density of human population increases.

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Appendix A

To establish the local stability behaviour of , we consider the following positive definite function

(A1)

Where are small perturbations about . The constants are all positive to be chosen appropriately.

Differentiating (A1) with respect to , we get

Now using the linearized system of (1) – (3), we get,

Choosing and , will be negative definite provided the condition (13) is satisfied and hence the theorem.

Appendix B

To establish the nonlinear stability behaviour of , we consider the following positive definite function

(B1)

where are positive constants to be chosen appropriately.

Differentiating (B1) with respect to along the model system (1) – (3), we get,

Now substituting the values of , and from the model system (1) – (3) about , we get

Choosing and , will be negative definite inside the region of attraction provided the condition (14) is satisfied and hence the theorem.

Appendix C

To establish the local stability behaviour of , we consider the following positive definite function

(C1)

Where are small perturbations about . The constants and are positive to be chosen appropriately.

Differentiating (C1) with respect to , we get

Now using the linearized system of (1) – (3), we get,

Choosing and , will be negative definite without any condition and hence the theorem.

Appendix D

To establish the nonlinear stability behaviour of , we consider the following positive definite function

(D1)

where and are positive constants to be chosen appropriately.

Differentiating (D1) with respect to along the model system (1) – (3), we get,

Now substituting the values of , and from the model system (1) – (3) about, we get

Choosing and , will be negative definite inside the region of attraction without any condition and hence the theorem.