Mathematical modeling of mitigation of carbon dioxide emissions by controlling the population pressure

Abstract

The anthropogenic carbon dioxide $(CO _2)$ emission from the burning of fossil fuels is the prime cause behind the menace of global warming. Over the past few decades, fossil fuel consumption has increased drastically to fulfill the energy demand of the growing population and economy. The population pressure has not only contributed to the increase in fossil fuel consumption but also accelerated the deforestation for industrial, agricultural, and infrastructure expansion. This paper presents a nonlinear mathematical model to study the effect of an increase in fossil fuel use and deforestation due to population pressure on atmospheric carbon dioxide concentration. Further, the effect of economic efforts applied to reduce the population pressure over the control of atmospheric $CO _2$ levels is explored. The model analysis shows that an increase in the fossil fuel consumption rate causes an increase in the equilibrium level of carbon dioxide. Further, it is found that an increase in the deforestation rate coefficient has a destabilizing effect on the stability of positive state of the system. If the deforestation rate crosses a critical threshold, the positive state of the system loses stability and the periodic solutions arise via Hopf-bifurcation. It is shown that at high deforestation rates, an increase in the implementation rate of economic efforts applied to reduce the population pressure may cause reduction in the amplitude of periodic oscillations. The periodic oscillations may disappear if the implementation rate of economic effort increased beyond a critical threshold and the concentration of carbon dioxide gets stabilized.

Appendices

Appendix

A Proof of Lemma 1

From the fourth equation of the model system (1), we have

$$\begin{aligned} \frac{{\textrm{d}}B}{dt}\le u B\left( 1-\frac{B}{M}\right) . \end{aligned}$$

By comparing the above differential inequality with the differential equation

$$\begin{aligned} \frac{{\textrm{d}}B}{dt}=uB\left( 1-\frac{B}{M}\right) , \end{aligned}$$

and using a standard comparison theorem [37, 38], we get

$$\begin{aligned} B(t)\le \frac{M}{1+\left( \frac{M}{B(0)}-1\right) e^{-ut}}. \end{aligned}$$

Let $\epsilon >0$ be given. Then $\exists$ $t_0>0$ such that

$$\begin{aligned} B(t)\le M+\epsilon , \quad for \quad all \quad t\ge t_0. \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup B(t)\le M. \end{aligned}$$

From the third equation of model system (1), we have

$$\begin{aligned} \frac{dF_s}{dt} \le \eta (F_{s_0}-F_s). \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup F_s(t)\le F_{s_0}. \end{aligned}$$

From the second equation of the model system (1), we get

$$\begin{aligned} \frac{dN}{dt} \le \left( r+\pi _1\phi _{11}M+\pi _2\eta _{11}F_{s_0} \right) N -\frac{r}{L}N^2, \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup N(t) \le \frac{L}{r}(r+\pi _1\phi _{11}M+\pi _2\eta _{11}F_{s_0})=N_m \ (say). \end{aligned}$$

From the first equation of model system (1), we have

$$\begin{aligned} \frac{dC}{dt} \le Q-\alpha C+\lambda N_m+e_1\eta _{11}F_{s_0}N_m, \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup C(t) \le \frac{Q+\lambda N_m+e_1\eta _{11}F_{s_0}N_m}{\alpha }={\hat{C}}_m \ (say). \end{aligned}$$

This proves the lemma.

B Proof of Lemma 2

From the fourth equation of the model system (55), we have

$$\begin{aligned} \frac{{\textrm{d}}B}{dt}\le uB\left( 1-\frac{B}{M}\right) . \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup B(t)\le M. \end{aligned}$$

The third equation of model system (55) gives

$$\begin{aligned} \frac{dF_s}{dt} \le \eta (F_{s_0}-F_s). \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup F_s(t)\le F_{s_0}. \end{aligned}$$

From the second equation of the model system (55), we have

$$\begin{aligned} \frac{dN}{dt} \le \left( r+\pi _1\phi _{11}M+\pi _2\eta _{11}F_{s_0} \right) N -\frac{r}{L}N^2. \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup N(t) \le \frac{L}{r}(r+\pi _1\phi _{11}M+\pi _2\eta _{11}F_{s_0})=N_m \ (say). \end{aligned}$$

From the fifth equation of model system (55), we have

$$\begin{aligned} \frac{dP}{dt} \le sN_m-s_0P. \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup P(t) \le \frac{s}{s_0}\frac{L}{r}(r+\pi _1\phi _{11}M+\pi _2\eta _{11}F_{s_0})=P_m \ (say). \end{aligned}$$

From the sixth equation of system (55), we get

$$\begin{aligned} \frac{dI}{dt} \le \psi _1P_m-\psi _0I. \end{aligned}$$

This implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup I(t) \le \frac{\psi _1}{\psi _0}P_m=I_m \ (say). \end{aligned}$$

From the first equation of model system (55), we have

$$\begin{aligned} \frac{dC}{dt} \le Q-\alpha C+\lambda N_m+e_1\eta _{11}F_{s_0}N_m+e_1\eta _{12}F_{s_0}P_m, \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{t \rightarrow \infty }\sup C(t) \le \frac{1}{\alpha }(Q+\lambda N_m+e_1\eta _{11}F_{s_0}N_m+e_1\eta _{12}F_{s_0}P_m)=C_m \ (say). \end{aligned}$$

This proves the lemma.