Mathematical modelling and analysis to study the impact of mining on natural interactive dynamics between plants and carbon dioxide

Abstract

A mathematical model is formulated to delve into the changes that have been produced on the natural interactive dynamics between plants and carbon dioxide when mining intervenes between them. The model is represented as a nonlinear system of ordinary differential equations comprising dynamic interactions among plants, carbon-dioxide, and mining. In the model formulation, it is assumed that mining activities enhance carbon dioxide in the atmosphere and cause plants’ ruination. The qualitative analysis of the model is executed to obtain properties like boundedness, identification of equilibrium solutions, and their stabilities. Besides qualitative investigation, the formulated model is also quantitatively analyzed from the obtained numerical solutions. The numerical values of input model parameters are estimated from global observational data of plants and carbon dioxide. However, the influential parameters towards the model outcomes are identified through a global sensitivity analysis technique that combines the evaluation of Partial Rank Correlation Coefficients (PRCCs) with Latin Hypercube Sampling (LHS). To better elucidate the role of mining over the interactive dynamics, results obtained from the proposed model are compared with the model that does not involve mining activities as one of the dynamic variables. The overall mathematical analyses reveal that the equilibrium level of carbon-dioxide concentration exhibits a significant increase, and stability of the system gets harder to achieve in the presence of mining. It is also being deduced that system outcomes are more sensitive towards mining-related parameters than plant density and carbon dioxide concentration parameters.

Appendix 1 (Proof of Lemma 3.1)

Taking

$$\mathop {\lim }\limits_{t \to \infty } \sup (C(t)) \le C_{M}$$

(44)

Then, the Eq. (3) of model system (II) implies

$$\begin{gathered} \frac{{{\text{d}}P}}{{{\text{d}}t}} \le rP - r_{0} \frac{{P^{2} }}{L} + \theta \beta CP \hfill \\ \, \le \left( {r + \theta \beta C_{M} } \right)P - r_{0} \frac{{P^{2} }}{L} \hfill \\ \end{gathered}$$

Comparing the obtained differential inequality with the differential equation:

$$\frac{{{\text{d}}P}}{{{\text{d}}t}} = \left( {r + \theta \beta C_{M} } \right)P - r_{0} \frac{{P^{2} }}{L}$$

and applying comparison theorem (Freedman and So 1985; Hale 1969), boundedness of $P$ is obtained as:

$$0 \le P(t) \le P_{M}$$

(45)

where

$$P_{M} = \frac{L}{{r_{0} }}(r + \theta \beta C_{M} )$$

(46)

Similarly, the Eq. (4) implies

$$0 < A(t) \le A_{M}$$

(47)

where

$$A_{M} = \frac{{A_{0} }}{{\mu_{0} }}$$

(48)

The Eq. (5) of system (II) gives

$$\begin{gathered} \frac{{{\text{d}}C}}{{{\text{d}}t}} \le C_{0} + \lambda A + \gamma P - \delta_{0} C \hfill \\ \, \le C_{0} + \lambda A_{M} + \gamma P_{M} - \delta_{0} C \hfill \\ \end{gathered}$$

Then, by using comparison theorem, we get

$$0 < C(t) \le C_{M}$$

(49)

where

$$C_{M} = \frac{{C_{0} + \lambda A_{M} + \gamma P_{M} }}{{\delta_{0} }}$$

(50)

Now, substituting the value of $P_{M}$ in the above equation, the following value is obtained

$$C_{M} = \frac{{C_{0} r_{0} + r_{0} \lambda A + r\gamma L}}{{\delta_{0} r_{0} - \gamma \theta \beta L}}$$

(51)

However, for positivity of $C_{M} ,$ the following condition should true:

$$\delta_{0} r_{0} > \gamma \theta \beta L$$

(52)

From the obtained $C_{M} ,$ $P_{M}$ can be obtained from Eq. (46).