Mathematical Models of how the Transpiration Affects Rainfall through Agriculture Crops

Abstract

The water vapor in the atmosphere is mainly formed through the vaporization of water from oceans, open land water sources, and transpiration through forests and agricultural crops. In the present study, our aim was to formulate a mathematical model to see the effect of transpiration from agricultural crops on rainfall. In the model formulation, it is assumed that clouds are formed in the atmosphere through the transpiration of agricultural crops proportional to the crops apart from the oceans and other sources (assumed constant). It is observed that in the presence of agriculture crops, the average rainfall increases. In the presence of clouds, cloud seeding techniques (use of conducive aerosols) are used to stimulate rainfall, which increases the agricultural crops with saturated type functional form. To capture the impact of environmental fluctuations, the stochastic version of the proposed model is also studied. The analysis of the model reveals that cloud seeding not only increases the rainfall, but it also increases the agricultural crops and clouds in the atmospheric environment.

Appendices

Appendix A

Proof

Since the right-hand side of the Eq. (7) are locally Lipschitz continuous, hence system Eq. (7) possesses a unique local solution in $[-\tau , \tau _l]$, Now from the third equation of Eq. (7)

$$\begin{aligned} {dA(t)}=&A(-\tau ) \exp \int _{-\tau }^t\left\{ s(u)\left( 1-\frac{A(u)}{L}\right) +\frac{w(u)C_r(u)}{m+C_r(u)}\right\} \\ &du>0 \ \ \forall \ t\ge -\tau . \end{aligned}$$

(14)

While from the first equation of model Eq. (7), $\frac{dC_d(t)}{dt}\ge -\lambda _0 C_d(t), \ \ \forall \ t\ge -\tau ,$ which gives $C_d(t)\ge C_d(-\tau ) e^{-\lambda _0t}>0.$ Similarly, the second equation of model Eq. (7) gives $C_r\ge C_r(-\tau ) e^{-r_0 t}>0$.

Appendix B

Proof

Suppose that $(C_d(t), C_r(t), A(t))$ be a solution of the system Eq. (7) with initial conditions Eq. (8), then from the third equation of the system Eq. (7), we can write

$$\begin{aligned} \frac{dA}{dt}\le & {} s(t) A(t)\left( 1-\frac{A(t)}{L}\right) +w^u A(t)\le A(t)\left( s^u+w^u-\frac{s^l A(t)}{L}\right) . \end{aligned}$$

(15)

Let $M_1>\frac{(s^u+w^u)L}{s^l}$ then if $A(0)<M_1,$ we have $A(t)\le M_1$ $\forall \ t\ge 0$. But if $A(0)>M_1$, then $-e_1=M_1(s+w^u-\frac{sM_1}{L^l})<0,$ so we have $\frac{dA(t)}{dt}\le -e_1$. Therefore, there exists $T_1>0$ such that $A(t)\le M_1 \ \forall \ t\ge T_1.$ Let $M_2>\frac{Q_d^u+\kappa ^u M_1}{\lambda _0}$ then from the first equation of Eq. (7)

$$\begin{aligned} \frac{dC_d(t)}{dt}\le & {} Q_d^u+\kappa ^u M_1-{\lambda _0}C_d(t). \end{aligned}$$

So if $C_d(0)<M_2,$ then $C_d(t)\le M_2$ $\forall \ t\ge T_1+\tau .$ In case $C_d(0)>M_2$, $\frac{dC_d(t)}{dt}<-e_2,$ where $e_2=(Q_d^u+\kappa M_1 -\lambda _0 M_2),$ therefore there exists $T_2>T_1+\tau ,$ such that $C_d(t)\le M_2$$\forall t\ge T_2>T_1+\tau$. Similarly, we can show that there exists $T_3>T_2>T_1+\tau ,$ such that $C_r(t)\le M_3$ $\forall$ $t\ge T_3$, $M_3>rM_2/r_0.$

Now $\frac{dC_d(t)}{dt}\ge Q_d^l-\lambda _0 C_d(t),$ let $0<m_2< \frac{Q_d^l}{\lambda _0},$ then $C_d(t)>\frac{Q_d^l}{\lambda _0}$ $\forall t\ge T_4.$ Same way if $m_3< \frac{r^lQ_d^l}{\lambda _0r_0},$ then there exists a $T_5>0,$ such that $C_r(t)>m_3$ $\forall \ t \ge T_5$ and if $m_1< \frac{\left( s^l+\frac{w^l m_3}{m+M_3}\right) L}{s^u},$ then there exists $T_6>0,$ such that $A(t)\ge m_1 \ \forall \ t\ge T_6.$

Now to prove global stability Let $(C_d, C_r, A)$ and $(\tilde{C_d}, \tilde{C_r}, \tilde{A})$ be any two solutions of the system Eq. (7) with initial conditions Eq. (8). Define $W_1(t)=\vert C_d(t)-\tilde{C}_d(t)\vert ,$ $W_2(t)=\vert C_r(t)-\tilde{C}_r(t)\vert ,$ $W_3(t)=\vert A(t)-\tilde{A}(t)\vert .$ The right upper derivative of $W_1, \ W_2$ and $W_3$ can be obtained as

$$\begin{aligned} D^+W_1(t)= & \ (-\lambda _0(C_d(t)-\tilde{C}_d(t))+\kappa (A(t)-\tilde{A}(t)))sgn(C_d(t)-\tilde{C}_d(t)),\nonumber \\ & {} \le -\lambda _0^l\vert C_d(t)-\tilde{C}_d(t)\vert +\kappa (t)\vert A(t-\tau )-\tilde{A}(t-\tau )\vert .\end{aligned}$$

(16)

$$\begin{aligned} D^+W_2(t)= & \ (r(C_d(t)-\tilde{C}_d(t))-r_0( C_r(t)-\tilde{C}_r(t)))sgn(C_r(t)\\&-\tilde{C}_r(t)),\nonumber {} \le r^u\vert C_d(t)-\tilde{C}_d(t)\vert -r_0^l\vert C_r(t)-\tilde{C}_r(t)\vert .\\ &D^+W_3(t)\le & \ s(t)\vert A(t)-\tilde{A}(t)\vert -\frac{s(t)(A(t)+\tilde{A}(t))\vert A(t)-\tilde{A}(t)\vert }{L}\nonumber \\&+(2w^u+mw^u) \vert A(t-\tau )-\tilde{A}(t-\tau )\vert \\&+ w^u\tilde{A}(t)\vert C_r(t)-\tilde{C}_r(t)\vert .\nonumber \end{aligned}$$

(17)

Let $W_4(t)=\int _{t-\tau }^t\kappa (u+\tau )\vert A(u)-\tilde{A}(u)\vert du,$ then

$$\begin{aligned} D^+W_4(t)=\kappa (t+\tau )\vert A(t)-\tilde{A}(t)\vert -\kappa (t) \vert A(t-\tau )-\tilde{A}(t-\tau )\vert . \end{aligned}$$

So if $W=W_1+W_2+W_3+W_4,$ then

$$\begin{aligned} D^+W(t)\le & {} -(\lambda _0-r_u)\vert C_d(t)-\tilde{C}_d(t)\vert -(r_0-w^um_1\vert C_r(t)-\tilde{C}_r(t)\vert \nonumber \\&-(-\kappa ^u-s^u-w^um+\frac{s^u}{L^l}(A(t)+\tilde{A}(t)))\vert A(t)-\tilde{A}(t)\vert ,\nonumber \\&\le & {} -C_1\vert C_d(t)-\tilde{C}_d(t)\vert -C_2\vert C_r(t)-\tilde{C}_r(t)\vert -C_3\vert A(t)-\tilde{A}(t)\vert , \end{aligned}$$

(18)

where $C_1=\lambda _0-r_u,$ $C_2=r_0-w^um_1$ and $C_3=\frac{2m_1s^u}{L^l}$$-\kappa ^u-s^u-w^um.$ Thus for $t\ge T,$ we get $D^+W(t)\le -C_1$$\vert C_d(t)-\tilde{C}_d(t)\vert -C_2\vert C_r(t)-\tilde{C}_r(t)\vert -C_3\vert A(t)-\tilde{A}(t)\vert .$

Integrating the above inequality from T to t, we have

$$\begin{aligned} &\int _T^t (C_1\vert C_d(u)-\tilde{C}_d(u)\vert +C_2\vert C_r(u)-\tilde{C}_r(u)\vert + C_3\vert A-\tilde{A}\vert )\\ &<W(T)-W(t)<\infty . \end{aligned}$$

Appendix C

Proof

By comparison theorem, the fourth equation of model Eq. (12) gives $0\le A\le L\left( 1+\frac{w}{s}\right)$. Now we define a function $V_4(x(t))=\sum _{i=1}^3x_i(t)$, then Itô’s formula gives

$$\begin{aligned} dV_4(x(t))= & \ (Q_d-\lambda _0x_1-\lambda _1x_1x_3+\kappa x_4+rx_1-r_0x_2+\pi _1\lambda _1x_1x_3+\phi x_1\\&-\delta _0x_3-\lambda _1x_1x_3)dt+\sum _{i=1}^3(x_i-x_i^*)\alpha _idB_i\\ &\le & {} \left( \left( Q_d+L \; \left( 1+\frac{w}{s}\right) \right) -(\lambda _0-r-\phi )x_1-r_0x_2-\delta _0x_3\right) dt\\&+\sum _{i=1}^3(x_i-x_i^*)\alpha _idB_i. \end{aligned}$$

Multiplying by $e^{\beta t}$ both sides, we get

$$\begin{aligned} e^{\beta t}dV_4(x(t))\le & \ \left( \left( Q_d+L\left( 1+\frac{w}{s}\right) \right) -(\lambda _0-r-\phi )\right.\\ &\left.x_1-r_0x_2-\delta _0x_3\right) e^{\beta t}dt\\&+\sum _{i=1}^3e^{\beta t}(x_i-x_i^*)\alpha _idB_i. \end{aligned}$$

Some algebraic manipulation gives

$$\begin{aligned} d(e^{\beta t}V_4(x(t)))\le & \ \biggl( \left( Q_d+L\left( 1+\frac{w}{s}\right) \right) e^{\beta t}-(\lambda _0-r-\phi -\beta )x_1 \\ &- (r_0-\beta )x_2 -(\delta _0-\beta )x_3\bigg) dt \\ &+\sum _{i=1}^3e^{\beta t}(x_i-x_i^*)\alpha _idB_i. \end{aligned}$$

(19)

Let $c=\min \{(\lambda _0-r-\phi -\beta ), (r_0-\beta ), (\delta _0-\beta )\}$, then

$$\begin{aligned} d(e^{\beta t}V_4(x(t)))\le & \ \left( \left( Q_d+L\left( 1+\frac{w}{s}\right) \right) e^{\beta t}-cV_4(x(t))\right) dt \\ & +\sum _{i=1}^3e^{\beta t}(x_i-x_i^*)\alpha _idB_i e^{\beta t}E V_4(x(t)) \\ & \le & \ EV_4(0)+\frac{\left( Q_d+L\left( 1+\frac{w}{s}\right) \right) }{\beta }e^{\beta }t. \end{aligned}$$

Therefore, $\lim _{t\rightarrow \infty } EV_4(x(t))\le \frac{\left( Q_d+L\left( 1+\frac{w}{s}\right) \right) }{\beta }.$ Thus $x_i\le$$\frac{\left( Q_d+L(1+\frac{w}{s})\right) }{\beta },$ for $i=1,2,3$ and $x_4\le L\left( 1+\frac{w}{s}\right) .$