Mitigation and Adaptation Strategies for Global Change

, Volume 19, Issue 7, pp 927–945

Modeling the impact of mitigation options on methane abatement from rice fields

Authors

    • Department of Mathematics, Faculty of ScienceBanaras Hindu University
  • Maitri Verma
    • Department of Mathematics, Faculty of ScienceBanaras Hindu University
Original Article

DOI: 10.1007/s11027-013-9451-5

Cite this article as:
Misra, A. & Verma, M. Mitig Adapt Strateg Glob Change (2014) 19: 927. doi:10.1007/s11027-013-9451-5

Abstract

The enhanced concentration of methane (CH4) in the atmosphere is significantly responsible for the ominous threat of global warming. Rice (Oryza) paddies are one of the largest anthropogenic sources of atmospheric CH4. Abatement strategies for mitigating CH4 emissions from rice fields offer an avenue to reduce the global atmospheric burden of methane and hence the associated menace of climate change. Projections on population growth suggest that world rice production must increase to meet the population’s food energy demand. In this scenario, those mitigation options are advocated which address both the objectives of methane mitigation and increased production of rice simultaneously. In this paper, we have formulated a nonlinear mathematical model to investigate the effectiveness and limitations of such options in reducing and stabilizing the atmospheric concentration of CH4 while increasing rice yield. In modeling process, it is assumed that implementation rate of mitigation options is proportional to the enhanced concentration of atmospheric CH4 due to rice fields. Model analysis reveals that implementation of mitigation options not always provides “win-win” outcome. Conditions under which these options reduce and stabilize CH4 emission from rice fields have been derived. These conditions are useful in devising strategies for effective abatement of CH4 emission from rice fields along with sustainable increase in rice yield. The analysis also shows that CH4 abatement highly depends on efficiencies of mitigation options to mitigate CH4 emission and improve rice production as well as on the implementation rate of mitigation options. Numerical simulation is carried out to verify theoretical findings.

Keywords

Mathematical modelMethane gasRice paddiesSensitivity analysisStability analysis

1 Introduction

The elevated level of methane (CH4) in the Earth’s atmosphere is a matter of great concern due to its large impact on climate change. Methane is a potent greenhouse gas with global warming potential of 25 over a period of 100 years. It contributes nearly 18 % to overall global increase in radiative forcing since the mid eighteenth century (IPCC 2007a). In addition to this, CH4 acts as a precursor to tropospheric ozone (\(\textmd{O}_3\)) which is another greenhouse gas. Due to the high global warming potential of atmospheric CH4, reduction in CH4 emission produces substantial and fast response towards the alleviation of climate changes. This makes CH4 mitigation a vital part of climate policies. Rice (Oryza) agriculture has contributed significantly to the elevated level of atmospheric CH4. Estimates reveal that worldwide rice production is responsible for nearly 20 % of global anthropogenic CH4 emissions (Cao et al. 1996). As rice fields are the dominant source of CH4, attenuating CH4 emission from rice fields is crucial to pursuit the climate change mitigation objective.

Methane emission from rice fields is a result of anaerobic decomposition of soil organic matter by methanogenic bacteria. Methane emission is affected by agricultural practices as well as edaphic and climate factors. The agricultural practices include cultivation method, water management, cultivar selection, cropping patterns, fertilization practices, etc.; while, edaphic and climate factors include physical and chemical properties of soil (soil texture (Sass et al. 1994), soil temperature (Schütz et al. 1990), pH, redox potential, etc.), amount and timing of rain, wind speed, sky cover, etc. All of these factors have varying degrees of influence over methane emission from rice fields. Agricultural practices are the most curial factors; by manipulating these practices rationally, CH4 flux from rice fields can be reduced (Wassmann et al. 2009). Studies show that the water management practices like mid season drainage, intermediated irrigation, etc., effectively curtail methane emission from rice fields (Husin et al. 1995; Khosa et al. 2011; Sass et al. 1992; Shin et al. 1996; Singh et al. 2003; Tyagi et al. 2010; Yagi et al. 1996, 1997; Wassmann et al. 2000). Other agriculture practices like fertilizer application, selection of low methane emitting cultivars, organic abatements, etc., are also found to be promising mitigation options (Aulakh et al. 2001; IPCC 2007b; Linquist et al. 2012; Lu et al. 2000; Minami 1995; Rath et al. 1999; Setyanto et al. 2004; Shin et al. 1996; Singh et al. 2003; Xie et al. 2010). But the importance of rice in food security and poverty alleviation of developing world requires sustainable increase in rice production along with reduction in CH4 emission from rice fields.

Rice is a staple food for nearly 3 billion people residing in Asia and certain parts of Africa. It provides a major part of the world’s dietary energy supply. It is estimated that the world’s rice production must increase by 50 % till 2030 to stand with population’s food demand (Osman et al. 2012). Thus, a sustainable increase in rice production is requisite along with significant reduction in CH4 emission from rice fields. This can be achieved by implementation of those options which have potential to improve the rice production while curtailing CH4 emission from rice fields. Adoption of such mitigation options is crucial, not merely for food security but also in economic perspective. Implementation of mitigation options involves some extra expenditure and this pose a negative effect on the income of farmers. Those mitigation strategies, which increase rice production, definitely improve their income and thus are economically beneficial. All these facts drive the attention of policymakers and researchers towards those strategies which are useful in pursuing both the goals of methane abatement and increased rice production simultaneously (Rosenzweig and Tubiello 2007). Various research projects have been conducted by the International Rice Research Institute (IRRI), the United States Environmental Protection Agency (US-EPA), the Fraunhofer Institute for Atmospheric Environmental Research (IFU), etc., to investigate the potential effectiveness of mitigation options in reducing CH4 emission from rice fields with an increment in rice production (Matthews and Wassmann 2003 and references therein). These projects provide crucial information, which help in identifying suitable mitigation options for CH4 mitigation from rice fields. Water management practices are found to be extremely effective for attaining both the aforesaid goals. A case study of Bohol Island at Philippines shows that adoption of a water-saving technology called ‘alternate wetting and drying (AWD)’, developed by IRRI, not only increase the rice productivity but also curtails CH4 emission from rice fields (Wassmann et al. 2009). Another case study of China rice fields shows that mid-season drainage leads to drop in the annual rice paddy methane flux from 8.6–16.0 Tg CH4 per year to 3.5–11.6 Tg CH4 per year over a 20 year period with a significant increase in rice yield (Li et al. 2002). Some other studies have also mentioned that water management practices like mid-season drainage and intermittent irrigation improve the rice yield along with significant reduction in methane emission (Itoh et al. 2011; Tyagi et al. 2010; Wang et al. 2000; Xu et al. 2007). Application of soil amendments and mineral fertilizers are also possible options for the same. A case study of Bangladesh shows that soil amendment applications like silicate fertilization with urea and silicate in combination with sulfate of ammonia increase rice production and reduce methane efflux effectively (Ali et al. 2012). Use of nitrification inhibitors is also beneficial in mitigating CH4 gas and improving crop quality (Ghosh et al. 2007). Thus, there are various mitigation options which not only reduce methane emission but also produce positive effect on yield. Through a careful selection of mitigation options, the dual objective of reduced CH4 emission and increased production of rice may be achieved.

The deciding factors in selection and implementation of mitigation technologies is their effectiveness in reducing and stabilizing the net CH4 efflux from paddy fields while increasing the rice production. Therefore, in this paper, we propose a nonlinear mathematical model to study the effectiveness of mitigation options in reducing CH4 emission from rice field significantly with a sustainable increase in rice yield.

2 Formulation of mathematical model and its analysis

2.1 Mathematical model

In any region under consideration, let C and R denote the atmospheric concentration of CH4 and the rice yield at any time t, respectively. Let M be a measure of mitigation options, which are applied to reduce CH4 emission from rice fields as well as to increase rice yield at time t. These mitigation options can be measured in terms of labor and other costs involved in their implementation. The emission rate of CH4 from rice fields is assumed to be dependent on the amount of rice paddies and hence on the rice yield (Bouwman 1991). Since mitigation options reduce the emission rate of CH4 from rice fields, therefore we have taken the emission rate coefficient of CH4 from rice fields as a decreasing function of mitigation options. Under these assumptions, the dynamics of atmospheric CH4 can be modeled as
$$\label{eqd1} \dot{C}= -\alpha (C-C_0)+ \left(\gamma -\frac{\gamma_1 M}{K_1+M}\right)R, $$
(1)
where \(\dot{C}\) stands for dC/dt. In the above equation, C0 is the constant input CH4 concentration in the atmosphere from the sources other than rice paddies. The constant ‘α’ is the natural depletion rate coefficient of atmospheric CH4 and the constant ‘γ’ is the emission rate coefficient of CH4 from rice fields in absence of mitigation options. The constant γ1 represents the efficiency of mitigation options to reduce CH4 emission from rice fields and K1 is a half saturation constant which represents mitigation options at which the reduction in CH4 emission is half of its maximum possible reduction that can be ever achieved through mitigation options. The constant K1 limits the effect of mitigation options in reducing CH4 emission from rice fields.
It is observed that yield growth pattern for major cereal crops is logistic (Harris and Kennedy 1999); therefore, it is assumed that, in absence of mitigation options, rice yield grows logistically with intrinsic growth rate ‘r’ and carrying capacity ‘L’. The mitigation options are assumed to further enhance its growth. Since the capacity of mitigation options to increase rice production must have some upper ceiling, we have taken that mitigation options increase the growth rate of rice yield at a rate ‘γ2MR/(K2 + M)’. The benefit of taking such interaction is that for large values of M (M > > K2), M/(K2 + M) saturates to 1, this limits the effect of mitigation options in increasing rice yield. This type of interaction can be viewed as analogous to the Holling type II interaction for predator prey system (Holling 1959). Thus, the differential equation describing the dynamics of rice yield is given by
$$\label{eqd2} \dot{R}= r R \left(1-\frac{R}{L}\right)+\frac{\gamma_2 M R}{K_2+M}, $$
(2)
where, γ2 represents the efficiency of mitigation options to increase rice yield and K2 is a half saturation constant which represents the mitigation options at which the increase in rice yield is half of its maximum possible increase.
The implementation rate of mitigation options is assumed to be proportional to the net emission of methane from rice fields. Some of the mitigation efforts will diminish due to their ineffectiveness in mitigating CH4 emission from rice fields. Thus, the differential equation governing dynamics of mitigation options is
$$\label{eqd3} \dot{M}=\nu(C-C_0)-\delta_0 M. $$
(3)
In the above equation, the constants ν and δ0 are implementation and depletion rate coefficients of the mitigation options, respectively.
Thus, we have the following system of nonlinear ordinary differential equations governing the dynamics of the problem:
$$ \begin{array}{rll}\label{eqd4} \dot{C} &=& -\alpha(C-C_0)+\left(\gamma-\frac{\gamma_1 M}{K_1+M}\right)R, \\ \dot{R}&=&r R \left(1-\frac{R}{L}\right)+\frac{\gamma_2 M R}{K_2+M},\\ \dot{M} &=& \nu(C-C_0)-\delta_0 M , \end{array} $$
(4)
where, C(0) > C0, R(0) ≥ 0, M(0) ≥ 0.
The region of attraction (Freedman and So 1985) for model system 4 is given by the set:
$$ \Omega =\left\{(C, R, M):C_0< C \leq C_m; 0\leq R \leq R_m; 0\leq M \leq \frac{\nu (C_m-C_0)}{\delta_0} \right\}, $$
where Cm = C0 + (γRm/α), \(\displaystyle R_m = L(1+\gamma_2/r)\), and it attracts all solutions initiating in the interior of the positive octant.

2.2 Equilibrium states

Due to nonlinearity of model system 4, it is not possible to find exact solutions to the system. Instead, we settle for determining the long-term behavior of the system. In general, a nonlinear system either gravitates towards an equilibrium point or it blows up. The equilibrium points are those states of dynamical system at which system does not move. Once the system reaches at an equilibrium state, it freeze at this state for all future times. These points can be obtained by putting the growth rate of different variables of model system equal to zero.

The model system 4 exhibits the following two non-negative equilibria:
  1. (i)

    The axial equilibrium, E0(C0, 0, 0) always exists.

     
  2. (ii)
    The interior equilibrium, E*(C*, R*, M*) exists provided the following condition is satisfied:
    $$\label{eqd5} \alpha-\frac{\gamma \gamma_2 \nu L}{\delta_0 r K_2}>0. $$
    (5)
     
The existence of equilibrium E0 is trivial.
In the following, we discuss the existence of interior equilibrium E*. This equilibrium may be obtained by solving the following set of algebraic equations:
$$ -\alpha(C-C_0)+\left(\gamma-\frac{\gamma_1 M}{K_1+M}\right)R =0,\label{eqd6} $$
(6)
$$ r \left(1-\frac{R}{L}\right)+\frac{\gamma_2 M}{K_2+M}=0,\label{eqd7} $$
(7)
$$ \nu(C-C_0)-\delta_0 M =0.\label{eqd8} $$
(8)
From Eq. 7, we get
$$\label{eqd9} R= \frac{L}{r}\left(r+ \frac{\gamma_2 M}{K_2+ M}\right). $$
(9)
From Eq. 8, we get
$$\label{eqd10} M= \frac{ \nu (C-C_0)}{\delta_0}. $$
(10)
Using Eqs. 9 and 10 in Eq. 6, we obtain the following equation in C:
$$ \begin{array}{rll}\label{eqd11} f(C)&=& -\alpha(C-C_0)+\frac{L}{r}\left(\gamma-\frac{\gamma_1 \nu (C-C_0)}{\delta_0 K_1 +\nu (C-C_0)}\right) \\ &&\times\,\left(r+\frac{\gamma_2 \nu (C-C_0)}{\delta_0 K_2 +\nu (C-C_0)}\right)=0. \end{array} $$
(11)
It can be checked that:
  1. (i)

    f(C0) = γL > 0,

     
  2. (ii)

    f(Cm) < 0 and

     
  3. (iii)

    f′(C) < 0 if condition 5 is satisfied.

     
Thus, a unique positive root C = C* of Eq. 11 lies in the interval (C0, Cm) under the condition 5. Using this value of C = C* in Eqs. 9 and 10, we get the positive values of R = R* and M = M*, respectively.

Remark

It can be easily shown that \(d C^*/d \gamma_1 <0\) and \(d M^*/d \gamma_1 <0\). This implies that as the efficiency of mitigation options to reduce CH4 emission increases, the equilibrium levels of atmospheric CH4 and mitigation options decrease (see Fig. 1). Also, it is found that \(d C^*/d \gamma_2 >0\) and \(d M^*/d \gamma_2>0\). This implies that as the efficiency of mitigation options to increase rice yield increases, the equilibrium levels of atmospheric CH4 and mitigation options increase (see Fig. 2).
https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig1_HTML.gif
Fig. 1

Variations in atmospheric concentration of methane and mitigation options w.r.t. time for different values of γ1

https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig2_HTML.gif
Fig. 2

Variations in atmospheric concentration of methane and mitigation options w.r.t. time for different values of γ2

Remark

It is found that dR*/dν > 0 i.e., increase in the implementation rate of mitigation options leads to increase in the equilibrium level of rice yield. Also, it is found that dC*/dν < 0 if
$$\label{eqdd} \frac{\gamma_1 K_1 R^*}{ (K_1+M^*)^2}> \frac{ \gamma_2 K_2 L}{ r (K_2+M^*)^2}\left(\gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right). $$
(12)
This shows that an increase in implementation rate of mitigation options reduces the equilibrium level of CH4 under condition 12. From this condition, it is clear that the efficiencies of mitigation options to curtail CH4 emission and increase rice yield play a crucial role in deciding whether or not implementation of more mitigation options leads to reduction in atmospheric concentration of CH4 (see Fig. 3).
https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig3_HTML.gif
Fig. 3

Effect of increasing the implementation rate of mitigation options on atmospheric concentration of methane for different values of γ1 and γ2

It may be noted that the above condition is automatically satisfied for small values of γ2 and large values of γ1. This suggest that one possible strategy for the significant reduction in CH4 emission while increasing yield is the implementation of those mitigation options that have high efficiency of CH4 abatement and low efficiency of increasing rice production.

2.3 Stability analysis

2.3.1 Local stability of equilibria

In this section, we perform the local stability analysis of the equilibria E0 and E*. This analysis provides excellent information about the behavior of a dynamical system. The local stability analysis characterizes whether or not the system settles to the equilibrium point if its state is initiated close to, but not precisely at a given equilibrium point. The equilibrium point is said to be locally asymptotically stable if there is a neighborhood of the equilibrium point such that for all initial starts in this neighborhood, the system approaches to the equilibrium point as t→ ∞. The local stability of an equilibrium can be investigated by determining the sign of the eigenvalues of Jacobian matrix evaluated at the equilibrium (Perko 2000).

The Jacobian matrix for the model system 4 is given as follows:
$$ \label{eqd12} P=\left( \begin{array}{ccc} -\alpha & \gamma-\dfrac{\gamma_1 M}{K_1+M} & -\dfrac{\gamma_1 K_1 R}{(K_1+M)^2}\\[10pt] 0 & r\left(1-\dfrac{2R}{L}\right)+\dfrac{\gamma_2 M}{K_2+M} & \dfrac{\gamma_2 K_2 R}{(K_2+M)^2}\\[10pt] \nu & 0 & -\delta_0\\ \end{array} \right).\\ $$
Let P0 and P* be the Jacobian matrices evaluated at equilibria E0 and E*, respectively.

It is found that the eigenvalues of matrix P0 are -α, r and -δ0. Since P0 has two negative eigenvalues and one positive eigenvalue. This implies that the equilibrium E0 is a saddle point with stable manifold locally in C − M- plane and unstable manifold locally in R-direction (Perko 2000). Thus, the system will never settle down to the equilibrium E0.

To study the local stability behavior of equilibrium E*, we apply the Routh-Hurwitz criterion (Rao 1981). The Routh-Hurwitz criterion provides necessary and sufficient conditions for all roots of a polynomial to lie in the left half of the complex plane. The characteristic equation for the matrix P* is given as:
$$\label{eqd13} \psi^3+ A_1 \psi^2+ A_2 \psi+A_3=0, $$
(13)
$$ \begin{array}{rll} \text{where,} \\ A_1 &=& \alpha+\frac{r R^*}{L}+\delta_0, \\ A_2&=& \alpha\left(\frac{r R^*}{L}+\delta_0\right)+ \frac{r R^* \delta_0}{L} + \frac{\nu \gamma_1 K_1 R^*}{(K_1+M^*)^2}, \\ A_3&=&\frac{\alpha r R^* \delta_0}{L}+ \frac{ \nu r \gamma_1 K_1 {R^*}^2}{L (K_1+M^*)^2}- \frac{\nu \gamma_2 K_2 R^*}{(K_2+M^*)^2}\left( \gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right). \end{array} $$
(14)
It may be easily noted that A1 > 0 and A3 > 0. Also,
$$ \begin{array}{rll}\label{eqd14} A_1 A_2-A_3 &=& \alpha^2 \left(\frac{r R^*}{L}+\delta_0\right)+\frac{\alpha r R^*}{L} \left(\frac{r R^*}{L}+\delta_0\right)+ \frac{r^2 {R^*}^2 \delta_0}{L}+\alpha \delta_0 \left(\frac{r R^*}{L}+\delta_0\right)\\ &&+ \frac{r R^*\delta_0^2}{L}+(\alpha+\delta_0)\frac{\nu \gamma_1 K_1 R^*}{(K_1+M^*)^2}+\frac{\nu \gamma_2 K_2 R^*}{(K_2+M^*)^2}\left( \gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right)\\ &&>0. \end{array} $$
Thus, all the conditions of Routh-Hurwitz criterion are satisfied, this implies that the eigenvalues of matrix P* will be either negative or with negative real part. Now, we have the following result:

Theorem 1

The interior equilibrium E*, if exists, is locally asymptotically stable.

This theorem tells that if the initial state of system 4 is near the equilibrium point E*, solution trajectories not only stay near E* for all t > 0 but also approaches to E* as t → ∞. Thus, if the initial value of state variables C, R and M are close to C*, R* and M*, respectively, system 4 will eventually get stabilized.

2.3.2 Nonlinear stability of interior equilibrium

In this section, we extend our stability analysis beyond the small region near equilibrium point to the whole region of attraction using Liapunov’s second method (LaSalle and Lefschetz 1961). The basic idea of this technique for verifying nonlinear stability of equilibrium point is to seek an energy function that decreases with time along the trajectories of the system.

Consider the following positive definite function:
$$\label{eqd17} V= \frac{1}{2}(C-C^*)^2+m_1 \left(R-R^*-R^*\ln \frac{R}{R^*}\right)+ \frac{1}{2} m_2 (M-M^*)^2, $$
(15)
where m1 and m2 are positive constants to be chosen appropriately.
Differentiating ‘V’ with respect to ‘t’ along the solution of model system 4, we get
$$ \begin{array}{rll}\label{eqd18} \frac{dV}{dt}&= &-\alpha (C-C^*)^2-\frac{m_1 r}{L}(R-R^*)^2-m_2 \delta_0 (M-M^*)^2 \\ &&+ m_2 \nu (C-C^*)(M-M^*)+\left(\gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right)(C-C^*)(R-R^*) \\ &&-\frac{\gamma_1 K_1 R}{(K_1+M)(K_1+M^*)}(C-C^*)(M-M^*) \\ &&+ \frac{m_1 \gamma_2 K_2}{(K_2+M)(K_2+M^*)}(M-M^*)(R-R^*). \end{array} $$
dV/dt will be negative definite inside the region of attraction ‘Ω’ if the following inequalities are satisfied:
$$ m_2< \frac{4 \alpha\delta_0}{9 \nu^2},\label{eqd19} $$
(16)
$$ m_1> \frac{3L}{2 \alpha r}\left(\gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right)^2\label{eqd20} $$
(17)
$$ m_2> \frac{9 \gamma_1^2 L^2}{4 \alpha \delta_0 (K_1+M^*)^2}\left(1+\frac{\gamma_2}{r}\right)^2,\label{eqd21} $$
(18)
$$ m_2 > \frac{3 L \gamma_2^2}{2 r \delta_0 (K_2+M^*)^2} m_1. \label{eqd22} $$
(19)
From inequalities 16 and 18, we can choose m2 > 0 provided
$$\frac{4 \alpha\delta_0}{9 \nu}(K_1+M^*)>\gamma_1 L \left(1+\frac{\gamma_2}{r}\right).$$
After choosing m2, we can choose m1 > 0 from the inequalities 17 and 19 provided
$$ \frac{4 \alpha r \delta_0 }{9 \nu \gamma_2 L}(K_2+M^*)>\left(\gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right). $$
Thus, we have following result for the nonlinear stability of the interior equilibrium E* which is stated in the form of following theorem:

Theorem 2

The interior equilibrium E*, if exists, is nonlinearly stable provided the following conditions hold:
$$ s_1 = \frac{4 \alpha\delta_0}{9 \nu}(K_1+M^*)-\gamma_1 L \left(1+\frac{\gamma_2}{r}\right)>0,\label{eqd15} $$
(20)
$$ s_2 = \frac{4 \alpha r \delta_0 }{9 \nu \gamma_2 L}(K_2+M^*)-\left(\gamma-\frac{\gamma_1 M^*}{K_1+M^*}\right)>0\label{eqd16}. $$
(21)

If the above conditions are satisfied, then it guarantees that for every initial start within the region of attraction Ω, solution trajectories will reach to the equilibrium state E*(i.e. the concentration of atmospheric CH4 will get stabilized).

Remark

From the above conditions, it may be noted that the parameters γ1 and γ2 have destabilizing effect on the dynamics of system 4 (see Figs. 4 and 5).
https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig4_HTML.gif
Fig. 4

Impact of γ1 on the stability conditions s1 and s2

https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig5_HTML.gif
Fig. 5

Impact of γ2 on the stability conditions s1 and s2

3 Numerical simulation

To confirm the analytically obtained results and to illustrate the dynamical behavior of the system, numerical simulation has been carried out using MATLAB 7.0.5. We have taken the set of parameter values in model system 4 as given in Table 1.

For the above set of parameter values, condition 5 for existence of interior equilibrium E* is satisfied. Components of interior equilibrium E* are obtained as:
$$ \text{$C^* =2145.7332$ ppb, $R^* = 1027.4465$ tons, $M^* = 189.1466$ dollars.} $$

Eigenvalues of Jacobian matrix corresponding to the equilibrium E* for model system 4 are −0.01716, −0.014278 and −0.008831. Since all the eigenvalues are negative, this implies that the interior equilibrium E* is locally asymptotically stable. The nonlinear stability conditions stated in Theorem 2 are also satisfied for the foregoing set of parameter values.

In analysis of model, we have found that implementation rate of mitigation options, and efficiencies of mitigation options to curtail CH4 emission and increase rice yield, have crucial effects on the equilibrium concentration of atmospheric CH4. These results are shown in Figs. 13. Figures 1 and 2 show the variations in atmospheric concentration of CH4 and mitigation options with respect to time, for different values of γ1 and γ2 respectively. It is apparent from Fig. 1 that if the efficiency of mitigation options to reduce CH4 emission is high, atmospheric concentration of CH4 settles to low level and the equilibrium level of mitigation options is also low. Figure 2 illustrates that as the efficiency of mitigation options to increase rice yield increases, the equilibrium levels of atmospheric CH4 and mitigation options increase. Thus the implementation of those mitigation options, which are highly efficient to increase rice production may not be able to reduce the atmospheric level of CH4. Moreover, expenditure on CH4 mitigation will also be high. Thus, while selecting a mitigation strategy, one should be very careful about its efficiencies of curtailing CH4 emission and increasing rice yield (i.e., γ1 and γ2). Different values of γ1 and γ2 give rise to different scenarios, as demonstrated in Fig. 3. In this figure, we have shown the effect of increase in the implementation rate coefficient of mitigation options ‘ν’ for three different set of parameter values of γ1 and γ2. It is clear that for low value of γ2 (= 0.001), an increase in value of ν reduces the equilibrium level of CH4; but if γ2 is taken to be 0.01, then for the same value of γ1, increase in value of ν leads to increase in the equilibrium level of CH4. Now if we increase value of γ1 from 0.01 to 0.018, increase in the implementation rate of mitigation options first enhance atmospheric concentration of CH4 and then reduce it. Thus, implementation of more mitigation options not always reduces atmospheric concentration of CH4. It happens only if the condition 12 is satisfied.

For the set of parameter values given in Table 1, solution trajectories of system 4 have been drawn in Fig. 6 with different initial starts. We see that all trajectories initiating inside the region of attraction are approaching towards equilibrium values (C*, R*, M*). This depicts the non-linear stability behavior of interior equilibrium E*(C*, R*, M*) in C − R − M space. The variation of nonlinear stability conditions s1 and s2 with respect to the parameter γ1 is demonstrated in Fig. 4. This figure clearly shows that when the value of γ1 is below 0.046, the stability condition 20 is satisfied but for γ1 > 0.046, it is no longer satisfied. Moreover, on increasing the value of parameter γ1, value of s2 decreases. This shows that parameter γ1 has destabilizing effect on the dynamics of the system 4. Similarly, Fig. 5 depicts the destabilizing effect of γ2. It is apparent from this figure that the stability condition 20 is not satisfied for γ2 > 0.073, and the stability condition 21 fails for γ2 > 0.02. Thus for very high values of γ1 or γ2, the system may not settle down to the equilibrium state. This means that the concentration of atmospheric CH4 may not get stabilized if the efficiencies of mitigation options to increase rice yield and reduce CH4 emission are very high.
Table 1

Parameter values in model system 4

Parameter

Value

Unit

α

0.02

year − 1

C0

1200

ppb

γ

0.02

ppb (ton year) − 1

γ1

0.01

ppb (ton year) − 1

K1

1000

ton

r

0.01

year − 1

L

1000

ton

γ2

0.001

ton (year) − 1

K2

500

dollar

ν

0.002

ton (ppb year) − 1

δ0

0.01

year − 1

https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig6_HTML.gif
Fig. 6

Nonlinear stability of (C*, R*, M*) in C-R-M space

4 Sensitivity analysis

To study the manner in which model behavior depends upon model parameterization, we present the basic sensitivity analysis of the differential equations in model system 4 (Bortz and Nelson 2004 and references therein). The sensitivity systems with respect to parameters γ1, γ2 and ν are given by
$$ \begin{array}{rll} \dot{C}_{\gamma_1}(t,\gamma_1)&=& -\alpha C_{\gamma_1}(t,\gamma_1)+\gamma R_{\gamma_1}(t,\gamma_1) -\frac{\gamma_1 M (t,\gamma_1)}{K_1+M(t,\gamma_1)}R_{\gamma_1}(t,\gamma_1)\\ &&-\frac{\gamma_1 K_1 R(t,\gamma_1)}{(K_1+M(t,\gamma_1))^2}M_{\gamma_1}(t,\gamma_1) -\frac{M(t,\gamma_1) R(t,\gamma_1)}{K_1+M(t,\gamma_1) }, \end{array} $$
$$ \begin{array}{rll} \dot{R}_{\gamma_1}(t,\gamma_1)&=& r {R}_{\gamma_1}(t,\gamma_1)-\frac{2 r}{L} R(t,\gamma_1)R_{\gamma_1}(t,\gamma_1) +\frac{\gamma_2 M(t,\gamma_1)}{K_2+M(t,\gamma_1)}R_{\gamma_1}(t,\gamma_1) \\ &&+ \frac{\gamma_2 K_2 R(t,\gamma_1)}{(K_2+M(t,\gamma_1))^2}M_{\gamma_1}(t,\gamma_1), \\ \dot{M}_{\gamma_1}(t,\gamma_1)&=& \nu C_{\gamma_1}(t,\gamma_1)- \delta_0 M_{\gamma_1}(t,\gamma_1), \end{array} $$
$$ \begin{array}{rll} \dot{C}_{\gamma_2}(t,\gamma_2)&=& -\alpha C_{\gamma_2}(t,\gamma_2)+\gamma R_{\gamma_2}(t,\gamma_2) -\frac{\gamma_1 M (t,\gamma_2)}{K_1+M(t,\gamma_2)}R_{\gamma_2}(t,\gamma_2)\\ &&-\frac{\gamma_1 K_1 R(t,\gamma_2)}{(K_1+M(t,\gamma_2))^2}M_{\gamma_2}(t,\gamma_2),\\\dot{R}_{\gamma_2}(t,\gamma_2)&=& r {R}_{\gamma_2}(t,\gamma_2)-\frac{2 r}{L} R(t,\gamma_2)R_{\gamma_2}(t,\gamma_2) +\frac{\gamma_2 M(t,\gamma_2)}{K_2+M(t,\gamma_2)}R_{\gamma_2}(t,\gamma_2)\\ &&+ \frac{\gamma_2 K_2 R(t,\gamma_2)}{(K_2+M(t,\gamma_2))^2}M_{\gamma_2}(t,\gamma_2) +\frac{M(t,\gamma_2) R(t,\gamma_2)}{K_2+M(t,\gamma_2) },\\ \dot{M}_{\gamma_2}(t,\gamma_2)&=& \nu C_{\gamma_2}(t,\gamma_2)- \delta_0 M_{\gamma_2}(t,\gamma_2), \end{array} $$
and
$$ \begin{array}{rll} \dot{C}_{\nu}(t,\nu)&=& -\alpha C_{\nu}(t,\nu)+\gamma R_{\nu}(t,\nu) -\frac{\gamma_1 M (t,\nu)}{K_1+M(t,\nu)}R_{\nu}(t,\nu)\\ &&-\frac{\gamma_1 K_1 R(t,\nu)}{(K_1+M(t,\nu))^2}M_{\nu}(t,\nu),\\ \dot{R}_{\nu}(t,\nu)&=& r {R}_{\nu}(t,\nu)-\frac{2 r}{L} R(t,\nu)R_{\nu}(t,\nu) +\frac{\gamma_2 M(t,\nu)}{K_2+M(t,\nu)}R_{\nu}(t,\nu)\\ &&+ \frac{\gamma_2 K_2 R(t,\nu)}{(K_2+M(t,\nu))^2}M_{\nu}(t,\nu),\\ \dot{M}_{\nu}(t,\nu)&=& \nu C_{\nu}(t,\nu)- \delta_0 M_{\nu}(t,\nu)+C(t,\nu), \end{array} $$
respectively.
To access the impact of parameters γ1, γ2 and ν on the dynamics of the state variables, semi-relative as well as logarithmic sensitivity solutions have been calculated for the set of parameter values given in Table 1. Figure 7 depicts the semi-relative sensitivity solutions for all the three state variables with respect to each of the three parameters of interest γ1, γ2 and ν. The semi-relative sensitivity solution provides information about how much the state of a variable changes when the value of a parameter is doubled. The first plot in Fig. 7 clearly shows that if the implementation rate coefficient of mitigation options ‘ν’ is doubled, atmospheric concentration of CH4 drop by 56 ppb in a period of 100 years. Similarly, a doubling of parameter γ1 reduces atmospheric concentration of CH4 by 46 ppb; whereas, doubling of parameter γ2 yields an increase of nearly 7ppb in atmospheric concentration of CH4 by the time 100 years. The second plot of Fig. 7 shows that the impact of doubling of parameter γ1 on rice yield is negligibly small; while, the parameters γ2 and ν posses significant impact on rice yield. The third plot of Fig. 7 shows that doubling of parameters γ1 and γ2 posses very small impact on the state variable M, whereas this state variable is highly influenced by perturbations of the parameter ν. The logarithmic sensitivity solutions for all the state variables with respect to parameters γ1, γ2 and ν are illustrated in Fig. 8. The logarithmic sensitivity solutions interpret the percentage changes in the solutions in response to a doubling of the parameter value. From the first plot of Fig. 8, it can be easily noted that a doubling of parameter ν causes nearly 25 % decrease in atmospheric concentration of CH4 by the time of 100 years. It is clear from Fig. 7 and 8 that perturbations of γ1 have negative impact on the solutions (i.e., values of C, R and M decrease with increase in γ1); whereas, perturbations of γ2 have positive impact on the solutions. The changes in values of γ1 and γ2 have largest impact on the variable C and R respectively; moreover, these impacts increase with time. The perturbations of ν have negative impact on C, whereas it has positive impact on other variables. It is clear from this sensitivity analysis that atmospheric concentration of CH4 is highly sensitive to the changes in values of parameters γ1, γ2 and ν.
https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig7_HTML.gif
Fig. 7

Semi-relative sensitivity solutions for the state variables with respect to parameters γ1, γ2 and ν

https://static-content.springer.com/image/art%3A10.1007%2Fs11027-013-9451-5/MediaObjects/11027_2013_9451_Fig8_HTML.gif
Fig. 8

Logarithmic sensitivity solutions for the state variables with respect to parameters γ1, γ2 and ν

5 Discussion

Attenuating CH4 emission from rice fields and sustainable increase in rice yield both are crucial in the present scenario. This suggests for the implementation of those mitigation options which pursuit the dual goal of CH4 abatement and increased rice production. Successful implementation of these options requires a full understanding about the effectiveness and limitations of these options in performing both jobs simultaneously. In this regard, we have proposed a mathematical model which explores the effect of mitigation options in curtailing CH4 emission from rice paddies along with increase in rice production. The proposed model has two equilibria: an axial equilibrium and an interior equilibrium. The axial equilibrium is always unstable, whereas the interior equilibrium is locally asymptotically stable whenever exists. The model analysis explores the trade offs of CH4 mitigation from rice fields. It is found that if mitigation options are highly efficient to increase rice yield (i.e., γ2 is high), the equilibrium levels of atmospheric CH4 and mitigation options are high. On the other hand, high efficiency of mitigation options to curtail methane emission (i.e., γ1 is high) leads to low equilibrium levels of atmospheric CH4 and mitigation options. Also, it is found that an increase in the implementation rate of mitigation options leads to decrease in the equilibrium level of atmospheric CH4 provided condition 12 holds. This condition suggests various possible strategies for reduction of CH4 emission from rice fields. One such strategy is the implementation of those mitigation options which are less efficient to increase rice yield but highly efficient to curtail CH4 emission. Along with the condition of reduction of CH4 emission, some sufficient conditions (condition 20 and 21) under which the system settles down to the positive equilibrium state are derived. It is shown that the parameters γ1 and γ2 have destabilizing effects on the dynamics of the system under consideration. This pose a restriction on the selection of those mitigation options which are extremely efficient towards CH4 abatement and/or improving rice yield.

The obtained results drive attention towards the key factors which decide the potential effectiveness of mitigation options in achieving the dual objective of CH4 mitigation and sustainable increase in rice yield. One of the most important factor is the efficiency of mitigation options to curtail CH4 emission ‘γ1’. Analysis clearly shows that an increase in the value of γ1 reduce the atmospheric level of CH4 but for large value of γ1, the atmospheric level of CH4 may not get stabilized. Apart from γ1, the parameters γ2 and ν also have significant effect on the dynamics of CH4. Sensitivity analysis clearly demonstrates that the atmospheric level of CH4 is highly affected by changes in these three parameters i.e., γ1, γ2 and ν. While devising any strategy regarding the control of CH4 emission from rice field these parameters, and hence a mitigation option or a combination of mitigation options, should be selected in such a fashion that the condition 12 along with nonlinear stability conditions 20 and 21 are satisfied. Such a selection of mitigation options will result in the reduction of CH4 emission from rice fields along with improvement in rice yield. The conditions 12, 20 and 21, which provide criterions for reduction and stabilization of concentration of methane are very helpful in devising different successful mitigation strategies. Suppose one wants to implement those mitigation options which increase rice production; then, by knowing the value of parameters, it can be estimated with the help of condition 12 that whether or not the implementation of these options will reduce the CH4 emission from rice fields. Moreover, if conditions 20 and 21 are also satisfied then it ensures that the applied mitigation option will stabilize the CH4 emission from rice fields. The stabilized level of atmospheric CH4, rice yield and mitigation option can be evaluated by using the Eqs. 911. This provides an estimation to the level of mitigation options which should be maintained in order to keep the atmospheric concentration of CH4 and rice production at the corresponding equilibrium levels. Nevertheless, in implementation of mitigation options, significant barriers exist. These can be economical, institutional, governmental, social or behavioral. These barriers further limit the choice of mitigation options. The model analysis provides useful information about these limitations. For instance, if there are financial constraints in implementation of mitigation options; then, by fixing the values of implementation rate of mitigation options and efficiency of mitigation options to increase the rice production (i.e., ν and γ2) to desired values and by knowing the values of other parameters, one can easily identify those mitigation strategies which are optimal in the sense of their efficiency towards CH4 abatement. The implementation of these strategies suits economically as well as fulfills the dual objective of CH4 abatement and increased rice production.

Acknowledgements

The authors are grateful to the handling editor and the anonymous reviewers for their useful comments, which have improved the quality of this paper. The second author thankfully acknowledges the University Grants Commission, New Delhi, India for providing financial assistance in the form of Senior Research Fellowship (20-12/2009(ii) EU-IV).

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