Modeling the impact of mitigation options on methane abatement from rice fields
Authors
- First Online:
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 (CH_{4}) 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 CH_{4}. Abatement strategies for mitigating CH_{4} 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 CH_{4} while increasing rice yield. In modeling process, it is assumed that implementation rate of mitigation options is proportional to the enhanced concentration of atmospheric CH_{4} 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 CH_{4} emission from rice fields have been derived. These conditions are useful in devising strategies for effective abatement of CH_{4} emission from rice fields along with sustainable increase in rice yield. The analysis also shows that CH_{4} abatement highly depends on efficiencies of mitigation options to mitigate CH_{4} 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 analysis1 Introduction
The elevated level of methane (CH_{4}) 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, CH_{4} acts as a precursor to tropospheric ozone (\(\textmd{O}_3\)) which is another greenhouse gas. Due to the high global warming potential of atmospheric CH_{4}, reduction in CH_{4} emission produces substantial and fast response towards the alleviation of climate changes. This makes CH_{4} mitigation a vital part of climate policies. Rice (Oryza) agriculture has contributed significantly to the elevated level of atmospheric CH_{4}. Estimates reveal that worldwide rice production is responsible for nearly 20 % of global anthropogenic CH_{4} emissions (Cao et al. 1996). As rice fields are the dominant source of CH_{4}, attenuating CH_{4} 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, CH_{4} 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 CH_{4} 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 CH_{4} emission from rice fields. This can be achieved by implementation of those options which have potential to improve the rice production while curtailing CH_{4} 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 CH_{4} 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 CH_{4} 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 CH_{4} 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 CH_{4} per year to 3.5–11.6 Tg CH_{4} 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 CH_{4} 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 CH_{4} 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 CH_{4} 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 CH_{4} emission from rice field significantly with a sustainable increase in rice yield.
2 Formulation of mathematical model and its analysis
2.1 Mathematical model
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.
- (i)
The axial equilibrium, E_{0}(C_{0}, 0, 0) always exists.
- (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)
- (i)
f(C_{0}) = γL > 0,
- (ii)
f(C_{m}) < 0 and
- (iii)
f′(C) < 0 if condition 5 is satisfied.
Remark
Remark
It may be noted that the above condition is automatically satisfied for small values of γ_{2} and large values of γ_{1}. This suggest that one possible strategy for the significant reduction in CH_{4} emission while increasing yield is the implementation of those mitigation options that have high efficiency of CH_{4} 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 E_{0} 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).
It is found that the eigenvalues of matrix P_{0} are -α, r and -δ_{0}. Since P_{0} has two negative eigenvalues and one positive eigenvalue. This implies that the equilibrium E_{0} 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 E_{0}.
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.
Theorem 2
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 CH_{4} will get stabilized).
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.
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 CH_{4} emission and increase rice yield, have crucial effects on the equilibrium concentration of atmospheric CH_{4}. These results are shown in Figs. 1–3. Figures 1 and 2 show the variations in atmospheric concentration of CH_{4} and mitigation options with respect to time, for different values of γ_{1} and γ_{2} respectively. It is apparent from Fig. 1 that if the efficiency of mitigation options to reduce CH_{4} emission is high, atmospheric concentration of CH_{4} 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 CH_{4} 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 CH_{4}. Moreover, expenditure on CH_{4} mitigation will also be high. Thus, while selecting a mitigation strategy, one should be very careful about its efficiencies of curtailing CH_{4} emission and increasing rice yield (i.e., γ_{1} and γ_{2}). Different values of γ_{1} and γ_{2} 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 γ_{1} and γ_{2}. It is clear that for low value of γ_{2} (= 0.001), an increase in value of ν reduces the equilibrium level of CH_{4}; but if γ_{2} is taken to be 0.01, then for the same value of γ_{1}, increase in value of ν leads to increase in the equilibrium level of CH_{4}. Now if we increase value of γ_{1} from 0.01 to 0.018, increase in the implementation rate of mitigation options first enhance atmospheric concentration of CH_{4} and then reduce it. Thus, implementation of more mitigation options not always reduces atmospheric concentration of CH_{4}. It happens only if the condition 12 is satisfied.
Parameter values in model system 4
Parameter | Value | Unit |
---|---|---|
α | 0.02 | year^{ − 1} |
C_{0} | 1200 | ppb |
γ | 0.02 | ppb (ton year)^{ − 1} |
γ_{1} | 0.01 | ppb (ton year)^{ − 1} |
K_{1} | 1000 | ton |
r | 0.01 | year^{ − 1} |
L | 1000 | ton |
γ_{2} | 0.001 | ton (year)^{ − 1} |
K_{2} | 500 | dollar |
ν | 0.002 | ton (ppb year)^{ − 1} |
δ_{0} | 0.01 | year^{ − 1} |
4 Sensitivity analysis
5 Discussion
Attenuating CH_{4} 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 CH_{4} 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 CH_{4} 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 CH_{4} mitigation from rice fields. It is found that if mitigation options are highly efficient to increase rice yield (i.e., γ_{2} is high), the equilibrium levels of atmospheric CH_{4} and mitigation options are high. On the other hand, high efficiency of mitigation options to curtail methane emission (i.e., γ_{1} is high) leads to low equilibrium levels of atmospheric CH_{4} 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 CH_{4} provided condition 12 holds. This condition suggests various possible strategies for reduction of CH_{4} 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 CH_{4} emission. Along with the condition of reduction of CH_{4} 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 γ_{1} and γ_{2} 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 CH_{4} 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 CH_{4} mitigation and sustainable increase in rice yield. One of the most important factor is the efficiency of mitigation options to curtail CH_{4} emission ‘γ_{1}’. Analysis clearly shows that an increase in the value of γ_{1} reduce the atmospheric level of CH_{4} but for large value of γ_{1}, the atmospheric level of CH_{4} may not get stabilized. Apart from γ_{1}, the parameters γ_{2} and ν also have significant effect on the dynamics of CH_{4}. Sensitivity analysis clearly demonstrates that the atmospheric level of CH_{4} is highly affected by changes in these three parameters i.e., γ_{1}, γ_{2} and ν. While devising any strategy regarding the control of CH_{4} 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 CH_{4} 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 CH_{4} emission from rice fields. Moreover, if conditions 20 and 21 are also satisfied then it ensures that the applied mitigation option will stabilize the CH_{4} emission from rice fields. The stabilized level of atmospheric CH_{4}, rice yield and mitigation option can be evaluated by using the Eqs. 9–11. This provides an estimation to the level of mitigation options which should be maintained in order to keep the atmospheric concentration of CH_{4} 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 γ_{2}) 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 CH_{4} abatement. The implementation of these strategies suits economically as well as fulfills the dual objective of CH_{4} 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).