Modeling the impact of mitigation options on abatement of methane emission from livestock

Mitigation of methane emission from livestock sector is crucial to combat the menace of global warming. In the present paper, a nonlinear mathematical model is proposed to investigate the impact of mitigation options for curtailing livestock methane emission on the reduction of atmospheric concentration of methane. In modeling process, it is assumed that the mitigation options are applied at a rate proportion to the livestock population. The conditions for reduction and stabilization of atmospheric methane have been obtained. Numerical simulation has been performed to verify the analytical findings by taking the secondary data of atmospheric concentration of methane, human and livestock populations. Sensitivity analysis is carried out to explore the impact of the key parameters of the model system.


Introduction
Global warming is presently one of the most gravest threats to human environment. Global temperatures are at much higher levels than they are in past millions of years and are expected to rise in the future. This warming is expected to have many adverse consequences including melting of glaciers and ice caps, increase in frequency and intensity of extreme weather events, increase in incidence of vector-borne infections, change in rain fall pattern, etc. The main culprit behind global warming is the excessive emission of greenhouse gases by human activities. The prime human-influenced greenhouse gases are carbon dioxide (CO 2 ), methane (CH 4 ), nitrous oxide (N 2 O) and chlorofluorocarbon (CFC). Methane is the most prevalent anthropogenic greenhouse gas after carbon dioxide. anticipated to rise by 45.3 kg per year till 2030. The per capita milk consumption has also increased from 73.9 kg per year in 1964-1966 to 78.1 kg per year in 1997-1999 and is anticipated to rise by 89.5 kg per year till 2030 [29]. Due to the high consumption and demand of livestock products, livestock farming has increased around the globe and taken the form of livestock revolution. This livestock revolution is also important for rural development. A major portion of rural population in developing countries depends on livestock for their livelihood and nutritional requirements. Thus, expansion in livestock framing aids in poverty alleviation in developing countries. Government in many countries have taken steps to promote the livestock farming. But, the expansion in livestock sector must be accomplished with substantial reduction in livestock's environmental impact. To devise strategies for attaining such scenario, it is crucial to investigate the effectiveness of the available mitigation options for livestock methane emission on the reduction of atmospheric methane. Mathematical modeling using differential equations may be an effective tool for such investigations. In recent years, differential equation models have played a crucial role in exploring the impact of various factors on the reduction and stabilization of greenhouse gases and pollutants in the atmosphere [4, 11-15, 17, 18, 20, 22, 23]. The objective of the present study is to formulate a nonlinear mathematical model to study the impact of mitigation options for livestock methane emission on the atmospheric level of methane in a scenario, where effort are made to increase the livestock population to meet food demand.

Model formulation
Let N (t) and P (t) be the global human and livestock population, respectively. Also, let C(t) be the atmospheric concentration of CH 4 and S(t) be a measure of mitigation options, which are applied to reduce the livestock methane emission at the time t. These mitigation options can be measured in terms of the cost involved in their implementation. It is assumed that concentration of atmospheric methane is increasing due to its emission from livestock farming as well as other human activities. Since the amount of methane emission from livestock depends in large on the number of livestock, the livestock methane emission is assumed to be proportional to livestock population. Methane emission from other human activities is assumed to be proportional to human population. The emission rate of methane from non-anthropogenic (i.e., natural) sources is assumed to be constant. Since mitigation options reduce the methane emission from livestock, therefore we have taken the emission rate coefficient of methane from livestock as a decreasing function of mitigation options. Under these assumptions, the dynamics of atmospheric methane can be modeled asĊ whereĊ stands for dC/dt. In the above equation, C 0 is the constant input methane concentration in the atmosphere from natural sources. The constant α 0 is the natural depletion rate coefficient of atmospheric methane. The constants λ 1 is the emission rate coefficient of methane from human activities other than livestock farming, whereas λ 2 is emission rate coefficient of methane from livestock. The constant η 2 represents the efficiency of mitigation options to reduce the livestock methane emission, and k 2 is a half saturation constant, which represents the mitigation options at which the reduction in livestock methane emission is half of its maximum possible reduction, which can be ever achieved through mitigation options. The constant k 2 limits the effect of mitigation options in reducing the livestock methane emissions. As the reduction in methane emission rate cannot exceed the emission rate itself, so η 2 < λ 2 . It is assumed that human and livestock population both grow logistically. The demand of livestock product increases with the increase in human population and this leads to expansion in livestock sector [24,31]. Also, the increase in livestock population boosts the growth of human population directly via consumption of livestock products and indirectly via promoting the economical state of the people involved in livestock farming [9,26]. Thus, human and livestock share a mutualistic relationship. In this view, it is assumed that human population and livestock population both facilitate each other's growth. Since global warming due to the increased concentration of methane has adverse impact on human as well as livestock population, we assume that human and livestock populations decrease due to elevated level of atmospheric methane. Under these assumptions, the differential equations describing the dynamics of human and livestock populations arė respectively. In the above equations, the constants s and L are the intrinsic growth rate and carrying capacity of human population in the absence of livestock population and the adverse impacts of elevated level of methane. The constants s 1 and L 1 are the intrinsic growth rate and carrying capacity of livestock population in absence of human population and the adverse impacts of elevated level of methane. The constants θ 1 and θ 2 are respectively the decline rate coefficients of human and livestock populations due to the adverse effects of elevated level of atmospheric methane. The constant β 1 is the growth rate coefficient of human population due to increase in livestock population. The constant β 2 is the growth rate coefficient of livestock population due to human efforts. The implementation rate of mitigation options is assumed to be proportional to the livestock population. Some of these mitigation efforts diminish with the passage of time due to their inefficacy in reducing livestock methane emission or economical barriers. Thus, the differential equation governing dynamics of mitigation options iṡ S = νP − δS.
In the above equation, the constants ν and δ are the implementation and depletion rate coefficients of mitigation options, respectively.
Here it may be noted that condition (2) implies ss 1 /(LL 1 ) − β 1 β 2 > 0, which is the condition for boundedness of solutions of the classical two-species models of mutualism [16].

Equilibria and their stability
Since it is not possible to find out the exact solutions to the nonlinear model system (1), we determine the long-term behavior of the system by using stability theory of differential equations. In the following, we determine the equilibrium points of the model system and then perform stability analysis of the equilibria.

Equilibria
The model system (1) has four non-negative equilibria, which are listed as below: (i) E 1 (C 0 , 0, 0, 0) always exists. This equilibrium implies that human and livestock populations both are absent and thus not contributing to the methane emission, and in this case, atmospheric methane is at its natural level C 0 .
(ii) E 2 (C 2 , N 2 , 0, 0), where C 2 = C 0 +λ 1 sL/(sα 0 +θ 1 λ 1 L) and N 2 = sα 0 L/(sα 0 + θ 1 λ 1 L), always exists. This equilibrium states that only human population is present however livestock population is absent. In this case, the concentration of methane will be more than its natural level due to its emission from human activities. As livestock population is absent, so the mitigation options to control the emission of methane from livestock are zero.
(iii) The equilibrium E 3 (C 3 , 0, P 3 , S 3 ) always exists. This equilibrium states that livestock population is present and human population is absent. In this case, also the concentration of methane will be more than its natural level due to its emission from livestock. As the mitigation options are applied to control the emission of methane from livestock population, which is present, and so in this case, mitigation options are also present.
(iv) The interior equilibrium E * (C * , N * , P * , S * ) exists if the following conditions hold: The existence of equilibria E 1 and E 2 is obvious. In the following, we show the existence of equilibria E 3 and E * . In the equilibrium E 3 (C 3 , 0, P 3 , S 3 ), the values C 3 , P 3 and S 3 may be obtained by solving the following set of algebraic equations: νP − δS = 0.
From equation (8), we have Using equation (9) in equation (6), we get Now, using equation (10) in equation (7), we get following quadratic equation in P : where a = ν( Here a > 0 and c < 0, this implies that there exists a unique positive root, say P 3 , of equation (11). Using this value of P in (9) and (10), we get positive values of S = S 3 and C = C 3 , respectively.
The values of C * , N * , P * and S * in the equilibrium E * may be obtained by solving the following set of algebraic equations: From equation (15), we have From equation (12), we have Using equation (17) in equation (13), we get Using equations (16), (17) and (18) in equation (14), we get the following quadratic equation in P :ã Now equation (19) has a unique positive root, say P * , ifã andc are of opposite signs. Thus, here arises two possibilities for unique positive root of (19): Possibility (ii) is dropped out as it is not biologically feasible. The term s 1 + β 2 L − θ 2 λ 1 L/α 0 represents the net intrinsic growth rate in livestock population and hence should be positive. This givesc < 0. Thus, equation (19) has a unique positive root ifã > 0 andc < 0. Now,ã > 0 if condition (3) holds, andc < 0 if condition (4) holds. This implies that equation (19) has a unique positive root if conditions (3) and (4) are satisfied.
Condition (3) is analogous to the condition of existence of interior equilibrium in basic two-species model of mutualism. It says that the self effects in human and livestock population must be greater than the effect of their interactions. Using the value of P = P * in (16), we get the positive value of S = S * . Using P = P * and S = S * in (18), we get the positive value of N = N * provided condition (5) is satisfied. Finally, using the above positive values of N , P and S in equation (17), we get positive value of C = C * . Remark 1. From equations (12) to (15), we find that Since A > 0 (from condition (3)), dC * /dν < 0 and dC * /dη 2 < 0 if This shows that an increase in the implementation rate of mitigation option or efficiency of mitigation options leads to reduction in atmospheric level of methane under condition (20).

Stability of equilibria
In this section, we study the stability behavior of the obtained equilibria. The results regarding the stability of equilibria are stated in the following theorems.
The interior equilibrium E * , if exists, is locally asymptotically stable if and only if the following conditions are satisfied: where A i s (i = 1, 2, 3, 4) are the coefficients of characteristic equation of Jacobian matrix evaluated at E * and are defined in the proof.
Proof. The Jacobian matrix for model system (1) is given as follows: Let J i be the Jacobian matrix J evaluated at E i (i = 1, 2, 3). From the matrix J 1 , it is found that its eigenvalues are −α 0 , s, s 1 and −δ. Thus, E 1 always has a stable manifold locally in C,S-plane, whereas it has unstable manifold locally in N,P -plane. Thus, E 1 is always unstable.
From the Jacobian matrix J 2 , it is found that one of the eigenvalues of the matrix J 2 is s 1 + β 2 N 2 − θ 2 (C 2 − C 0 ). This eigenvalue can be written as Since s(s 1 + β 2 L − θ 2 λ 1 L/α 0 ) + s 1 λ 1 θ 1 L/α 0 > 0 if E * exists (condition (4)), therefore The other three eigenvalues of J E2 are either negative or with negative real part. Thus, E 2 has stable manifold locally in C, N, S-space whereas it has an unstable manifold locally in P -direction whenever E * exists. From the Jacobian matrix J 3 , it is found that one of its eigenvalues is s + β 1 P 3 − θ 1 (C 3 − C 0 ) and the other three eigenvalues are either negative or with negative real part. Thus, the local stability of equilibrium E 3 depends on the sign of this eigenvalue. The equilibrium E 3 is stable or unstable according to the sign of s To determine the sign of eigenvalues of the Jacobian matrix J evaluated at the equilibrium E * , we make use of Routh-Hurwitz criterion. The characteristic equation for Jacobian matrix J evaluated at the equilibrium E * is given as where Here it is apparent that A 1 and A 4 are positive. Using Routh-Hurwitz criterion, it is inferred that all the eigenvalues of the Jacobian matrix J E * will be either negative or with negative real part iff the conditions stated in (21) are satisfied.

Theorem 2.
The equilibrium E * , if exists, is globally asymptotically stable provided the following inequalities are satisfied: Proof. Consider the following positive definite function: where m 1 , m 2 and m 3 are positive constants to be chosen appropriately. Now differentiating V with respect to t along the solution of model system (1), we get Since SP/(k 2 + S) − S * P * /(k 2 + S * ) = S(P − P * )/(k 2 + S) + k 2 P * (S − S * )/ ((k 2 + S)(k 2 + S * )), we get Choosing m 1 = λ 1 /θ 1 and m 2 = λ 2 /θ 2 , we get https://www.mii.vu.lt/NA Now, dV /dt can be made negative definite provided Now, we can choose m 3 from inequalities (30) and (33) provided the following condition holds: Since S * = νP * /δ, the above inequality can be rewritten as Combining the above inequality with (29), we get inequality (25). Thus, dV /dt can be made negative definite provided inequalities (23)-(25) are satisfied. Remark 2. From the above theorem, it may be noted that the parameters β 1 and β 2 have destabilizing effect on the dynamics of system (1). This implies that if the growth in human population due to livestock product consumption or growth in livestock population due to human efforts is large, the atmospheric concentration of methane may not get stabilized.

Numerical simulation 4.1 Parameter estimation
To estimate the model parameters, the secondary data of atmospheric concentration of methane, human population and livestock population is used. The initial time is taken to be the year 1961. The annual time series data for average atmospheric concentration of methane for the period 1961-2011 is taken from European Environment Agency [28]. The data for world population for the period 1961-2011 is obtained from United nations population division [27] and world's livestock population data for the period of 1961-2006 is taken from [32]. Since the life time of methane in the atmosphere is 12.5 years [25], the value of α 0 is taken as 1/12.5 = 0.08 year −1 . The value of C 0 is taken to be 700 ppb (parts per billion), the pre-industrial time methane concentration [5]. While fitting the data, it is assumed that there are no mitigation options in the system (i.e., η 2 = 0, k 2 = 0, ν = 0 and δ = 0). In the absence of appropriate data regarding other parameters, the model system (1) is calibrated for different values of parameters λ 1 , λ 2 , s, L, s 1 , L 1 , θ 1 , β 1 , β 2 and find the best fit for λ 1 = 0.0095, λ 2 = 0.01, s = 0.03, L = 10000, s 1 = 0.01, L 1 = 6000, θ 1 = 10 −7 , β 1 = 4 × 10 −7 , β 2 = 5 × 10 −7 . Thus, we have the following estimated parameter values:

Sensitivity analysis
Sensitivity analysis assess the response of model variables to change in parameter values.
To determine the effect of changes in the values of parameters β 2 , η 2 and ν on the state of the model system (1), the basic sensitivity analysis of the model system (1) is performed with respect to these parameters following Bortz and Nelson [3]. The sensitivity systems with respect to parameters β 2 , η 2 and ν are given bẏ respectively. Here C β2 (t, β 2 ) denote the rate of change of state variable C with respect to parameter β 2 and is called sensitivity function of C with respect to the parameter β 2 . The semi-relative sensitivity solutions have been calculated with respect to each of the three parameters of interest β 2 , η 2 and ν. The semi-relative sensitivity solution is obtained by multiplying the sensitivity solution with the parameter. These solutions provides information about the change in the state of a variable when a parameter value is doubled. These solutions are depicted in Fig. 6. The first plot of this figure clearly shows that on doubling the parameter η 2 , the atmospheric level of methane is decreased by 150.2 ppb in a period of 100 years and 273.5 ppb in 200 years. On doubling the parameter ν, the atmospheric level of methane is decreased by 72.6 ppb in 100 years and 120.3 ppb in 200 years. The doubling of the parameter β 2 brings a rise of 169 ppb and 297 ppb in atmospheric concentration of methane over the periods of 100 years and 200 years, respectively. Thus, the atmospheric concentration of CH 4 is highly affected by the changes in values of parameters η 2 , ν and β 2 . It can be noted that doubling of the efficiency of mitigation options η 2 brings more reduction in atmospheric concentration of methane than that of the implementation rate of mitigation options ν. From the second and third plots of Fig. 6, it can be seen that the doubling of the growth rate coefficient of livestock population due to human efforts β 2 brings large increase in livestock and human populations; while the impact of doubling of the efficiency and the implementation rate of mitigation options on livestock and human populations is very small.

Conclusion
Enhanced level of atmospheric methane is one of the main culprits for global warming. Livestock farming is the largest source of methane, thus, it is crucial to mitigate methane emission from livestock sector. But, the livestock farming is increasing around the globe to meet the food demand of the growing population. In this scenario, it is important to investigate the effectiveness of mitigation options for methane emission from livestock over the reduction of atmospheric level of methane. For this purpose, we have proposed a nonlinear mathematical model. In the modeling process, it is assumed that human population makes efforts to increase the livestock population and the increase in livestock population facilitates the growth of human population. Further, it is assumed that the mitigation options are implemented to reduce the methane emission from livestock farming at a rate proportional to livestock population. The proposed nonlinear model is analyzed by using stability theory of differential equations. The model system exhibits four non-negative equilibria. The conditions for local and global stability of interior equilibrium have been derived. In the analysis of model, it is found that increase in the implementation rate of mitigation options ν and efficiency of mitigation options η 2 reduces the atmospheric level of methane under condition (20). Sensitivity analysis is performed with respect to the key parameters η 2 , ν and β 2 , which clearly demonstrates the impact of these parameters on the atmospheric level of methane and other model variables.
The findings of this paper suggest various strategies for the reduction of methane emission from livestock farming. Sensitivity analysis shows that η 2 is more influential parameter than ν towards the reduction of methane emission. This suggest that the implementation of high efficient options with low implementation rate would be a better policy than the implementation of low efficient options with high rate. For instance, use of propionate precursors is more effective option than use of probiotics to reduce methane emission from enteric fermentation in dairy cows [30]. The parameter β 2 , which represents the growth in livestock population due to human efforts, is found to have large impact on the dynamics of the system. For very high values of β 2 , condition (20) for reduction of atmospheric level of methane via mitigation options will not satisfy. Also, the parameter β 2 has destabilizing effect on the dynamics of the system. Thus, for attaining sustainable scenario, the livestock farming should not be promoted beyond a critical level. This critical level can be obtained with help of condition (20). Overall, the present paper provides a basic framework to access the impact of mitigation options for livestock methane emission on the reduction of atmospheric concentration of methane and addresses the constraints that exist in achieving the dual goal of reduction in atmospheric level of methane and increase in livestock production.