|| Sarvam Krishna Arpanam ||
Model based approach to Study the Impact of Biofuels on the Sustainability of an Integrated System
Prakash Kotecha1, Urmila Diwekar1*, Heriberto Cabezas2,
1 Center for Uncertain Systems: Tools for Optimization and Management, Vishwamitra Research Institute, Clarendon Hills, Illinois, USA
2U.S. Environmental Protection Agency, Office of Research and Development, National Risk Management Research Laboratory, Sustainable Technology Division, Cincinnati, Ohio, USA
* Corresponding author. Tel: +630 886 3047, E-mail:
In this supporting material, we present the necessary mathematical details of the fourteen compartmental model that we have proposed to study the sustainability of an ecosystem. For better readability, we have presented the mathematical details for each compartment separately. However, it should be noted that all these compartments are interconnected and the flows need to be appropriately comprehended. We start with the details on industrial sector as it fixes the wages of the human labor for all the other industries.
Table 1 lists the various state variables whereas Table 2 lists the various model outputs.
Industrial Sector
The industrial sector producing the IS product from RP and P1 sets the wage rate for all other industries. In addition to the constant terms, the wage rate is dependent on two time varying factors namely the demand supply gap of the IS product and the amount of labor force contributed by the human households in the market. At any given time step, the wages are determined as shown in the following equation
(1)
In the above equation, and are constant parameters. The last two parameters are dependent on the level of the IS stock and the human population.indicates the amount of IS deficit at the given time point t, denotes the amount of IS available at the current time instant whereas is the end-of-period inventory target. denotes the amount of product P1 necessary to produce a single unit of IS and is the amount of RP required to produce a single unit of IS. represents the number of human population at the current time instant. From the above equation, it can be seen that the wages decrease as the difference between the stock and target of IS decreases. Also, an increase in the number of human population indicates the availability of more labor and thereby decreases the wage rates. The determination of wage rates enables the determination of the price and production levels of various other products.
The amount and price of IS produced depends on the demand supply gap and the wages of the labor. The amount of IS produced is given by the following equation
(2)
In the above equation, and are constant parameters. The last two parameters are dependent on the wage rates and the stock of the IS product. From the above equation, it can be seen that an increase in the wage rates decreases the amount of IS produced. Additionally, an increase in the end-of-period inventory target increases the production of IS. The price of the IS product is governed by the following equation
(3)
In equation(3), and are constant parameters. The last two parameters are dependent on the wage rates and the stock of the IS product. From the above equation, it can be seen that an increase in the wage rates increases the price of IS whereas the price of the IS product decreases with an increase in the stock of the IS product.
As explained in the main article, the consumption of IS by the human household directly contributes to the increase in mass of the inaccessible resource pool (IRP) and hence represents the waste that is generated by industries. Hence, the equations (2) and (3) used determining the price and production of IS are replaced by the following two equations if the discharge fee on the industry is considered.
(4)
(5)
In the above two equations, and are constant parameters affecting the price and production of the IS product whereas is the discharge fee. It can be seen that the addition of the discharge fee term increases the price of IS. The term refers to the demand of IS by the human households and is determined by the following equation.
(6)
where are constant parameters. The constant parameters are based on the price of the three products P1, H1 and IS. The terms anddenote the demands of P1 and H1. The terms and indicate the price of P1, H1 and IS. The determination of the demands and prices of P1 and H1 is shown in the later part of this supporting material. The above equation for determining demand of IS by HH is to be modified into the following if a energy producer is included
(7)
where is the demand of EE and is the price of energy and is a constant parameter related to the price of energy. It can be seen in equation (6) and (7) that an increase in the price of IS decreases its demand whereas an increase in the price of other commodities like P1, H1 or EE increases its demand. It is to be noted that denotes the per capita demand of IS.
The product IS is produced from RP and P1. Having determined the production of IS from equation (3) or(5), the amount of P1 and RP required for producing IS is given by the equations (8) and (9) respectively.
(8)
(9)
As explained earlier, denotes the amount of product P1 necessary to produce a single unit of IS and is the amount of RP required to produce a single unit of IS. The mass balance for IS at any time instant is given by the following equation
(10)
The above equation shows that the transfer of mass from P1 and RP increases the mass of the IS compartment whereas the flow of mass to the human households is captured by the term . However, it should be noted that the flow of mass from IS to human households does not increase the mass of the HH compartment but instead increases the mass of the IRP compartment. The deficit of IS for the next time instant is calculated using equation (11) where is the deficit of IS at the current time instant while is the amount of mass flowing from IS to IRP (use of IS product by the human households) and is the amount of IS required by HH.
(11)
Having discussed the dynamics of the industrial sector, we will now detail the dynamics of the primary producer.
Table 1: List of State Variables
Sl. No / State Variable / Sl. No / State Variable1 / Mass of P1 / 11 / Mass of RP
2 / Mass of P2 / 12 / Mass of IRP
3 / Mass of P3 / 13 / Mass of P1 deficit for H1
4 / Mass of H1 / 14 / Mass of P1 deficit for IS
5 / Mass of H2 / 15 / Mass of P1 deficit for HH
6 / Mass of H3 / 16 / Mass of H1 deficit for HH
7 / Mass of C1 / 17 / Mass of IS deficit for HH
8 / Mass of C2 / 18 / Number of Humans
9 / Mass of HH / 19 / Mass of Energy Source (ES)
10 / Mass of IS / 20 / Per capita human mass
Table 2: Model Outputs
Sl. No / Output Variable / Sl. No / Output Variable1 / Price of P1 / 20 / Production of H1
2 / Production of P1 / 21 / Mass of H1 to HH
3 / Mass of P1 to H1 / 22 / Mass of H1 to C1
4 / Mass of P1 to H2 / 23 / Mass of H1 to RP
5 / Mass of P1 to IS / 24 / Mass of H2 to RP
6 / Mass of P1 to HH / 25 / Mass of H2 to C1
7 / Mass of RP to P1 / 26 / Mass of H2 to C2
8 / Mass of P1 to RP / 27 / Mass of H3 to RP
9 / Mass of IRP to P2 / 28 / Mass of H3 to C2
10 / Mass of RP to P2 / 29 / Mass of C1 to RP
11 / Mass of P2 to RP / 30 / Mass of C2 to RP
12 / Mass of P2 to H1 / 31 / Price of IS
13 / Mass of P2 to H2 / 32 / Production of IS
14 / Mass of P2 to H3 / 33 / Mass of RP to IS
15 / Mass of IRP to P3 / 34 / Mass of IS to IRP
16 / Mass of RP to P3 / 35 / Mass of RP to IRP
17 / Mass of P3 to RP / 36 / Mass of IRP to RP
18 / Mass of P3 to H3 / 37 / Mass of HH to RP
19 / Price of H1 / 38 / Per capita Births
Table 2 (Contd): Model Outputs
Sl. No / Output Variable / Sl. No / Output Variable39 / Mortality of HH / 51 / Mass of P3
40 / Wages / 52 / Mass of H1
41 / Coefficient in the determination of per capita birth rate / 53 / Mass of H2
42 / Coefficient in the determination of per capita birth rate / 54 / Mass of H3
43 / Weighted price of the products / 55 / Mass of C1
44 / Price of EE / 56 / Mass of C2
45 / Production of EE / 57 / Mass of HH
46 / Mass of Energy Source to IRP / 58 / Number of Human
47 / Mass of Energy Source / 59 / Per capita mass of humans
48 / Mass of IS / 60 / Mass of RP
49 / Mass of P1 / 61 / Mass of IRP
50 / Mass of P2 / 62 / Mass of P1 transferred to IRP
Primary Producers
The initial discussion is focused on the P1 followed by the discussion on P2 and P3
Producer P1
The P1 industry produces its product by using mass from the RP compartment and labor from the human households (HH). The price of its product depends on the wage rates set by the IS industry and hence can be calculated only after the wage rates are determined as shown earlier. The price and production of P1 are calculated using equation (12) and (13)
(12)
(13)
In the above equations, and are constant parameters influencing the pricing of the product P1. The parameters and are also constant parameters which influence the amount of P1 being produced at any time instant. The constant parameters and affecting the price and production of P1 are related to the wage rates whereas the constant parameters and are constant parameters affecting the price and production levels of P1 are related to the stock of the P1 product. In the above two equations, the term is used to denote the deficit of P1 at the given time instant, denotes the stock of P1 at the given time instant and is the amount of end-of-period inventory target for P1. Equation (12) and (13) are similar to equations (2) and (3). With an increase in the wages, the price of P1 increases whereas the production of P1 decreases. Similarly, an increase in the demand of P1 increases the price and production of P1. However, there is no discharge fee associated with the P1 compartment and is reflected by the absence of additional terms in the price and production of the P1 industry.
The product P1 is consumed by four other compartments of the ecosystem (H1, H2, IS and HH). In addition, P1 can also be used by the energy producer to produce energy and represents the energy produced from the biofuels. If P1 is used for producing energy, the amount of energy produced is set to 30% i.e., thirty percent of the energy demand of the ecosystem is met by the energy generated from the biofuels. In the absence of sufficient quantity of P1 to produce 30% of the energy, whatever amount of P1 is available is used to produce the energy. The rest of the energy demand is met by the conventional energy source. As done with the IS industry, the mass of P1 used for the production of energy is assumed to be transferred to the inaccessible resource pool. The flow of P1 to each of these compartments is given below. The amount of P1 consumed by H1 is given by the following equation
(14)
In the above equation, ,, and are constant parameters. The constant parameters and are related to the wage rates and price of product P1. In particular, denotes the growth of H1 due to the consumption of P1. From the above equation, it can be seen that an increase in either the wages or an increase in the price of P1 decreases the demand of P1 by H1. Additionally, an increase in the end-of-period inventory target of H1, , increases the demand of P1 by H1. and denote the deficit and stock of H1 at any given time instant. The demand of P1 by H2 is given by the following equation
(15)
In the above equation, and are constant parameters and denote the growth of P1 due to RP and mortality of P1 respectively. and denote the current stock of P1 and RP. The first term on the RHS indicates the natural growth of P1 and the second term indicates the amount of P1 produced by the P1 industry while the last term indicates the natural death of P1. Thus, the flow of P1 to H2 is equal to difference between the growth and death of P1 in addition to the production of P1 by the P1 industry.
The demand of P1 by human households depends on the prices and demands of various other commodities. The demand of P1 by HH in the presence of energy producer can be represented by the following equation
(16)
In the above equation, and are constant parameters. In particular, are parameters related to the price of P1, H1, IS and energy respectively. The terms and indicate the demands of P1, H1, IS and EE by the human households. The terms and indicate the price of P1, H1, IS and EE respectively. In the absence of EE, the above equation can be rewritten by eliminating the terms involving EE
(17)
The demand of P1 by IS has been described earlier and can be determined by (8). The amount of P1 deficit for each of the consumer (H1, IS and HH) is given by the following three equations
(18)
(19)
(20)
In the above equations, the deficit at any instant of time is determined by the summation of the deficit in the previous time instant and the deficit at the current time instant. The deficit at the current time instant is determined by the difference between the demand of the product and the amount of the product available to that particular compartment. The total deficit of P1 at any given time instant is determined by the summation of the deficits acquired from each of the three consumers and is shown in the following equation
(21)
The amount of P1 at the current time instant is given by the following mass balance for P1.
(22)
In the above equation, denotes the growth of P1 whereas denotes the death of P1. The terms and indicate the flow of P1 to H1, H2, HH and IS respectively. We will now discuss the dynamics of the primary producer P2.
Producer P2
The growth of P2 is dependent on the mass of the resource pool (RP) whereas P2 is consumed by all the three herbivores animals H1, H2 and H3. The demand of P2 by H1 is controlled by the government and is a constant as given in the following equation.
(23)
where is a constant value specified by the government. The consumption of P2 by H2 and H3 are given by the following equations
(24)
(25)
where and are the growth factor of H2 and H3 (respectively) dependent on P2 whileand is the amount of H2 at the given time instant. Additionally, the amount of IRP that is getting converted to P2 as recycle is given by
(26)
where and are the recycle factors for P2 and P3 respectively, is the factor governing the conversion of IRP to RP. The term indicates the stock of IRP at the given time instant while indicates the direct conversion of RP to IRP. The stock of P2 at any time instant is given by the following mass balance over P2
(27)
where and indicate the recycle of IRP to P2 and the natural growth of P2 due to RP. The term corresponds to the natural death of P2 whereas the other outflow terms and represent the consumption of P2 by H1, H2 and H3 respectively.It should be noted that P2 and P3 denote have any direct contribution to the economy and hence do not involve any price or production levels.
Producer P3
The dynamics of P3 is described in the following discussion and is similar to P2. The growth of P3 is dependent on the resource pool and P3 is consumed by only the herbivores H3. The consumption of P2 by H3 is given by the following equation
(28)
where is the growth factor of H3 dependent on P3 and is the amount of H3 at the given time instant. Additionally, the amount of IRP that is getting converted to P3 is given by
(29)
where and are the recycle factors for P2 and P3 respectively, is the factor governing the conversion of IRP to RP. The term indicates the stock of IRP at the given time instant while indicates the direct conversion of RP to IRP. The stock of P3 at any time instant is given by the following mass balance over P2
(30)
where and indicate the recycle of IRP to P3 and the natural growth of P3 due to RP. The term corresponds to the natural death of P3 whereas is the consumption of P3 by H3 respectively.
Herbivores
The initial discussion is focused on H1 followed by the discussion on H2 and H3.
Herbivores H1
The price of H1 product depends on the wage rates set by the IS industry and hence can be determined only after determining the wage rates. The price and production of H1 are calculated using equation (31) and (32)
(31)
(32)
In the above equations, and are constant parameters influencing the pricing of the product H1. The parameters and are also constant parameters which influence the amount of H1 being produced at any time instant. The terms and used in the determination of price and production levels of H1 are related to the wages whereas the other two terms and are related to the stock of H1. Though the stock of H1 may change with time, the parameters themselves are kept constant during the simulation period. The termdenotes the deficit of H1 at any given time, is the stock of H1 at any given time instant and is the amount of end-of-period inventory target for H1. Equation (31) and (32) are similar to equations (12) (also(2)) and (13) (also (3)). With an increase in the wages, the price of H1 increases whereas the production of H1 decreases. Similarly, an increase in the demand of H1 increases the price and production of H1.
H1 is consumed by the carnivores (C1) and human households (HH). The flow of H1 to each of these compartments is given below. The demand of H1 by human households depends on the prices and demands of various other commodities. The demand of H1 by HH in the presence of energy producer can be given by the following equation
(33)
In the above equation, and are constant parameters. The constant parameters and are dependent on the price of P1, H1, IS and EE respectively. The terms and indicate the demands of P1, H1, IS and energy by the human households. The terms and indicate the price of P1, H1, IS and EE respectively. In the absence of energy production, the above equation can be rewritten by eliminating the terms involving energy prices and demands
(34)
In the absence of an energy producer, equations (6), (17), and (34) constitute the demands of IS, P1 and H1.These three equations are in terms of per capita are also simultaneous equations. The explicit forms of these three equations are given as below.
Explicit form of equation (6)
(35)
Explicit form of equation (17)
(36)
Explicit form of equation (34)
(37)
The explicit form of the price of IS with discharge fee (equation (4)) is given by the following equation
(38)
The explicit form of the equations (7), (16), (33) will be discussed after presenting the expression of
The amount of H1 deficit is given by the following equation
(39)
In the above equations, the deficit at next instant of time is determined by the summation of the deficit in the current time instant and the deficit at the current time instant. The deficit at the current time instant is determined by the difference between the demand of H1 and the amount of H1 available. The flow of H1 to C1 is given by
(40)
where and are the growth of H1 due to P1 and P2, corresponds to the natural death of H1 and indicates the production of H1. The amount of H1 at the next time instant is given by the following mass balance for H1.