Online Resource
Table of Contents
1. Introduction 1
2. Section 1: Model description 1
3. Section 2: Sensitivity analysis 4
4. Section 3: Non-linear programming problem 5
5. Section 4: Six Parameter Multivariable Optimal Control 8
6. References 8
Introduction
This supplementary information is organized as follows. In Section 1, we extend the description of the ecosystem model presented in this paper. In Section 2, the methodology for the sensitivity analysis is presented followed by Section 3, where the non-linear programming is described. Finally, the last Section presents the multivariable optimal control for six parameters of the model studied in this paper.
Section 1: Model description
The model presented in this paper mimics a general ecosystem with very rudimentary social and ecological interactions. It is regulated through mass flows of biological resources such as biomass, nutrients, water, etc. In addition, this model involves aspects related to the production and utilization of energy and the industrial sector through a price-setting model. The production of energy considers various types of energy sources and the industrial sector represents a very elementary implementation of the generic human industrial activities in the model. As shown in Figure 1, the model contains three plants (P1, P2, and P3), three herbivores (H1, H2, and H3), two carnivores (C1 and C2), human households (HH), an industrial sector (IS), energy source (ES), energy producer (EP), resource pool (RP) and inaccessible resource pool (IRP), resulting in 14 compartments. The description of the compartments is listed by tropic level in Table 1.
The compartments HH, IS, EP, P1 and H1 describe the economic perspective of this model. The price-setting macroeconomic model determines flows of economic goods and labor governs the dynamic (e.g. decisions) of these five compartments. This price-setting model attempts to maximize the well-being of IS, EP, P1, H1 and HH. For instances, the energy source (ES) represents a finite non-renewable resource. The energy producer is an industry that uses labor to transform the (ES) into a usable form of energy. This energy is supplied to HH and IS. EP is also capable of producing energy using P1, and this would represent the production of energy using biomass, for example, sugar cane to produce bio-ethanol or soybeans to produce biodiesel. The IS produces products valuable to HH using P1 and RP.
Fig. 1 Flow of mass between various compartments in an economic-ecological model (Kotecha et al. 2012)
In Figure 1, the dashed lines indicate mass flows that occur under anthropogenic influence. The dotted lines indicate the flow of energy from EP to HH and IS. The square dotted lines between P2, P3 and IRP indicate slow transfers of mass as a result of microbial activity. Note that the compartments were differentiated in terms of human control over them by identifying species as being either domesticated or non-domesticated. The original formulation was presented by Cabezas et al. (2005) where only a generalized industrial sector is presented. However, this formulation was extended to the inclusion of biofuel production from biomass (Kotecha et al. 2010; Kotecha et al. 2011). Thus, the latter papers assumed that 30% of the total energy demand by the integrated system is being provided by the biomass. If sufficient amount of biomass is not available, the maximum available biomass is used for the production of energy and the remaining energy is produced from the non-renewable energy source. Other important parameters involved in the formulation of this model are the demands for goods and labor, the wages rate governed by the industrial sector, demands of various products by human households, and the prices of production target.
Table 1: Description of compartments used in the model (Cabezas et al. 2003)
Tropic Level / Compartment / DescriptionPlants / P1 / Domesticated plants eaten by H1 and H2
P2 / Non-domesticated plants eaten by H1, H2, and H3
P3 / Non-domesticated plants eaten by H3
Herbivores / H1 / Domesticated herbivore eaten by HH and C1
H2 / Non-domesticate herbivore eaten by C1 and C2
H3 / Non-domesticate herbivore eaten by C2
Carnivores / C1 / Non-domesticated carnivore
C2 / Non-domesticated carnivore
Humans / HH / Human households that consume energy from ES and consume mass from P1 and H1
IS / Industry that consumes mass from RP and P1. Industry creates biological inaccessible materials for IRP.
ES / Energy source with a finite amount of conventional fossil fuel
EP / Energy producer that converts mass from ES into energy
Resource Pools / RP / Accessible resource pool, from which all plants must get mass and nutrients, and to which all living compartments contribute mass through death.
IRP / Inaccessible resource pool, representing the mass and nutrients locked up (e.g., landfills, paving, etc.) due to human activity. As domestication activity increases, so does mass contributions to this compartment.
Section 2: Sensitivity analysis
The compartment model has 98 parameters within the algebraic and difference equations that comprise the system. Out of them 12 parameters were selected as potential candidates to control the system. Sensitivity analysis was used as a tool to decide which parameters shall be most effective in control. The stepwise algorithm for sensitivity analysis is presented here:
Step#1: Select one parameter (suppose kP1) at a time among the 12 potential candidates. Ensure that its value matches to the base case.
Step#2: Run the model for the base case value (kP1) and record the Fisher Information (FI) profile
Step#3: Increase the parameter value by 5% relative to its base value (kP1+5%). Run the model for the increased parameter value and record the FI profile
Step#4: Now decrease the parameter value by 5% relative to its base value (kP1-5%) and run the model for the decreased value and record the FI profile
(2.1)
Step#5: Plot the FI profiles for the base case, increased and decreased kP1 values on the same graph. Analyze the extent of impact and decide on selection or rejection of the parameter for the control procedure.
Step#6: Repeat steps 1 to 5 for the rest of the parameters
Sensitivity Analysis for Animal Consumption (lH1)
Figure 2 shows the sensitivity analysis results for the animal consumption parameter (i.e. lH1). Decreasing lH1 will decrease the price of H1, increasing the demand for H1 by HH, and therefore increasing the consumption of H1. The vice versa holds true. When H1 is depleted at an accelerated rate as a result in decreasing lH1 by 5%, as seen in the red profile in Figure 2B, C1 also depletes faster (Figure 2C) while human per capita mass increases (Figure 2D). This suggests that when humans consume more herbivores, carnivore populations will be adversely affected. However, P1 is unaffected by human patterns of animal consumption (Figure 2A).
Fig. 2 Sensitivity analysis results for animal consumption (lH1) without consumption increase. lH1 is increased and decreased by 5% to observe changes in P1, H1, C1 and per capita mass. The base case lH1 value is used as a standard of comparison
Sensitivity Analysis for Biomass to Biofuel Conversion Rate (λbiom)
Recently, biofuel has risen in popularity because of rising oil prices and the promise of energy security. The biomass to biofuel conversion rate parameter λbiom dictates how much biomass (P1) is needed per unit of biofuel. In theory, increasing λbiom would indicate that more mass is required for biofuel production, which is why P1 would deplete faster. The vice versa holds true as well. Figure 3 shows the tradeoff between allocating mass for biofuel and food.
The green profile from Figure 3A depicts how P1 depletes faster when biofuel production is inefficient. On the other hand, green profiles of Figure 3B and 3C show how allocating more mass for biofuel accelerates the reduction of H1 and C1, meaning that biofuels limit the food source available for wild species. Furthermore, the green profile of Figure 3D indicates a decrease in per capita human mass, which may be due to less plant-based food available for human needs and wild populations.
Fig. 3 Sensitivity analysis results for the biomass to biofuel conversion rate parameter (λbiom) without consumption increase. λbiom is increased and decreased by 5% to observe changes in P1, H1, C1 and per capita mass. The base case is used as a standard of comparison
Section 3: Non-linear programming problem
The non-linear programming (NLP) optimization problem can be divided in to two parts: the first part is to (i) rewrite the objective function in terms of the state and optimization variables, and the second part is to (ii) determine the algorithm for the discretized NLP.
(i) Conversion of the objective function into a usable form
From the time average expressions for FI we have;
FIt=1Tc0Tca(t)2v(t)4dt (3.1)
v(t)=i=1ndxidt2 (3.2)
a(t)=1v(t)i=1ndxidt d2xidt2 (i=1,2,…,n) (3.3)
However, the model, in terms of algebraic and difference equations, cannot provide the values of v(t) and a(t) directly. We need to use the finite difference approximations to evaluate the values of v(t) and a(t) at all time points (Gupta 2009).
The finite difference approximations for dxidt and d2xidt2 are listed as per the position of the evaluation point on the timeline;
Fig. 4 The timeline showing the different nodes for finite difference approximation
3.1. Approximations for the extreme left point (x1): The forward difference approximation formulae used are shown in Equation 3.4 and Equation 3.5
dxdti≈ xi+1-xi∆x (3.4)
d2xdt2i≈ xi+2-2xi+1+xi∆x2 (3.5)
3.2. Approximations for the extreme right point (xN): The backward difference approximation formulae used are shown in Equation 3.6 and Equation 3.7
dxdti≈ xi-xi-1∆t (3.6)
d2xdt2i≈ xi-2xi-1+xi-2∆t2 (3.7)
3.3. Approximations for the central points (x2, x3… xN-1): The central difference approximation formulae used are shown in Equation 3.8 and Equation 3.9
dxdti≈ xi+1-xi-12∆t (3.8)
d2xdt2i≈ xi+1-2xi+xi-1∆t2 (3.9)
Thus, Equations 3.4-3.9 can be used to rewrite the expressions for FI in a usable for, thus making the problem solvable for this system.
(ii) Algorithm for the discretized NLP (Yenkie 2014)
As mentioned in the paper, the total time is divided into 10 bins. The stepwise algorithm for single parameter ‘kP1’ NLP optimization is illustrated below in Figure 5.
Fig. 5 Optimal control general strategy. At each successive time step, the controlled parameter varies and influences the system’s FI. The parameter value is solved to satisfy the objective function to the greatest extent possible. By satisfying the objective function, the FI of the consumption increase with controlled parameter scenario will try to mimic the FI of the base case. Therefore, when examining consumption increase, the controlled parameter scenario will follow a more sustainable trajectory than the uncontrolled parameter scenario
Step#1: The decision variable is a vector consisting of the values of the parameter ‘kP1’ in the 10 equispaced bins.
Step#2: The objective function is the minimization of the summation of the FI variance values in the 10 bins. The part (i) describes the conversion of the objective function in terms of state and decision variables. Thus, the similar finite difference strategy is applied in each of the bins.
Step#3: This problem is subjected to the satisfaction of the model equations and bounds on the decision variables.
Step#4: The optimization problem is solved in each bin using the fmincon algorithm in Matlab where the optimizer attempts to satisfy optimality conditions known as Karush-Kuhn-Tucker conditions (Diwekar 2008).
The NLP optimizer’s methodology is shown in Figure 6. The optimizer calculates a vector of new values for the decision variable and this iterative sequence continues until the optimization criteria (Karush-Kuhn-Tucker conditions) are met.
Fig. 6 Nonlinear programming optimization process used to determine decision variable values (Diwekar, 2008)
Step#5: The optimized values of the parameter ‘kP1’ are then plugged into the model and the FI values are evaluated for the controlled scenario. These results are then compared with the base case and uncontrolled FI values, to see the impact of the systems behavior.
Note: For performing the optimal control for the 6 policy parameters together, the length of the decision vector increases to 60 [6 (# of policy parameters) x 10 (# of time divisions)] values. The other part of the NLP algorithm remains the same.
Section 4: Six parameter multivariable optimal control
Figure 7 depicts how six parameter multivariable optimal control affects each of the individual mass compartments in the model.
Fig. 7 Effects on all compartments due to six parameter multivariable optimal control
Based on the green profiles of Figure 7, all eight ecological mass compartments avoid extinction for two hundred years during the consumption increase scenario when controlled using six model parameters. Figure 7 proves how optimal control increases the sustainability of not only the domesticated mass compartments (P1, H1, C1), but also the wild mass compartments (P2, P3, H2, H3, C2) as well.
Looking at years 150 to 200 in Graph C, it may appear that the controlled scenario contains less P3 mass than the uncontrolled scenario. The same situation can be observed in the majority of mass compartments for certain time periods. Keep in mind that the objective was to minimize FI variance from the base case profile, not maximize the FI and the amount of mass in every individual mass compartment. Graph C suggests that a smaller amount of P3 results in a more stable mass compartment, especially since the sustainable base case profile has less P3. Therefore when considering ecosystem management, sacrifices in a few ecosystem populations for certain time periods is necessary in order to promote the sustainability of the overall system.
References
Cabezas H, Pawlowski C, Mayer A, Hoagland T (2005) Simulated experiments with complex
sustainable systems: ecology and technology. Resourc Conserv Recycling 44: 279-291. doi: 10.1016/j.resconrec.2005.01.005
Cabezas H, Pawlowski CW, Mayer AL, Hoagland NT (2003) Sustainability: Ecological, Social,
Economic, Technological, and Systems Perspectives. Clean Technol Environ Policy 5:167-180. doi: 10.1007/s10098-003-0214-y
Diwekar U (2008) Introduction to Applied Optimization. New York City, New York
Gupta SK (2009) Numerical Methods for Engineers. New Delha, India
Kotecha P, Diwekar U, Cabezas H (2013) Model based approach to study the impact of biofuels