# Automobile Technological Transition Scenarios Based on Environmental Drivers
## 2 Materials and Methods  
The starting point of the analysis presented in this document is the mathematical model of Equation (4). This system of first-order ordinary differential equations has been obtained from system dynamics methodology (see for example [56]). The rules of causality are simple: (1) conventional automobiles generate emissions of CO\({}_{2}\)[22; 23; 57], (2) electric automobiles produce hazardous waste [36; 58], (3) emissions and hazardous waste generate nonconformity in vehicle consumers taking the option to change technology to reduce its environmental impact [32], (4) the enthusiasm to acquire one or another technology leads to the entry of automobiles into the system. The hazardous waste of conventional vehicles is out of the scope of this research work, since it is not as important a selection criterion for buyers as the hazardous waste of electric automobiles. From these causality rules, we built the Forrester diagram shown in Figure 1. Conventional and electric automobiles are the level variables of the system. The equations presented below are an interpretation of the Forrester diagram. The rules used were [56]: (1) the change over time in level variables, represented in boxes, is the difference between the inflow and outflow variables, symbolized as valves; (2) the flow variables are the products between the variables connected with arrows, and (3) the auxiliary variables (represented as circles), which are functions of the variables connected through arrows, are defined according to the relationship between them. The conventional automobiles \(x\) change over time (represented by a point on the variable), given in \([automobiles]\), is the difference between automobiles entering the market \(CAI\) and those leaving \(CAO\), both given in \([automobiles/year]\). The electric automobiles \(y\) change over time, given in \([automobiles]\), is the difference between the automobiles that enter the market \(EAI\) and those leaving \(EAO\), both given in \([automobiles/year]\). So, the level equations are defined as follows:  
\[ \begin{split}\dot{x}&=CAI-CAO\\ \dot{y}&=EAI-EAO\end{split}  \tag{1}\]  
Figure 1: Stocks and Flows diagram of the automobile technology transition from environmental drivers. The change ratio equations presented in Equation (1) show that the increase in the quantities of each automobile category depends on the disagreement that users have with the other category, and the decrease depends on an average depreciation, as follows:  
\[CAI=a\cdot x\cdot(1-CEA)\qquad CAO=b\cdot x \tag{2}\]  
\[EAI=c\cdot y\cdot(1-CCA)\qquad EAO=d\cdot y\]  
where \(a\), \(b\), \(c\), and \(d\) are exchange rates in the range \([0,1]\), given in \([year^{-1}]\), and the conformities with each category depend on the closeness of the emissions \(E\), given in \([\mathit{TonCO_{2}}/year]\), or hazardous waste disposal \(W\), given in \([\mathit{Ton}/year]\), with an allowable limit value (emissions limit \(N\), given in \([\mathit{TonCO_{2}}/year]\), and hazardous waste limit \(W\), given in \([\mathit{Ton}/year]\)) that could be agreed through national or international environmental commitments. The equations for emissions and hazardous waste are defined as follows:  
\[CEA=W/L\qquad CCA=E/N \tag{3}\]  
\[E=\epsilon\cdot x\qquad\qquad W=\rho\cdot y\]  
where \(\epsilon\geq 0\) and \(\rho\geq 0\) are exchange rates given in \([\mathit{TonCO_{2}}/(\mathit{automobile}\cdot year)]\) and \([\mathit{Ton}/(\mathit{automobile}\cdot year)]\), respectively. Finally, by replacing the equations, and considering \(\alpha=a-b\), \(\beta=c-d\), \(\delta=a\rho/L\), and \(\lambda=c\epsilon/N\), we obtain the following mathematical model:  
\[\dot{x}=x(\alpha-\delta y) \tag{4}\]  
\[\dot{y}=y(\beta-\lambda x)\]  
Note that if the system of Equation (4) is studied under the variation of \(\alpha\) and \(\beta\), different scenarios of exchange rates will be considered. The model validation was performed using the tests reported in [59].