# The Evolution of Service Ecosystems Based on the Lotka-Volterra Model
## 6 Real-World Comparative Analysis and Discussion  
To further investigate and validate the impact of interaction coefficients between original services on the development of new services, this section employs real-world comparative analysis, studying two innovative composite services: Tesla and Google Glass. Tesla not only represents an automotive service but also integrates cutting-edge technology from the new-energy-material services, creating a novel comprehensive service product. Similarly, Google Glass combines personal goods services with advanced information technology services, attempting to open up the emerging market for smart wearable devices. To obtain complete data on the market development of different services in their early stages while ensuring data reliability and persuasiveness, the real-world comparison uses the rolling price-to-earnings ratio (PE-TTM-S) of the original services of Tesla and Google Glass as indicators of service development status. The rolling price-to-earnings ratio provides a real-time assessment of service profitability and is widely recognized as an effective indicator for evaluating service profitability and stock valuation, reflecting the current status and market expectations of services. Tesla became the first independent pure electric vehicle manufacturer in June 2010, with the first electric car delivered in June 2012. Google Glass had a shorter development period, with rapid growth in patent applications for smart glasses in 2011 and the official development of the Project Glass in April 2012. Therefore, the study uses the rolling price-to-earnings ratio data from 2010 to 2012 for automotive and new-energy-material services, and from 2011 to 2012 for personal goods and information technology services. The specific data are shown in the Figures 14 and 15:  
Figure 14: Change in PE.TTM.S for Tesla’s original services. Figure 13: Changes in the Simpson’s Index of service quantity with different \(\alpha\). The grey estimation method is currently the primary method for parameter estimation and fitting competition relationships for the Lotka-Volterra model. It is simple, requires less computation, is highly generalizable, and overcomes the difficulties caused by incomplete experimental data. The method will be used below to estimate the competition relationships between original services. The competition relationships between original services can be represented by the following Equations (27):  
\[ \begin{cases}\dfrac{ds_{1}(t)}{dt}=a_{10}s_{1}+a_{11}s_{1}^{2}+a_{12}s_{1}s_{2} \\ \dfrac{ds_{2}(t)}{dt}=a_{20}s_{2}+a_{21}s_{2}^{2}+a_{22}s_{1}s_{2}\end{cases}  \tag{27}\]  
Based on the grey derivative and paired logarithmic mapping relationships, the following relationships can be obtained:  
\[ \begin{cases}s_{1}(t+1)-s_{1}(t)=a_{10}\dfrac{s_{1}(t+1)+s_{1}(t)}{2}+a_{11} \bigg{(}\dfrac{s_{1}(t+1)+s_{1}(t)}{2}\bigg{)}^{2}\\ \hskip 14.226378 pt+a_{12}(\dfrac{s_{1}(t+1)+s_{1}(t)}{2})(\dfrac{s_{2}(t+1)+s_{ 2}(t)}{2})\\ \hskip 14.226378 pts_{2}(t+1)-s_{2}(t)=a_{20}\dfrac{s_{2}(t+1)+s_{2}(t)}{2}+a_{2 1}\bigg{(}\dfrac{s_{2}(t+1)+s_{2}(t)}{2}\bigg{)}^{2}\\ \hskip 14.226378 pt+a_{22}(\dfrac{s_{1}(t+1)+s_{1}(t)}{2})(\dfrac{s_{2}(t+1)+s_{ 2}(t)}{2})\end{cases}  \tag{28}\]  
Substituting data for \(t=1,2,\ldots,n\) into Equation (28), we obtain the matrix equations \(Y_{1}=w_{1}a_{1}\) and \(Y_{2}=w_{2}a_{2}\), where:  
\[w_{i}= \begin{bmatrix}\frac{s_{1}(2)+s_{1}(1)}{2}&\Big{(}\frac{s_{1}(2)+s_{1}(1 )}{2}\Big{)}^{2}&\Big{(}\frac{s_{1}(2)+s_{1}(1)}{2}\Big{)}\Big{(}\frac{s_{2}(2 )+s_{2}(1)}{2}\Big{)}\\ \frac{s_{1}(3)+s_{1}(2)}{2}&\Big{(}\frac{s_{1}(3)+s_{1}(2)}{2}\Big{)}^{2}& \Big{(}\frac{s_{1}(3)+s_{1}(2)}{2}\Big{)}\Big{(}\frac{s_{2}(3)+s_{2}(2)}{2} \Big{)}\\ \hskip 14.226378 pt\ldots\\ \frac{s_{1}(t+1)+s_{1}(t)}{2}&\Big{(}\frac{s_{1}(t+1)+s_{1}(t)}{2}\Big{)}^{2}& \Big{(}\frac{s_{1}(t+1)+s_{1}(t)}{2}\Big{)}\Big{(}\frac{s_{2}(t+1)+s_{2}(t)}{2 }\Big{)}\end{bmatrix}