# The Evolution of Service Ecosystems Based on the Lotka-Volterra Model
## 3 Service Competition-Cooperation Evolution Model
### Ternary Lokta-Volterra Improvement Model
#### 3.2.2 Service Dynamics  
In biology, population changes are usually affected by evolutionary forces such as gene mutation, natural selection, and gene flow, which take multiple generations to significantly change the characteristics of the population. Therefore, the impact of the population on the environment is relatively stable over a short time scale. In contrast, in a service ecosystem, even uniform types of services are constantly undergoing rapid change and adjustment due to multiple factors. The lifecycle of service with the change in market can be divided into germination, growth, maturity, and decline periods. To model the lifecycle dynamics of services, we employ the Weibull probability density function--a flexible continuous distribution widely used to characterize time-dependent phenomena like failure rates or adoption curves. In this section, the Weibull probability density function can be expressed as:  
\[f(t)=k\lambda^{2}te^{-(\lambda t)^{2}} \tag{6}\]  
where \(t\) represents the time, \(f(t)\) denotes the competitiveness of the service itself in the market, \(k\) and \(\lambda\) are the variable parameter weighting of various influencing factors of the competitiveness of the service. The extreme point \(\bigg{(}\dfrac{1}{\sqrt{2}\lambda},\frac{\sqrt{2}\lambda k_{t}e^{-\frac{1}{2}} }{2}\bigg{)}\) of the function can be calculated by taking the derivative of Equation (6), where \(\lambda\) determines the time and size of the service competitiveness reaching the peak, and k only determines the size of the service competitiveness peak. When \(k=3\) and \(\lambda=0.4\), the Weibull probability density function curve is shown in Figure 2.  
From the perspective of service ecosystem, the factors that affect the competitiveness of a class of services are quite diversified, and the common factors include service technology dependence \(T\), including the degree and popularity of new technologies or new business models adopted; the degree of service standardization \(Q\), whether the service can be replicated and expanded quickly; user acceptance of the service \(U\), whether the service meets user expectations; service compliance \(L\), whether the service meets relevant laws and regulations, etc. By combination with the common influencing factors and the analysis of the extreme value of Weibull curve, the following relationship can be shown: \(\lambda=TP_{T}+QP_{Q}+LP_{L}+OP_{O}\), \(k=QP_{Q}+UP_{U}+OP_{O}\), where \(P_{i}\) represents the influence weight of factor \(i\), and \(O\) represents other influence factors. The impact of service dynamics in a service ecosystem is not between services, but rather the competitiveness of the services themselves to change over time in the system. Therefore, combined with Equation (7), a ternary Lokta-Volterra improvement model can be obtained considering social environmental factors and service dynamics:  
\[\left\{ \begin{aligned} \frac{ds_{1}}{dt}&=r_{1}s_{1} \bigg{(}1-\delta_{1}\frac{s_{1}}{N_{1}(1+\theta_{1})}-\alpha_{12}\frac{s_{2}}{ N_{2}(1+\theta_{2})}-\alpha_{13}\frac{s_{3}}{N_{3}(1+\theta_{3})}\bigg{)}\\ \frac{ds_{2}}{dt}&=r_{2}s_{2}\bigg{(}1-\delta_{2} \frac{s_{2}}{N_{2}(1+\theta_{2})}-\alpha_{21}\frac{s_{1}}{N_{1}(1+\theta_{1}) }-\alpha_{23}\frac{s_{3}}{N_{3}(1+\theta_{3})}\bigg{)}\\ \frac{ds_{3}}{dt}&=r_{3}s_{3}\bigg{(}1-\delta_{3} \frac{s_{3}}{N_{3}(1+\theta_{3})}-\alpha_{31}\frac{s_{1}}{N_{1}(1+\theta_{1}) }-\alpha_{32}\frac{s_{2}}{N_{2}(1+\theta_{2})}\bigg{)}\end{aligned} \right. \tag{7}\]  
where \(\delta_{i}(\delta_{i}>0)\) represents the impact factor of the dynamism of class \(i\) service on that service; \(r_{i}\) represents the intrinsic growth rate of service \(i\); \(s_{i}\) represents the population size of service \(i\); \(N_{i}\) represents the carrying capacity of service \(i\); \(\theta_{i}\) represents the social-environmental impact on service \(i\).