# The Evolution of Service Ecosystems Based on the Lotka-Volterra Model
## 3 Service Competition-Cooperation Evolution Model
### Ternary Lotka-Volterra Service Competition-Cooperation Model  
In the traditional biological population theory, the Lokta-Volterra model is an extension of the Logistic model, and its process is mainly determined by three factors: the number of the population \(p\), the observation time of the evolutionary model \(t\), the environmental capacity of the population \(K\) and the growth rate of the population \(r\). Therefore, the detailed logistic regression is as follows:  
\[\frac{dp}{dt}=rp\Big{(}1-\frac{p}{K}\Big{)} \tag{1}\]  
In service ecosystems, the coexistence of competition and cooperation among services is known as \"competition-cooperation\". This phenomenon is increasingly prevalent in the modern business environment, where the key strategy lies in identifying collaborative opportunities even in highly competitive markets to achieve mutual benefits. Existing services compete for limited market space while simultaneously leveraging cooperation and integration to enhance their competitiveness, evolving into composite services better suited to current market demands [30]. Newly evolved composite services inevitably participate in the competitive-cooperative relationships within the service ecosystem. These new services may challenge the market positions of existing services, particularly when they offer more efficient, cost-effective, or superior user experiences, which inevitably intensifies the battle for market share. Conversely, new services can also expand their market reach through collaboration with existing services, such as through service bundling or cross-selling. This complementary relationship can provide a more comprehensive solution to consumers. From a comprehensive perspective, the competitive-cooperative relationships between existing and new services are dynamic, continuously evolving in response to changes in the market environment, technological advancements, customer needs, and strategic directions. Depending on the context and timing, these relationships may exhibit varying degrees of competition or cooperation. Based on the above analysis, the service competition-cooperation evolution model can be obtained as shown in Equation (2):  
\[\left\{ \begin{aligned} &\frac{ds_{1}}{dt}=r_{1}s_{1}\bigg{(}1-\frac{s_{1}}{N_{1}}- \alpha_{12}^{\prime}\frac{s_{2}}{N_{2}}-\alpha_{13}^{\prime}\frac{s_{3}}{N_{3} }+\alpha_{12}^{\prime\prime}\frac{s_{2}}{N_{2}}+\alpha_{13}^{\prime\prime} \frac{s_{3}}{N_{3}}\bigg{)}\\ &\frac{ds_{2}}{dt}=r_{2}s_{2}\bigg{(}1-\frac{s_{2}}{N_{2}}- \alpha_{21}^{\prime}\frac{s_{1}}{N_{1}}-\alpha_{23}^{\prime}\frac{s_{3}}{N_{3} }+\alpha_{21}^{\prime\prime}\frac{s_{1}}{N_{1}}+\alpha_{23}^{\prime\prime} \frac{s_{3}}{N_{3}}\bigg{)}\\ &\frac{ds_{3}}{dt}=r_{3}s_{3}\bigg{(}1-\frac{s_{3}}{N_{3}}- \alpha_{31}^{\prime}\frac{s_{1}}{N_{1}}-\alpha_{32}^{\prime}\frac{s_{2}}{N_{2} }+\alpha_{31}^{\prime\prime}\frac{s_{1}}{N_{1}}+\alpha_{32}^{\prime\prime} \frac{s_{2}}{N_{2}}\bigg{)}\end{aligned} \right. \tag{2}\]  
where \(s_{i}\) represents the size of class service \(i\) in the service ecosystem, \(s_{1}\), \(s_{2}\) are regarded as the original two types of atomic services, and \(s_{3}\) is regarded as the newly evolved combined services. The \(\alpha_{ij}^{\prime}\) indicates competition between services, and \(\alpha_{ij}^{\prime\prime}\) indicates the influence of class service \(i\) on \(i\), which is called the interaction coefficient among services. Moreover, the Equation (3) can be simply expressed as:  
\[ \begin{cases}\dfrac{ds_{1}}{dt}=r_{1}s_{1}\bigg{(}1-\dfrac{s_{1}}{N_{1}}-\alpha_{1 2}\dfrac{s_{2}}{N_{2}}-\alpha_{13}\dfrac{s_{3}}{N_{3}}\bigg{)}\\ \dfrac{ds_{2}}{dt}=r_{2}s_{2}\bigg{(}1-\dfrac{s_{2}}{N_{2}}-\alpha_{21}\dfrac{s _{1}}{N_{1}}-\alpha_{22}\dfrac{s_{3}}{N_{3}}\bigg{)}\\ \dfrac{ds_{3}}{dt}=r_{3}s_{3}\bigg{(}1-\dfrac{s_{3}}{N_{3}}-\alpha_{31}\dfrac{s _{1}}{N_{1}}-\alpha_{32}\dfrac{s_{2}}{N_{2}}\bigg{)}\end{cases}