# The Evolution of Service Ecosystems Based on the Lotka-Volterra Model
## 3 Service Competition-Cooperation Evolution Model
### Ternary Lokta-Volterra Improvement Model
#### 3.2.1 Social and Environmental Factors  
In the natural and scientific fields, entropy is an omnipresent concept. In physics, chemistry, biology, information theory, and other fields, entropy has its unique meaning and application, such as the commonly used loss function of cross-entropy in deep learning [31], statistical mechanical entropy from the perspective of microscopic particles [32], and value entropy in the natural social environment [33]. The second law of thermodynamics states that in an isolated system, spontaneous processes always increase the system's entropy, and the higher the entropy, the greater the complexity of the system. This theory also applies to the evolutionary processes of service ecosystems, where social environments can exert driving forces or resistance.  
According to the entropy theory, the service ecosystem will adapt to the current market environment with the entropy change effect [34; 35]. The types of services tend to be diversified and complicated gradually, and the uncertainty of the system increases. Meanwhile, changes in the social environment, such as economic fluctuations and technological innovation, may also lead to the increase of entropy in the system. Moreover, with the mature supervision mechanism and the stability of the social environment, the standardization of services will be improved, and the entropy in the system will be reduced.  
Based on the above analysis, entropy-increase (\(E_{I}\)) and -decrease (\(E_{R}\)) models applicable to service ecosystems affected by social and environmental factors can also be proposed:  
\[\left\{\begin{aligned} & E_{I}=\sum_{n=1}^{N}W_{n}\sum_{i=1}^{C}X_{i} logX_{i}\\ & E_{R}=-\sum_{n=1}^{N}W_{n}\sum_{i=1}^{C}X_{i}logX_{i}\end{aligned}\right. \tag{4}\]  
where \(n=1,2,\ldots,N\) represents the influencing factors of social environment on the entropy change of service ecosystem, such as economic factors, technological change factors, policy and regulation factors, etc, and \(W_{n}\) represents the influence weight of the influencing factors in the social environment on the service ecosystem. The \(i=1,2,\ldots,C\) represents a specific event in one of the influencing factors. For example, a series of game recharge policies have been recently announced, which restrict incentives such as first recharge and continuous recharge, which has caused a shock to the game industry,resulting in the cancellation of a large number of Internet companies researching games and a decline in the environmental capacity of the entire game industry. \(X_{i}\) represents the probability that the event will have an effect on entropy.  
The influence of social environment on the entropy increase and decrease of the service ecosystem will not change the interaction relationship between services themselves, but will affect the environmental capacity of some services. Therefore, according to the entropy theory, the three-way service competition-cooperation model under the influence of social environment can be obtained:  
\[\begin{cases}\dfrac{ds_{1}}{dt}=r_{1}s_{1}\bigg{(}1-\dfrac{s_{1}}{N_{1}(1+ \theta_{1})}-\alpha_{12}\dfrac{s_{2}}{N_{2}(1+\theta_{2})}-\alpha_{13}\dfrac{s _{3}}{N_{3}(1+\theta_{3})}\bigg{)}\\ \dfrac{ds_{2}}{dt}=r_{2}s_{2}\bigg{(}1-\dfrac{s_{2}}{N_{2}(1+\theta_{2})}- \alpha_{21}\dfrac{s_{1}}{N_{1}(1+\theta_{1})}-\alpha_{23}\dfrac{s_{3}}{N_{3}(1 +\theta_{3})}\bigg{)}\\ \dfrac{ds_{3}}{dt}=r_{3}s_{3}\bigg{(}1-\dfrac{s_{3}}{N_{3}(1+\theta_{3})}- \alpha_{31}\dfrac{s_{1}}{N_{1}(1+\theta_{1})}-\alpha_{32}\dfrac{s_{2}}{N_{2}(1 +\theta_{2})}\bigg{)}\end{cases} \tag{5}\]  
where \(\theta_{i}(\theta_{i}>0)\) represents the influence coefficient of this social environment on service \(i\).