# Predator-Prey Model Based Asymmetry Resource Allocation in Satellite-Terrestrial Network
## 3 System Model
If the services loads in the prey-satellites \(s_{1}\) are not offloaded to the predator-satellites \(s_{2}\), the dynamic variation of services loads in the predator-satellites \(s_{2}\) can be described as follows [32],  
\[\frac{dx_{2}(t)}{dt}=[-a_{2}(t)+\varepsilon_{2}u_{2}(t)]x_{2}(t), \tag{2}\]  
where \(-a_{2}(t)\) is the variation rate of the services loads in the predator-satellites \(s_{2}\). \(u_{2}(t)\) is the action of the predator-satellites \(s_{2}\) to attract the mobile users to access to the satellite network, and \(\varepsilon_{2}\) is the related weighted parameter to denote the conversion percent of the action. Assuming the computation abilities of the predator-satellites, \(s_{2}\) is lower than the services loads offloading rate of the prey-satellites \(s_{1}\), and then the services loads in the predator-satellites \(s_{2}\) is increased with the offloading process of the services loads from the prey-satellites \(s_{1}\), and the services loads in the prey-satellites \(s_{1}\) is decreased with the offloading process of the services loads to the predator-satellites \(s_{2}\). Then, we have the following functions to denote the dynamic variation of services loads in the prey-satellites \(s_{1}\) and the predator-satellites \(s_{2}\) based on the Volterra model, respectively,  
\[\frac{dx_{1}(t)}{dt}=x_{1}(t)[a_{1}(t)-b_{1}x_{2}(t)+\varepsilon_{1}u_{1}(t)], \tag{3}\]  
\[\frac{dx_{2}(t)}{dt}=x_{2}(t)[-a_{2}(t)+b_{2}x_{1}(t)+\varepsilon_{2}u_{2}(t)], \tag{4}\]  
where \(b_{1}\) is the proportion of services loads offloaded from the prey-satellites \(s_{1}\) to the predator-satellites \(s_{2}\), and \(b_{2}\) is the growth rate of services loads due to the offloading process from the prey-satellites \(s_{1}\) to the predator-satellites \(s_{2}\). As increased satellites are utilized to compose a hyper dense satellite network, the density of the satellite nodes in each satellite group is the influence factor need to be considered. Then the dynamic variation of the services loads in the prey-satellites \(s_{1}\) and the predator-satellites \(s_{2}\) should consider the effect within the satellite groups, and the dynamic variation of services loads in the prey-satellites \(s_{1}\) and the predator-satellites \(s_{2}\) can be re-formulated based on Lotka-Volterra model as follows,  
\[\frac{dx_{1}(t)}{dt}=x_{1}(t)[a_{1}(t)-b_{1}x_{2}(t)-c_{1}x_{1}(t)+\varepsilon_ {1}u_{1}(t)], \tag{5}\]  
\[\frac{dx_{2}(t)}{dt}=x_{2}(t)[-a_{2}(t)+b_{2}x_{1}(t)-c_{2}x_{2}(t)+\varepsilon _{2}u_{2}(t)], \tag{6}\]  
where \(c_{1}\) and \(c_{2}\) are the influence factors within the satellite group. In the satellite networks, due to the impact of the communication environments, the loads in each satellite group will be inevitably interfered by external stochastic factors, such as cosmic radiation, solar spot, rain attenuation, etc. Then, the stochastic disturbances need to be added to the dynamic variation of the services loads in the prey-satellites \(s_{1}\) and the predator-satellites \(s_{2}\) as follows,  
\[ \begin{array}{l}dx_{1}(t)=x_{1}(t)[a_{1}(t)-b_{1}x_{2}(t)-c_{1}x_{1}(t)+\varepsilon _{1}u_{1}(t)]dt\\ +\sigma_{1}(t)x_{1}(t)dB_{1}(t)\end{array} , \tag{7}\]  
\[ \begin{array}{l}dx_{2}(t)=x_{2}(t)[-a_{2}(t)+b_{2}x_{1}(t)-c_{2}x_{2}(t)+ \varepsilon_{2}u_{2}(t)]dt\\ +\sigma_{2}(t)x_{2}(t)dB_{2}(t)\end{array} , \tag{8}\]