# Predator-Prey Model Based Asymmetry Resource Allocation in Satellite-Terrestrial Network
## 4 Analysis and Algorithms  
In this section, let us consider the solution of the proposed model and also prove the existence and uniqueness of this solution. Definition 1 (Asymmetry Model): The relationships between the prey-satellites and the predator-satellites are asymmetry relations, where the predator-satellites can offload the services from the prey-satellites, and the prey-satellites cannot obtain the load from the predator-satellites. Under the predator-prey model, the mobile satellites can be divided two groups. The first group is the set of satellites denoted by predator-satellites. We assume that this kind of satellite cannot directly provide the access services to the mobile users. The second group is the set of satellites noted by prey-satellites. They can provide the access services to the mobile users with lower transmission latency. The predator-satellites can obtain the services from the prey-satellites, where the prey-satellites cannot obtain the services from the predator-satellites. Theorem 4.1 (Existing and Unique): _For any initial value \((x_{1}(0),x_{2}(0))\in R_{+}^{2}\), the solution of model (8) exists and is unique._  
Proof: According to the theory of stochastic differential equations, when the coefficients of Model (7) and (8) satisfy the local Lipschitz condition and the linear growth condition, the solutions of Model (7) and (8) satisfying the initial value condition are global existence and uniqueness. The coefficients of Model (7) and (8) satisfy the local Lipschitz condition, but do not satisfy the linear growth condition. Therefore, the solutions of Model (7) and (8) tend to be infinity in a finite time. Since the coefficients of Model (7) and (8) satisfy the local Lipschitz condition, Model (7) and (8) would have unique local solutions in the time interval \([0,\tau_{e})\) for the given initial conditions \((x_{1}(0),x_{2}(0))\in R_{+}^{2}\), where \(\tau_{e}\) is the explosion time. We only need to prove that the solutions of Model (7) and (8) are global when \(\tau_{e}=+\infty\). Note that in the form of Model (7) and (8), we can find the explicit solution in the interval solution \([0,\tau_{e})\) as follows,  
\[x_{1}(t)=\frac{\exp\Bigl{\{}\int_{0}^{t}[a_{1}(s)+\varepsilon_{1}u_{1}(s)-0.5 \tau_{1}(s)]ds-b_{1}\int_{0}^{t}x_{2}(s)ds+\int_{0}^{t}\sigma_{1}(s)dB_{1}(s) \Bigr{\}}}{{x_{1}(0)}^{-1}+c_{1}\int_{0}^{t}\exp\Bigl{\{}\int_{0}^{s}[a_{1}( \tau)+\varepsilon_{1}u_{1}(s)-0.5\sigma_{1}(\tau)]d\tau-b_{1}\int_{0}^{t}x_{2} (\tau)d\tau+\int_{0}^{t}\sigma_{1}(\tau)dB_{1}(\tau)\Bigr{\}}ds}, \tag{9}\]  
\[x_{2}(t)=\frac{\exp\Bigl{\{}\int_{0}^{t}[-a_{2}(s)+\varepsilon_{2}u_{2}(s)-0.5 \sigma_{2}(s)]ds+b_{2}\int_{0}^{t}x_{1}(s)ds+\int_{0}^{t}\sigma_{2}(s)dB_{2}(s) \Bigr{\}}}{{x_{2}(0)}^{-1}+c_{2}\int_{0}^{t}\exp\Bigl{\{}\int_{0}^{s}[-a_{2}( \tau)+\varepsilon_{2}u_{2}(s)-0.5\sigma_{1}(\tau)]d\tau-b_{2}\int_{0}^{t}x_{1 }(\tau)d\tau+\int_{0}^{t}\sigma_{2}(\tau)dB_{2}(\tau)\Bigr{\}}ds}, \tag{10}\]  
where \(x_{1}(t)>0\) and \(x_{2}(t)>0\). In order to prove the boundedness, we construct the following comparison equation,  
\[\left\{ \begin{array}{l}\frac{d\overline{x}_{1}(t)}{dt}=\overline{x}_{1}(t)[a _{1}(t)+\varepsilon_{1}u_{1}(t)-c_{1}\overline{x}_{1}(t)]\\ +\sigma_{1}(t)\overline{x}_{1}(t)dB_{1}(t)\\ \frac{d\overline{x}_{2}(t)}{dt}=\overline{x}_{2}(t)[-a_{2}(t)+\varepsilon_{2}u _{2}(t)+b_{2}\overline{x}_{1}(t)-c_{2}\overline{x}_{2}(t)]\\ +\sigma_{2}(t)\overline{x}_{2}(t)dB_{2}(t)\end{array} ,\right. \tag{11}\]where the initial conditions are \(x_{1}(0)=\overline{x}_{1}(0)\) and \(x_{2}(0)=\overline{x}_{2}(0)\).