随机环境中多类型接触过程的Hydrodynamic极限

The Hydrodynamic Limit for the Multitype Contact Processin a Random Environment

摘要: 本文首先构造随机环境中多类型接触过程,然后研究其Hydrodynamic行为,证明了该模型的Hydrody-namic 极限是下列偏微分方程的解:\left\\beginarrayl\frac\partial u_0\partial t=\frac12 \Delta u_0+\left(1-u_0\right)-u_0\left(\lambda_1 u_1+\lambda_2 u_2\right) \\ \frac\partial u_1\partial t=\frac12 \Delta u_1+\lambda_1 u_1\left(1-u_1-u_2\right)-u_1 \\ \frac\partial u_2\partial t=\frac12 \Delta u_2+\lambda_2 u_2\left(1-u_1-u_2\right)-u_2 \\ u_i(0, r)=g_i(r), \quad i=0,1,2 .\endarray\right.

Abstract: In this paper, the hydrodynamic behavior of the multitype contact process in a random environment wasstudied.First, the model was constructed by the graphical representation. Second, we have proved that it’shydrodynamic limit is a solution of the following partial differential equation: \left\\beginarrayl\frac\partial u_0\partial t=\frac12 \Delta u_0+\left(1-u_0\right)-u_0\left(\lambda_1 u_1+\lambda_2 u_2\right) \\ \frac\partial u_1\partial t=\frac12 \Delta u_1+\lambda_1 u_1\left(1-u_1-u_2\right)-u_1 \\ \frac\partial u_2\partial t=\frac12 \Delta u_2+\lambda_2 u_2\left(1-u_1-u_2\right)-u_2 \\ u_i(0, r)=g_i(r), \quad i=0,1,2.\endarray\right.