In this paper, we concentrate on the uniquencess of the positive solution for the general elliptic system $Delta$u+u($g_1$(u)-$g_2$(v))=0 $Delta$u+u($h_1$(u)-$h_2$(v))=0 in$R_{+}$ $ imes$ $Omega$, $umidpartialOmega = umidpartialOmega = 0$. This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.