Summary: The discrete dynamics for competing populations of Lotka-Volterra type modelled as \[ \begin{aligned} N_1 (t+1) &= N_1 (t) \exp [r_1 (1- N_1- b_{12} N_2)],\\ N_2 (t+1) &= N_2 (t) \exp [r_2(1- N_2- b_{21} N_1)], \end{aligned} \] is considered. In the case of non- persistence the attractive behavior of the model has been discussed. Especially, there are two attractive sets when \(b_{ij}>1\), and the attractive behaviors are more complicated than those of the corresponding continuous model. The attracted regions are given. We prove that the model is also persistent in the degenerate case of \(b_{ij}=1\). In the persistence case of \(b_{ij}<1\), the existence and uniqueness for two- period points of the model are studied at \(r_1= r_2\). The condition for multi-pairs of two-period points is indicated and their influence on population dynamical behavior is shown.