In this talk, we will firstly consider the monotonicity of the spectral bound of , where is a quasi-positive matrix describing population dispersal among patches in a heterogeneous environment and is a diagonal matrix encoding within-patch population dynamics. Then we will study a Lotka-Volterra competition patch model with asymmetric dispersal. The global dynamics of the model is classified under the assumptions of weak competition and the weighted digraph of the connection matrix is strongly connected and cycle-balanced. This talk is based on joint works with Shanshan Chen, Junping Shi, Zhisheng Shuai.