Many large-scale dynamical systems arising from different fields of science and engineering can be regarded as coupled systems on networks. Examples include biological and artificial neural networks, nonlinear oscillators on lattices, complex ecosystems and the transmission models of infectious diseases in heterogeneous populations. Of particular interest is to investigate in what degree and fashion the dynamical behaviors are determined by the architecture of the network encoded in the directed graph. We will address this from population dynamics perspectives. Specifically, many recent outbreaks and spatial spread of infectious diseases have been influenced by human movement over air, sea and land transport networks, and/or anthropogenic-induced pathogen/vector movement. These spatial movements in heterogeneous environments and networks are often asymmetric (biased). The effects of asymmetric movement versus symmetric movement will be investigated using several epidemiological models from the literature, and the analytical tools employed are from differential equations, dynamical systems to matrix theory and graph theory. These investigations provide new biological insights on disease transmission and control, and also highlight the need of a better understanding of dynamical systems on networks.