In this paper Australian domestic and international inbound travel are modelled by an anisotropic dynamic spatial lag panel Origin-Destination (OD) travel flow model. Spatial OD travel flow models have traditionally been applied in a single cross-sectional context, where the spatial structure is assumed to have reached its long run equilibrium and temporal dynamics are not explicitly considered. On the other hand, spatial effects are rarely accounted for in traditional tourism demand modelling. We attempt to address this dichotomy between spatial modelling and time series modelling in tourism research by using a spatial-temporal model. In particular, tourism behaviour is modelled as travel flows between regions. Temporal dependencies are accounted for via the inclusion of autoregressive components, while spatial autocorrelations are explicitly accounted for at both the origin and the destination. We allow the strength of spatial autocorrelation to exhibit seasonal variations, and we allow for the possibility of asymmetry between capital-city neighbours and non-capital-city neighbours. Significant temporal and spatial dynamics have been uncovered for both domestic and international tourism demand. For example we find strong seasonal temporal autocorrelations, significant trends and significant spatial autocorrelations at both the origin and the destination. Moreover, the spatial patterns are found to be most significant during peak holiday seasons. Understanding these patterns in tourist behaviour has important implications for tourism operators.