Understanding the long-term seasonal dynamics of alpine glacierized basins is essential to evaluating their relation to climatic forcing. We focus on process knowledge by following a minimalist approach, and propose a spatially lumped nonlinear differential model (MIAGE) to describe the link between the volume V of water that is stored on the basin and the river runoff Q at the seasonal scale. We formulate the model structure by mathematically describing the link relating precipitation P, temperature T, river runoff Q and stored volumes V. Beside reproducing some typical features of the catchment hydrology of glacierized basins, MIAGE offers an explanation of their seasonal hydroclimatic behaviour and of the origin of their dissipative properties from a dynamical system perspective. By studying the model nonlinear properties, characteristics, and performances, we show that climatic change has both direct and feedback effects on such basins. Eventually a synchronization of the runoffs with either the precipitation trend or the temperature trend may occur depending on the storage conditions. This model is subsequently used in a companion paper in order to investigate the potential impact of climatic change scenarios on basins of the Italian and Swiss Alps [Mechanistic interpretation of alpine glacierized environments: Part 2. Hydrologic interpretation and model parameters identification on case study, this issue].