Concepts and modeling of spatial diffusion

The appearance of S-curve in the treatment of a space-time data is very often associated to a spatial diffusion. This systematic assimilation of the S- curve to the logistic model is misleading, it leads to an amalgam which masks the diversity of mathematical models to capture really different phenomena of diffusion. The aims of this paper is to show some examples of diffusion which are all expressed by a Scurve but which correspond in fact to different models. Three models are presented: - logistic model is the most usually used by geographers despite his independency from space. We compare here a spatialized and probabilistic version of this model with its analytical and deterministic form. as opposed to the first one, the front diffusion model imposes a strong space constraint, the contact being prevalent in the diffusion. In this case, diffusion is not subjected to a global constraint but depends on the length and on the form of the front at any moment. - the third model is composite, it proceeds of the logistic growth forced by local constraints and of propagation of a front. These three models are studied in their analytical forms and are compared with their dynamic simulated. We show a continuity between these 3 models entering within the framework of a broader class of parameterized models. Moreover their combinations give an expansion of other models making it possible to apply to a great variety of problems.