A new model for limb development which incorporates both outgrowth due to cell growth and division. and interactions between morphogens produced in the zone of polarizing activity (ZPA) and the apical epidermal ridge (AER) is developed and analysed. The numerically-computed spatio-temporal distributions of these morphogens demonstrate the importance of interaction between the organizing regions in establishing the morphogenetic terrain on which cells reside, and because growth is explicitly incorporated, it is found that the history of a cell's exposure to the morphogens depends heavily on where the cell originates in the early limb bud. Because the biochemical steps between morphogen(s) and gene activation have not been elucidated, there is no biologically-based mechanism for translating the spatio-temporal distributions of morphogens into patterns of gene expression, but several theoretically plausible functions that bridge the gap are suggested. For example it is shown that interpretation functions based on the history of a cell's exposure to the morphogens can qualitatively account for observed patterns of gene expression. The mathematical model and the associated computational algorithms are sufficiently flexible that other schemes for the interactions between morphogens, and their effect on the spatio-temporal pattern of growth and gene expression, can easily be tested. Thus an additional result of this work is a computational tool that can be used to explore the effects of various mutations and experimental interventions on the growth of the limb and the pattern of gene expression. In future work we will extend the model to a three-dimensional representation of the limb and will incorporate a more realistic description of the theological properties of the tissue mass, which here is treated as a Newtonian fluid.