We present a dynamic optimisation model that helps to explain the expansion and shutdown of activities at the Gallo-Roman site of La Graufesenque (near Millau, France), and that requires us to solve a non-linear differential equation concerning the exploitation of the forest. Once the resolution of this equation has shown the existence and uniqueness of a trajectory, we can explain most of the findings of archaeologists and historians to date. One parameter to which the dynamics are very sensitive is the discount rate, which represents a greater or lesser consideration for the future. Using a Monte Carlo method, we show that if this parameter is not constant – which is very likely over the century and a half of site activity – but varies according to a plausible (autoregressive) process, the trajectories obtained do not differ very much from the set of trajectories obtained for constant values of the parameter. Ultimately, taking the long term into account in decision making delays the onset of a sudden decline, but does not prevent it. Any resource exploitation in our contemporary world that could be modelled by the same equations would suffer the same fate.