We present a theoretical model in tourism economics, assuming that the market for tourism is an oligopoly with differentiated products. Destinations (i.e., countries, regions, sites or even firms) can invest in order to improve their carrying capacity that can be interpreted as the stock of physical, natural or cultural resources. Tourism flows yield current revenues, but they are usually detrimental for the cultural or natural resource stock over time. We find the solution of the dynamic model, and in particular we find the open-loop Nash equilibrium of the game among the destinations, under alternative settings, depending on whether the arrivals are exogenous or endogenous, and depending on whether the degree of differentiation among destinations is exogenous or endogenous. The model is rather general, and it can provide answers to different specific questions, like the choice between mass- vs. elite-tourism development strategies; the effect of the number of competing products upon profits; the optimal degree of product differentiation.