The

Context.Thermally pulsating asymptotic giant branch (AGB) stars are the main producers of slow neutron capture (s-) process elements, but there are still large uncertainties associated with the formation of the main neutron source, 13C, and with the physics of these stars in general. Observations of s-process element enhancements in stars can be used as constraints on theoretical models. Aims.For the first time we have applied stellar population synthesis to the problem of s-process nucleosynthesis in AGB stars, in order to derive constraints on free parameters describing the physics behind the third dredge-up and the properties of the neutron source. Methods.We utilize a rapid evolution and nucleosynthesis code to synthesize different populations of s-enhanced stars, and compare them to their observational counterparts to find out which values of the free parameters in the code produce synthetic populations that fit the observed populations best. These free parameters are the amount of third dredge-up, the minimum core mass for third dredge-up, the effectiveness of 13C as a source of neutrons, and the size in mass of the 13C pocket. Results.We find that galactic disk objects are reproduced by a spread of a factor of two in the effectiveness of the 13C neutron source. Lower metallicity objects can be reproduced only by lowering the average value of the effectiveness of the 13C neutron source needed for the galactic disk objects by at least a factor of 3. Using observations of s-process elements in post-AGB stars as constraints we find that dredge-up has to start at a lower core mass than predicted by current theoretical models, that it has to be substantial $(\lambda\ga 0.2$) in stars with mass M $\la$ 1.5 $M_{\odot}$, and that the mass of the 13C pocket must be about 1/40 that of the intershell region