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tokens: from CO line emission ( Olofsson et al. 2002 ; González Delgado et al. 2003 ), as well as a linear fit to the observed values. These observed wind properties can be compared with wind properties derived from the model grid in order to evaluate what combinations of seed particle abundance and piston amplitude best reproduce the observational data. Constraining the pulsation period in the model data to less than 800 days to fit the periods of the observed sample, we found that models with log( n d ∕ n H ) = − 14.5 and u p = 2.0 give the best agreement with observations. The bottom panel of Fig. 12 shows the distribution of the differences between model mass-loss rates and the expected mass-loss rates from the linear fit to the observed values. The middle panel of Fig. 12 shows mass-loss rate as a function of pulsation period for the same observational sample as shown in the top panel. One way to asses how sensitive the mass-loss rates are to the pulsation mode is to look at the group of stars indicated by blue symbols in Fig. 12 . These stars are semiregular/irregular variables (SRVs), probably pulsating in the first overtone as indicated by their short pulsation periods. Even though these stars presumably pulsate in first overtone, they follow the same correlation between mass-loss rates and wind velocities as the fundamental mode pulsators (red symbols). This is clearly seen in the top panel of Fig. 12 , which shows the mass-loss rates and wind velocities for stars pulsating in different modes. Assuming that our reasoning about the stars with shorter periods being first overtone pulsators is correct, it seems that differences in the pulsation modes may not notably affect the mass-loss rates in M-type AGB stars. 5.2 Trends at the longest pulsation periods Figure 13 shows the mass-loss rate as a function of pulsation period for all models in the grid with a stellar wind. Also shown are observed mass-loss rates estimated from CO-line emission against pulsation periods for M-type AGB stars from Olofsson et al. (2002) and González Delgado et al. (2003) , as well as a sample of OH/IR stars taken from De Beck et al. (2010) and He Chen (2001) , with mass-loss rates estimated from CO-line and dust emission, respectively. In model data and in observations the mass-loss rates level off at longer pulsation periods, even if the models do not reach quite as high as the observed mass-loss rates. However, it is important to remember that the grid is currently biased against models with very high mass-loss rate, where dust formation and wind acceleration is very abrupt, resulting in non-converging dynamical structures (see Sect. 4.1 ). The exact level where the mass-loss rate flattens in the model values might also depend on the inner boundary conditions of the models, for example the value of f L or the phase-shift of the luminosity variation with respect to the radial variation of the surface layers ( Liljegren et al. 2016 , 2017 ). It might also be