The first set of coalescent simulations was run under FIM and SST only (Table 2 and S2 to check if simulated data could reproduce the pattern of genetic variability (both q estimators and TD) observed for C. melanopterusC.limbatus, and C. amblyrynchos. The simulated q values (excluding the equilibrium model) ranged between 0.001 and 0.003 per site, in line with the observed values (Table 1 and 2). TD follows a U-shaped distribution for each Nm value as a function of TCOL, being more positive at recent TCOL and at equilibrium and less positive (or negative for higher Nm) at intermediate values. Therefore, species demography with Nm ~10 (and higher) and TCOL within 15k and 50k generations B.P. will have negative TD values. In contrast, species with lower Nm and very recent or very ancient TCOL will have positive TD. This matches strikingly the TD observed for the three shark species and their estimated demographic parameters under SST (Table 1). We plot the normalized SFS and the stairwayplot for all scenarios presented in Table 2 (Figures 4, 5, S1, S2 and S3). First, we note that none of our scenarios, even those at equilibrium and with no variation in Nm through time, showed a normalized SFS compatible with a constant size population (Figures 4, 5, S1, S2 and S3). The normalized SFS and the reconstructed stairwayplot depend generally on the interaction between Nm and TCOL with a dynamic strikingly similar to TD (which is indeed a summary of the SFS). For Nm=1 we observed the signature of a recent decrease in Ne for all scenarios and independently of TCOL (Figure 4). The normalized SFS showed consistently a strong deficit of low frequency variants, typical of a demographic bottleneck and in agreement with the positive TD (Figure 4 and Table 1). Furthermore, the stairwayplot could never detect the ancestral expansion for any TCOL. For growing Nm, the interplay with TCOL becomes more complex. A general result is that, once again, all scenarios were characterized by a recent decrease of Ne when looking at the stairwayplot and a deficit of singletons compared to the other low frequency classes when looking at the normalized SFS (Figures 5, S1 and S2). However, a strong signature of ancestral expansion appeared for Nm >10 and TCOL between 15k and 50k generations B.P., mirroring the results of TD for which most of these scenarios displayed a negative value. Remarkably, the stairwayplot retrieved the ancestral expansion only slightly overestimating the simulated TCOL (Figures 5, S1 and S2).  Similar results were obtained for FIM (Figures S4 and S5).