Previous comparisons of experimental data with non-linear numerical simulations of density stratified Taylor-Couette flows revealed non-linear interactions of strato-rotational instability (SRI) modes that lead to periodic changes in the SRI spirals and their axial propagation. These pattern changes are associated to low-frequency velocity modulations that are related to the dynamics of two competing spiral wave modes propagating in opposite directions. In the present paper, a parameter study of the strato-rotational instability (SRI) is performed using Direct Numerical Simulations to evaluate the influence of the Reynolds numbers, the stratification, and of the container geometry on these SRI low-frequency modulations and spiral pattern changes. The results of this parameter study show that the modulations can be considered as a secondary instability that are not observed for all SRI unstable regimes. The findings are of interest when the Taylor-Couette odel is related to star formation processes in accretion disks.