rate the asymmetry in the difference between $\lambda$ and the corresponding fastest growing wavelength is increasing. In general, the smaller perturbation wavelengths in the disc centre can grow faster than the larger ones in the outer regime.\
The dispersion relation $\omega^2$ (Section \[subsec:Exact fastest growing perturbation wavelength\]) can be divided into a destabilizing term (negative), which is dominating the unstable regime $R_{\mathrm{u}}$, and stabilizing terms (positive), taking over in the stable region, see Fig. \[Results:fig:dispersion\_relation\_1\]. The stabilizing contribution in the disc centre is due to pressure (expressed by $c_{\mathrm{s}}$) and differential rotation (expressed by $\kappa^2$), while going outwards, the epicyclic frequency is more important than pressure.

![Minimum of the dispersion relation $\omega^2_{\mathrm{sech^2}} = \omega^2 (\lambda_{f_{\mathrm{sech^2}}})$ (equation \[eq:modified\_dispersion\_relation\_generalized\]) in absolute values (black lines) and its decomposition (coloured lines) at every radius. The negative gravitational term (green) of equation (\[eq:modified\_dispersion\_relation\_generalized\]) dominates the unstable regime (black dashed line), while the positive contributions become more important in the stable regime (black solid line). The stabilising terms (orange) consist of the epicyclic frequency term (red), with $\kappa^2$ and the pressure term (cyan), with $c_{\mathrm{s}}^2 \left( \frac{2 \upi}{\lambda_{f_{\mathrm{sech^2}}}} \right)^2$. The effects of the epicyclic frequency and the pressure are comparable in the inner part of the 