  We study numerically the vortex dynamics and vortex-lattice formation in a
rotating density-dependent Bose-Einstein condensate (BEC), characterized by the
presence of nonlinear rotation. By varying the strength of nonlinear rotation
in density-dependent BECs, we calculate the critical frequency,
$\Omega_{\text{cr}}$, for vortex nucleation both in adiabatic and sudden
external trap rotations. The nonlinear rotation modifies the extent of
deformation experienced by the BEC due to the trap and shifts the
$\Omega_{\text{cr}}$ values for vortex nucleation. The critical frequencies and
thereby, the transition to vortex-lattices in an adiabatic rotation ramp,
depend on conventional $\textit{s}$-wave scattering lengths through the
strength of nonlinear rotation, $\mathit{C}$, such that
$\Omega_{\text{cr}}(\mathit{C}>0) < \Omega_{\text{cr}}(\mathit{C}=0) <
\Omega_{\text{cr}}(\mathit{C}<0)$. In an analogous manner, the critical
ellipticity ($\epsilon_{\text{cr}}$) for vortex nucleation during an adiabatic
introduction of trap ellipticity ($\epsilon$) depends on the nature of
nonlinear rotation besides trap rotation frequency. The nonlinear rotation
additionally affects the vortex-vortex interactions and the motion of the
vortices through the condensate by altering the strength of Magnus force on
them. The combined result of these nonlinear effects is the formation of the
non-Abrikosov vortex-lattices and ring-vortex arrangements in the
density-dependent BECs.
