Longitudinal mode competition in (Al,In)GaN laser diodes at λ = 445nm and 515 nm with mode competition frequencies from 10 MHz to 150 MHz is observed. Up to two dozen lasing modes oscillate with the lasing mode rolling from the short wavelength edge to the long wavelength edge of the gain profile. The experimental results can be described very well with a set of multi-mode rate equations including self-, symmetric and asymmetric cross gain saturation. By tuning essential parameters of the gain saturation terms, mode competition disappears and single mode operation as well as mode clustering is found. This proves that the mechanisms of gain saturation have not only a profound impact on the complex temporal-spectral behavior but also explains mode clustering in (Al,In)GaN laser diodes, both in pulsed and continuous wave (cw) operation as a natural nonlinear effect without the necessity to add noise.