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We analyze the complex turn-on behavior of semiconductor quantum-dot (QD) lasers in terms of a nonlinear rate equation model for the electron and hole densities in the QDs and the wetting layer, and the photons. A basic ingredient of the model is the nonlinearity of the microscopic carrier-carrier scattering rates. With the framework of detailed balance, we analytically relate the microscopic in- and out-scattering rates. We gain insight into the anomalous nonlinear dynamics of QD lasers by a detailed analysis of various sections of the 5-D phase space, accounting for density-dependent carrier scattering times. We show that the strongly damped relaxation oscillations are characterized by a desynchronization of electron and hole dynamics in the dots. Analytic approximations for the steady-state characteristics are also derived.