Light-emitting diodes (LEDs) are widely used for data transmission in emerging optical wireless communications (OWC) systems. This paper analyzes the physical processes that limit the bandwidth and cause nonlinearities in the light output of modern, high-efficiency LEDs. The processes of carrier transport, as well as carrier storage, recombination, and leakage in the active region appear to affect the communications performance, but such purely physics-based models are not yet commonly considered in the algorithms to optimize OWC systems. Using a dynamic modeling of these phenomena, we compile a (invertable) signal processing model that describes the signal distortion and a parameter estimation procedure that is feasible in an operational communications link. We combine multiple approaches for steady-state and dynamic characterization to estimate such LED parameters. We verify that, for a high-efficiency blue GaN LED, the models become sufficiently accurate to allow digital compensation. We compare the simulation results using the model against optical measurements of harmonic distortion and against measurements of the LED response to a deep rectangular current modulation. We show how the topology of the model can be simplified, address the self-calibration techniques, and discuss the limits of the presented approach. The model is suitable for the creation of improved nonlinear equalizers to enhance the achievable bit rate in LED-based OWC systems and we believe it is significantly more realistic than LED models commonly used in communications systems.