Light is a key environmental factor for the growth of micro-algae, and optimizing the capture of light is critical for high efficiency production systems. As the density of the population of micro-algae increases, the availability of light decreases, leading to a reduction in the growth rate because of mutual shading, while other effects, such as photo-inhibition, might be especially frequent when the population density is low. Several models in the literature have been developed to take into account light phenomena and predict micro-algal growth, particularly in a mono-culture. With the help of a simple expression for the attenuation of the light, we propose and justify a new growth function that incorporates both photo-inhibition and photo-limitation. In agreement with the experimental data, this new formulation describes the micro-algal response to a wide range of situations of excessive or insufficient light intensities through an explicit dependence on both the incident light and the biomass concentration. While simple, the proposed expression can be satisfactorily applied to practical cases under nutrient replete conditions in photo-bioreactors with different sizes and geometries. It extends naturally to the growth of different species, providing a dynamic model which can simulate experiments in a mono-culture as well as in poly-cultures. The investigation of the competition for light-limited growth shows that the model predicts competitive exclusion, which has also been experimentally demonstrated. This leads to new perspectives for the control and optimization of mixed micro-algal cultures.