We introduce a nonlinear, nonlocal model to describe the range expansion of a population driven by growth and competition for space, with primary motivation drawn from the mathematical modeling of immotile cell colonies. We first explain why classical approaches, such as reaction-diffusion equations, fail to capture several key features of such phenomena. Through a singular limit analysis, we derive a free boundary problem that describes population range expansion as a result of growth, saturation, and dispersion, and we establish its principal mathematical properties. In particular, we characterize the evolution of a free boundary delimiting the saturated region, identify traveling wave solutions, and determine the asymptotic spreading speed of compactly supported solutions.