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Future launches are projected to significantly increase both the number of active satellites and aggregate collision risk in Low Earth Orbit (LEO). In this paper, a dynamical systems theory approach is used to analyze the effect of launch rate distribution on the stability of the LEO environment. A multi-shell, three-species source-sink model of the LEO environment, referred to as MOCAT-3 for MIT Orbital Capacity Assessment Tool 3 Species, is used to study the evolution of the species populations. The three species included in the model are active satellites, derelict satellites, and debris. The model's coefficients represent atmospheric drag, collision rate, mean satellite lifetime, post-mission disposal probability, and active debris removal rate. Solutions of the system of differential equations are computed, and an analysis of the stability of the equilibrium points is conducted for numerous launch rate distributions. The stability of the equilibrium points is used to test the sensitivity of the environment to run-away debris growth, known as Kessler syndrome, that occurs at the instability threshold. An analysis of the environment's response to perturbations in launch rate and debris population is conducted. The maximum perturbation in the debris population from the equilibrium state, for which the system remains in a stable configuration, is calculated. Plots of the phase space about the equilibrium points are generated. The results will help to better understand the orbital capacity of LEO and the stability of the space environment, as well as provide improved guidelines on future launch plans to avoid detrimental congestion of LEO.