The Taylor series approximation is often used to convert non-linear dynamical systems to linear systems, while the Hartman-Großman theorem analyzes the local qualitative behavior of the non-linear system around a hyperbolic equilibrium point. The global stability of an equilibrium point in the Lyapunov sense is based on the principle that if the equilibrium point is disturbed and the flow of the system is dissipative, then the system must be stable. This article applies these methods to an ecological Aedes aegypti model, whose local and global stability are characterized by a population growth threshold. In conclusion, the classical theory of dynamical systems, validated computationally, yields theoretical results in favor of controlling the local population of Aedes aegypti. It becomes usable if the proposed model is reinforced with the estimation of the parameters that describe the relationships between stages (aquatic and aerial) of the mosquito population and the inclusion of vector control strategies to protect people from the viruses transmitted by Aedes aegypti.