In this thesis, we consider population behavior models over time. First, we examine the behavior of a single population through the Malthusian and logistic growth models. Next, we consider the interactions of two populations using the predator-prey model. We conduct a qualitative analysis of the system over time without explicitly solving the equations. Throughout the work, we study the equilibria of linear and nonlinear systems, along with their corresponding phase portraits. The analysis of phase portraits of linear systems is based on the eigenvalues of the matrix of the system. We use the Hartman-Grobman theorem, which connects the trajectories of nonlinear systems and their linearizations in the case of hyperbolic equilibrium. We explain that for the predator-prey model, two equilibria exist: one is a saddle, and the other is a center. For the second equilibrium \(\left(\frac{c}{d},\frac{a}{b}\right) \) of the predator-prey model, the Hartman-Grobman theorem cannot be applied. We show that around second equlibrium solution trajectories are closed, which explains the periodic behavior of the predator and prey populations over time.