Population dynamics are used to give a mathematical explanation to changes in biological populations where they grow, compete, are limited by resources and affected by the environment. In this study, a framework for population analysis of temporal behavior is developed in the form of nonlinear governing equations of the form of differential equation. System variables that depend on population, assumptions used in the models, and equilibrium states and stability conditions are systematically formulated to describe long-term system behavior. Various population scenarios are represented by the model parameters and initial conditions. Numerical solution algorithm is used to approximate trajectories of populations for which analytical solutions are hard to come by. The suggested system measures changes in the size of the population, rate of growth, convergence to equilibrium and stability under varying parameter conditions. The examples show how differential equations can be used to model complex interactions in population dynamics and forecast changes in populations over time.