In this thesis, we compare the continuous and discrete logistic models for describing growth with an upper limit. We address the limitations of the standard discrete model, which can exhibit chaotic and unstable behavior for certain parameter values. As an alternative, we introduce an improved version, the Verhulst discrete model, which provides greater stability and has an explicit solution, representing a significant advantage. Through theoretical analysis and practical examples using real-world data on mobile subscriptions, Tokyo's population, and synthetic data on bacterial populations, we demonstrate that the Verhulst and continuous models fit the data more effectively than the standard discrete model. The main contribution of this thesis is demonstrating the practical utility of the Verhulst model and highlighting the benefits of its analytical approach in real-world scenarios.
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