Understanding the Discrete Logistic Map: A Key Model for Population Dynamics 🌱

We explore the discrete logistic map, a type of recursively defined sequence used as a discrete population model for a population with seasonal growth periods. The sequence is constructed using a seed value 𝑥0 and a parameter 𝑎, both constrained within specific ranges, to form the equation 𝑥(𝑛+1)=𝑎⋅𝑥(𝑛)⋅(1−𝑥(𝑛)). We note that this is also called a quadratic map because the recursion formula is a quadratic expression in 𝑥(𝑛).

We analyze the behavior of this sequence under various conditions, observing phenomena like convergence, periodic orbits, and chaos. In particular, we look at how changing a parameter can lead to different long-term population trends, and illustrate the idea of mathematical "chaos." Additionally, we interpret this model in the context of population dynamics, visualizing the sequence's behavior through web diagrams and population graphs.

(Mathematical Modeling Lecture 2.7)

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