- Create a logistic growth model to describe a real-world situation that follows a logistic pattern
- Evaluate a logistic growth model to determine if it accurately represents a situation
- Use a logistic growth model to make predictions or solve problems related to the growth of a population or system
The Main Idea
Logistic growth models are essential for understanding scenarios where growth is not just exponential but is limited by external factors, like resources or space. These models are characterized by an initial period of exponential growth, followed by a slowdown as the population approaches a maximum limit, known as the carrying capacity.
Key Concepts:
- Carrying Capacity ([latex]K[/latex]): The maximum population size that an environment can sustain indefinitely.
- Logistic Growth Equation: The logistic growth model can be represented by the equation [latex]{{P}_{n}}={{P}_{n-1}}+r\left(1-\frac{{{P}_{n-1}}}{K}\right){{P}_{n-1}}[/latex], where [latex]P_n[/latex] is the population at time [latex]n[/latex], [latex]r[/latex] is the growth rate, and [latex]K[/latex] is the carrying capacity.