• 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

Logistic Growth

We have seen that we often use exponential relationships to describe population growth. But populations do not grow without some limitation on their numbers provided by real constraints in the real world. Predators, disease, and limitations on resources all serve to limit the numbers of any living population.

You used the example of fish in a lake to explored the recursive and explicit forms of the logistic equation in the previous section on Logarithms and Logistic Growth and derived the following logistic growth model with carrying capacity [latex]K[/latex] and growth rate [latex]r[/latex]:

[latex]{{P}_{n}}={{P}_{n-1}}+r\left(1-\frac{{{P}_{n-1}}}{K}\right){{P}_{n-1}}[/latex].

While this formula will allow you to predict future population recursively for populations that grow discretely with one breeding time per year, it doesn’t present a closed form that you can use to make predictions about populations like fish or people that breed year round. For those situations, we can use a continuous logistic model in the form

[latex]P_{t}=\dfrac{c}{1+\left(\dfrac{c}{P_{0}}-1\right)e^{-rt}}[/latex]

where [latex]t[/latex] stands for time in years, [latex]c[/latex] is the carrying capacity (the maximal population), [latex]P_0[/latex] represents the starting quantity, and [latex]r[/latex] is the rate of growth.

When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without having to convert them to decimals, though in the example below, once it becomes clear that a quantity will reduce to a terminating decimal, it is done so for simplicity’s sake.

•  [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
•  [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex]
•  [latex]\dfrac{a}{b}=\dfrac{a\cdot c}{b \cdot c}[/latex]
•  [latex]\dfrac{a}{b}\pm \dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]
• If [latex]\dfrac{a}{b}=\dfrac{c}{d}[/latex], then [latex]ad = bc[/latex] and [latex]a=\dfrac{bc}{d}[/latex] and [latex]\dfrac{ad}{b}=c[/latex].