Problem 47 - College Algebra

Question

For the following exercises, refer to Table 11. $$\begin{array}{ccccccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ {f(x)} & {8.7} & {12.3} & {15.4} & {18.5} & {20.7} & {22.5} & {23.3} & {24} & {24.6} & {24.8}\end{array}$$ Use the LOGISTIC regression option to find a logistic growth model of the form \(y=\frac{c}{1+a e^{-b x}}\) that best fits the data in the table.

Step-by-Step Solution

Answer

Find the parameters \(a\), \(b\), and \(c\) using logistic regression to fit the data.

Step 1: Understand Logistic Regression

Logistic regression is used for modeling data where the growth accelerates rapidly at first and then slows down to approach a maximum limit. This type of curve often forms an S-shape, characteristic of logistic growth.

Step 2: Set Up the Logistic Equation

The logistic equation we're aiming to fit is \( y = \frac{c}{1 + a e^{-b x}} \). Here, \( c \) represents the carrying capacity, \( a \) and \( b \) are parameters that influence the shape and position of the curve on the graph. The task is to determine these parameters using the data.

Step 3: Apply Regression Software or Tool

To apply the logistic regression, you'll typically use statistical software or a graphing calculator capable of performing nonlinear regression. Enter the given data points \((x, f(x))\), and use the logistic regression feature to calculate the best fit parameters \(a\), \(b\), and \(c\).

Step 4: Analyze the Regression Output

Once you obtain the output, you'll have specific estimated values for \(a\), \(b\), and \(c\). These are determined by minimizing the error between the observed data and the values predicted by the logistic model. The exact values depend on computational analysis using the input data.

Step 5: Write the Logistic Growth Model

With the parameters found, insert them back into the logistic equation, creating the logistic growth model \( y = \frac{c}{1 + a e^{-b x}} \). This equation now represents the model that best fits the given dataset.

Key Concepts

Logistic Growth Model

The logistic growth model is a key concept when studying how populations grow over time. It describes a scenario where a population grows quickly initially but slows down as it approaches a certain limit, often due to environmental constraints. This creates an "S-shaped" curve, which is also known as a sigmoid curve.
One of the most important features of this model is the way it incorporates the idea of resource limitation. There's a limit to how large a population can get before resources become too scarce.
In the application of the logistic model, the equation takes the form \(y=\frac{c}{1+a e^{-b x}}\), where each component reflects a different aspect of growth:

• \(y\) represents the dependent variable, often population size or density.
• \(c\) stands for the carrying capacity, which is the maximum value \(y\) can approach.
• \(a\) and \(b\) are constants that shape the curve, influencing how steeply and quickly \(y\) changes.

This model is widely used in ecology to predict future population sizes, but also in various fields like economics and epidemiology.

Nonlinear Regression

Nonlinear regression is a statistical technique used for modeling where the data follows a non-linear trend. Unlike linear regression, which assumes a straight-line relationship between variables, nonlinear regression can fit more complex curves. It is ideal for scenarios where the relationship between variables involves exponential, logarithmic, or polynomial functions.
The logistic growth model is an example of nonlinear regression due to its S-shaped curve. To fit a logistic model to data, one must identify the best-fitting parameters (\(a, b, c\)) using nonlinear regression methods. This process typically involves:

• Choosing a mathematical model appropriate for the data.
• Using software tools or calculators that support nonlinear regression.
• Iteratively adjusting parameters to minimize the difference between the predicted and observed data points.

This sophisticated curve fitting allows researchers to accurately model complex real-world phenomena that simple linear models cannot.

Carrying Capacity

Carrying capacity is a fundamental concept in ecology but is also relevant in various scientific fields that involve growth models. It defines the maximum population size that an environment can sustain indefinitely, given the resources available.
In the logistic growth model \(\left( y = \frac{c}{1+a e^{-b x}} \right)\), the parameter \(c\) represents this carrying capacity. Essentially, it signifies the upper bound that the logistic curve approaches as time progresses.
The concept of carrying capacity is vital in planning and management across different domains:

• In wildlife management, understanding the carrying capacity helps gauge how many animals a habitat can support.
• For urban planning, it can inform decisions on infrastructure and resources needed to support human populations.
• In resource management, it aids in sustainable use strategies that prevent resource depletion.

Recognizing this limit allows for better prediction, planning, and application of growth models to real-world situations.