Problem 398 - Precalculus

Question

What is the \(y\) -intercept of the logistic growth model \(y=\frac{c}{1+a e^{-r x}} ?\) Show the steps for calculation. What does this point tell us about the population?

Step-by-Step Solution

Answer

The y-intercept is \( \frac{c}{1 + a} \). It represents the initial population size.

Step 1: Identify the Components of the Logistic Growth Model

The logistic growth model is given by the equation \( y = \frac{c}{1 + a e^{-rx}} \). The parameters are: \( c \) which represents the carrying capacity, \( a \) works as a scaling factor, \( r \) is the growth rate, and \( x \) is the independent variable often representing time.

Step 2: Understanding the Y-intercept

The \( y \)-intercept of a function is the value of \( y \) when \( x = 0 \). Thus, we need to substitute \( x = 0 \) into the logistic model to find the \( y \)-intercept.

Step 3: Calculate the Y-intercept

Substitute \( x = 0 \) into the equation: \( y = \frac{c}{1 + a e^{-r(0)}} = \frac{c}{1 + a \times 1} = \frac{c}{1 + a} \). The \( y \)-intercept is thus \( \frac{c}{1 + a} \).

Step 4: Interpreting the Y-intercept

The \( y \)-intercept \( \frac{c}{1 + a} \) represents the initial population size at time \( x = 0 \) in relation to how close it is to the carrying capacity \( c \). This helps us understand the starting position of the population before any logistic growth is considered.

Key Concepts

Y-intercept

In mathematics, the y-intercept of a function is the point where the graph of the equation hits the y-axis. For the logistic growth model, represented by the equation \( y = \frac{c}{1 + a e^{-rx}} \), finding the y-intercept gives us insight into the initial conditions of a population.

To calculate the y-intercept, you set \( x = 0 \). By substituting this into the model, you get:

• \( y = \frac{c}{1 + a e^{-r \cdot 0}} \)
• \( y = \frac{c}{1 + a \cdot 1} \)
• Thus, \( y = \frac{c}{1 + a} \)

This point tells us the initial population size relative to the carrying capacity and scaling factor, giving us a baseline for further analysis.

Carrying Capacity

The carrying capacity in the logistic growth model, denoted by \( c \), is essentially the maximum population size that the environment can sustainably support. It reflects the balance between the availability of resources and the needs of the population.

Understanding carrying capacity is vital because:

• It limits growth, ensuring that the population does not exceed what the environment can handle.
• As the population nears carrying capacity, the growth rate slows down and eventually stops.

In other words, if the resources are abundant, \( c \) would be large, depicting a higher chance of population growth. However, if resources are scarce, \( c \) will be small.

Growth Rate

The growth rate, represented by \( r \), indicates how quickly a population can grow over time under ideal conditions. It is a crucial factor in the logistic growth model, as it influences the speed at which the population approaches carrying capacity.

The growth rate under logistic conditions suggests:

• Higher values of \( r \) indicate a faster approach to the carrying capacity.
• Lower values of \( r \) mean the population increases gradually.

This parameter helps us predict population trends and understand how various factors can affect the speed of growth over different periods.

Initial Population Size

The initial population size in the context of the logistic growth model is signified by the y-intercept. When \( x = 0 \), the value of \( y = \frac{c}{1 + a} \) reveals the starting number of individuals in the population.

This initial value is significant because:

• It sets the starting point for any growth to occur within the conditions defined by the model.
• It helps understand the position of the population compared to its carrying capacity.

The relationship between the initial population and the carrying capacity influences how quickly the population can grow or how it stabilizes over time.