Problem 26 - College Algebra

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: Understand the Logistic Growth Model

The logistic growth model is given by the equation \( y = \frac{c}{1 + a e^{-rx}} \), where \( y \) represents the population size at time \( x \), \( c \) is the maximum carrying capacity, \( a \) and \( r \) are constants related to the growth rate and initial population conditions. Our task is to find the \( y \)-intercept of this equation.

Step 2: Identify the Y-Intercept Condition

The \( y \)-intercept of a function is the value of \( y \) when \( x = 0 \). Therefore, to find the \( y \)-intercept of the logistic growth model, we need to substitute \( x = 0 \) into the equation \( y = \frac{c}{1 + a e^{-rx}} \).

Step 3: Substitute \( x = 0 \) into the Equation

Substituting \( x = 0 \) into the logistic growth equation, we have:\[y = \frac{c}{1 + a e^{-r imes 0}}\]This simplifies to:\[y = \frac{c}{1 + a}\]This represents the \( y \)-intercept.

Step 4: Interpret the Y-Intercept

The \( y \)-intercept \( \frac{c}{1+a} \) represents the initial population size when \( x = 0 \), meaning it is the starting amount of the population before any growth according to the model. This point provides insight into the initial state of the population under the logistic growth model.

Key Concepts

Understanding the Y-Intercept

In the logistic growth model, the y-intercept is a crucial concept. It is the point where the graph of the function crosses the y-axis. This happens when the value of the independent variable, represented as \( x \), is zero. For the logistic equation given by \( y = \frac{c}{1 + a e^{-rx}} \), the y-intercept can be found by substituting \( x = 0 \) into the equation. This simplifies to:\[y = \frac{c}{1 + a}\]The y-intercept essentially gives us the initial population size at the very start of the time measurement, before any growth affects it. Since the logistic model considers limited resources, understanding this starting point is vital to predicting how the population will grow over time.

Population Size in Logistic Growth

Population size, denoted by \( y \) in the logistic model, tells us the number of individuals present in a population at any given time \( x \). Instead of growing indefinitely, this model simulates how populations initially grow rapidly when resources are abundant. As the population increases and resources become scarcer, the growth rate slows down significantly.

• Initial growth is fast when the population size is small.
• The growth tapers off as the population approaches a maximum sustainable size.

This model is more realistic for describing populations that face environmental constraints and is widely used in biology and ecology to predict how populations would behave over time.

Carrying Capacity

Carrying capacity, represented by \( c \) in the logistic growth equation, is the maximum population size that the environment can sustain indefinitely. This value sets the upper limit for how large a population can get given the available resources.

• It reflects the balance between the population size and the resources available.
• When the population size reaches the carrying capacity, the growth effectively stops as birth rates equal death rates.

The carrying capacity is a critical parameter in the logistic model. It influences how quickly populations reach their limits and how stable those populations remain once they reach this threshold. Changes in environmental conditions, such as food supply or space, can alter the carrying capacity over time.

Understanding Growth Rate

The growth rate, indicated by \( r \) in the logistic growth model, measures how quickly the population size changes over time. It combines the birth and death rates within a population to form a net growth rate.

• When \( r \) is high, the population size increases rapidly at the start.
• Conversely, a low \( r \) indicates slower initial growth.

The logistic growth model shows that growth rate is not constant; it changes as the population size changes. At the beginning, populations grow relatively rapidly, but as they near the carrying capacity, the growth rate slows almost to a halt. Understanding this concept is crucial for predicting and managing real-world populations, ensuring they do not exceed environmental limits.