Problem 9 - College Algebra

Question

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Find the carrying capacity.

Step-by-Step Solution

Answer

The carrying capacity is 150.

Step 1: Identify the Logistic Growth Equation

The logistic growth model generally takes the form \( f(x) = \frac{L}{1 + Ce^{-kx}} \), where \( L \) is the carrying capacity. This value represents the maximum population size that the environment can support.

Step 2: Determine the Carrying Capacity Parameter

In the given function \( f(x)=\frac{150}{1+8 e^{-2 x}} \), compare it to the standard form \( f(x) = \frac{L}{1 + Ce^{-kx}} \). Here, \( L = 150 \) represents the carrying capacity.

Key Concepts

Carrying Capacity

The concept of carrying capacity is a fundamental idea in learning about the logistic growth model. It represents the maximum population an environment can sustainably support over time without experiencing negative effects like resource depletion. This ceiling is controlled by limiting factors such as food, water, and space, along with environmental conditions.
To find the carrying capacity in a logistic function, simply identify the parameter \( L \) in its equation format \( f(x) = \frac{L}{1 + Ce^{-kx}} \). It stands as a horizontal line that the function approaches but never quite reaches. For example, in the equation \( f(x) = \frac{150}{1 + 8e^{-2x}} \), the carrying capacity is 150. This means the sustainable limit of the population described by this function is 150 individuals. Understanding this value helps ecologists and policymakers to predict and manage resources effectively.

Logistic Function

The logistic function is a type of mathematical model that is often used to describe population growth. It’s shaped like an 'S' or sigmoid curve and captures how populations increase rapidly at first before slowing down as they approach their carrying capacity. This is in contrast to exponential growth, which assumes unlimited resources and infinite growth.
A typical logistic function is expressed as \( f(x) = \frac{L}{1+Ce^{-kx}} \). Here:

• \( L \) is the carrying capacity, indicating the maximum population size.
• \( C \) and \( k \) are constants that determine the steepness and growth rate of the curve.

In our earlier example, \( f(x) = \frac{150}{1+8e^{-2x}} \), the growth starts slow, speeds up, and then tails off as it nears the carrying capacity of 150. This curve provides a realistic model for many populations in constrained environments where resources are limited.

Population Dynamics

Population dynamics is the study of how and why the number of individuals in a population changes over time. This study takes into account births, deaths, immigration, and emigration. The logistic growth model is an essential tool in understanding these dynamics, especially in scenarios where resources are limited.
The logistic model acknowledges that population growth is not infinite. Initially, a population may grow rapidly due to plenty of resources. However, as resources become scarce, growth slows, signaling a move toward the carrying capacity. This balance between birth and death rates at high population sizes is a key element of population dynamics.
Using the function \( f(x) = \frac{150}{1+8e^{-2x}} \) as a blueprint, one can predict how a population may change over a given period and how it might respond to alterations in environmental conditions. A core understanding of population dynamics can guide in sustainable resource management, conservation efforts, and predicting the impact of external factors like climate change on population sizes.