Problem 9 - College Algebra

Question

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: Introduction to the Logistic Growth Model

The logistic growth model is a common function used to model populations that stabilize over time. It is generally given in the form \(f(x) = \frac{L}{1+be^{-kx}}\), where \(L\) is the carrying capacity of the population, \(b\) is a constant related to the initial amount, and \(k\) is the growth rate.

Step 2: Identify the Carrying Capacity in the Model

In the logistic growth model \(f(x) = \frac{L}{1+be^{-kx}}\), the carrying capacity \(L\) is the value that the population approaches as time goes to infinity. From the given function \(f(x) = \frac{150}{1+8 e^{-2 x}}\), we can see that \(L = 150\). This is the horizontal asymptote of the function, representing the maximum population size.

Key Concepts

Carrying Capacity in the Logistic Model

In the realm of population dynamics, the concept of carrying capacity plays a vital role. It refers to the maximum population size that an environment can sustain indefinitely, given the available resources such as food, habitat, water, and other necessities. In mathematical models, particularly the logistic growth model, the carrying capacity is denoted by the parameter \(L\). This parameter represents the limit that a population is expected to reach as it grows over time. Understanding this is crucial because:

• The carrying capacity gives an estimate of how large a population can get before resources become too scarce to support any further growth.
• Changes in environmental conditions can affect \(L\), meaning it is not always a constant value across different scenarios.
• In our given logistic growth model \(f(x)=\frac{150}{1+8e^{-2x}}\), the carrying capacity is \(150\), indicating that the population will not exceed this number over time.

Recognizing \(L\) helps us understand the finite limits of resources and the importance of sustainable population management.

Understanding Horizontal Asymptotes

Horizontal asymptotes in the context of mathematical functions represent values that a function approaches as the input, often denoted as \(x\) (representing time in growth models), heads towards infinity. In the logistic growth function, the horizontal asymptote is equivalent to the carrying capacity \(L\). To break it down further:

• A horizontal asymptote is visualized like a line that the curve of the function gets closer and closer to, but may never actually reach.
• In the function \(f(x) = \frac{150}{1+8e^{-2x}}\), the horizontal asymptote is \(y=150\), highlighting the carrying capacity where the population stabilizes.
• This concept helps in understanding how populations stabilize as they approach their carrying capacity over time.

The idea of horizontal asymptotes emphasizes that populations tend to self-regulate when nearing environmental limits.

Population Stabilization in Logistic Growth

Population stabilization is a phenomenon where population size tends to remain constant over time, following a period of growth or fluctuation. In logistic growth models, this stabilization is achieved as populations near their carrying capacity.Here’s how this concept unfolds:

• Initially, populations grow rapidly when resources are abundant, depicted by the steep rise in the curve of the logistic function.
• As populations begin to approach the carrying capacity, resources become scarcer, leading to a slowdown in growth rate.
• The equation \(f(x) = \frac{150}{1+8e^{-2x}}\) shows that as \(x\) (time) increases, the growth slows and levels out close to \(y=150\).

Ultimately, population stabilization reflects the balance achieved when the birth rate equals the death rate, indicating dynamic equilibrium. This concept is vital for understanding long-term trends and planning resources sustainably.