Problem 378 - Precalculus

Question

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\) Find and interpret \(f(0) .\) Round to the nearest tenth.

Step-by-Step Solution

Answer

\(f(0) \approx 16.7\); this is the initial population size.

Step 1: Understand the Logistic Growth Model

The logistic growth model is given by the function \(f(x)=\frac{L}{1+be^{-kx}}\), where \(L\) is the carrying capacity, \(b\) and \(k\) are constants, and \(x\) is time. In the function provided, \(L=150\), \(b=8\), and \(k=2\).

Step 2: Substitute for x in the Function

To find \(f(0)\), substitute \(x=0\) into the function. This gives us:\[f(0)=\frac{150}{1+8e^{-2(0)}}\]

Step 3: Simplify the Expression inside the Exponent

Calculate the exponent: \(-2 \times 0 = 0\). So the function becomes:\[f(0)=\frac{150}{1+8e^0}\]

Step 4: Evaluate the Exponential Term

Since \(e^0 = 1\), substitute \(1\) into the equation:\[f(0)=\frac{150}{1+8 \times 1} = \frac{150}{1+8} = \frac{150}{9}\]

Step 5: Compute the Final Value

Now, divide \(150\) by \(9\):\[f(0) \approx 16.7\]

Step 6: Interpret f(0)

The value \(f(0) \approx 16.7\) represents the initial population size or the population size at time \(x=0\) according to the logistic growth model.

Key Concepts

Carrying Capacity

The carrying capacity of a population refers to the maximum number of individuals that an environment can sustainably support. This concept is crucial in the logistic growth model, where carrying capacity is denoted by the parameter \(L\). In our example, \(L=150\), meaning that the environment can support up to 150 individuals. This limit can be due to resources like food, space, or other environmental factors. Carrying capacity is a dynamic value that can change with conditions; for example, if resources increase, so can the carrying capacity. Understanding this concept helps in predicting how populations grow and what limits that growth.

Exponential Function

An exponential function describes a process where a quantity grows or decays at a constant rate relative to its current value. In mathematics, this is often represented as \(e^{kx}\), where \(e\) is Euler's number, approximately 2.718, and \(k\) is the rate constant. In the logistic growth model, the term \(be^{-kx}\) depicts how the population changes over time. Initially, if there are fewer individuals, the growth rate can seem exponential, indicating rapid growth. However, as time progresses and the population nears carrying capacity, this growth slows down due to the logistic model's sigmoid shape, unlike pure exponential growth which would continue indefinitely.

Population Dynamics

Population dynamics explore how populations change over time, encompassing factors that affect growth, such as birth rates, death rates, immigration, and emigration. In our logistic growth model, population dynamics are represented within the formula \(f(x)=\frac{150}{1+8e^{-2x}}\). Here, \(x\) represents time, and the equation models how the population evolves. Initially, the population grows rapidly because of abundant resources and low competition. Over time, as the population size increases, the growth rate decreases, approaching the carrying capacity \(L=150\). This reflects how real-world populations are not able to grow indefinitely due to limiting factors within their environment.