Question

Population Growth The population of the United States was 76 million people in 1900 and reached 300 million people in 2006. From 1900 through 2006 , assume the population of the United States can be modeled by logistic growth with a limit of \(839.1\) million people. (Source: U.S. Census Bureau) (a) Write a differential equation of the form \(\frac{d y}{d t}=k y\left(1-\frac{y}{L}\right)\) where \(y\) represents the population of the United States (in millions of people) and \(t\) represents the number of years since 1900 . (b) Find the logistic growth model \(y=\frac{L}{1+b e^{-k t}}\) for this population. (c) Use a graphing utility to graph the model from part (b). Then estimate the year in which the population of the United States will reach 400 million people.

Short answer

The differential equation for logistic growth of population in the US is \(\frac{dy}{dt} = ky(1 - \frac{y}{839.1})\). The logistic growth model for this population is \(y=\frac{839.1}{1+b e^{-k t}}\) with the values of \(b\) and \(k\) based on the historical data. Using the model, we can estimate the year in which the population will reach 400 million.

Step-by-step solution

Write the Differential Equation

The differential equation for logistic growth is given by \(\frac{dy}{dt} = ky(1 - \frac{y}{L})\). Here, \(y\) is the population in millions, \(t\) is the time in years since 1900, \(k\) is the relative growth rate, and \(L\) is the carrying capacity (max population). Since \(L\) is given as 839.1 million, the differential equation for the population growth becomes \(\frac{dy}{dt} = ky(1 - \frac{y}{839.1})\).

Finding Logistic Growth Model

The logistic growth model is given by \(y=\frac{L}{1+be^{-kt}}\). The values of \(k\) and \(b\) can be computed using the initial conditions. We know that in 1900 (t=0), the population was 76 million, and in 2006 (t=106), the population was 300 million. These can be used to form two equations to solve for \(b\) and \(k\). In 1900, we get 76 = \(\frac{839.1}{1+b}\) and we also have 300 = \(\frac{839.1}{1+b \cdot e^{-106k}}\) in 2006. We can solve these equations to find the values of \(b\) and \(k\).

Espresso Extraction Calculations

After getting the value of \(k\) and \(b\), we can substitute them into our logistic growth model equation. So, the logistic growth model for this population becomes \(y=\frac{839.1}{1+b e^{-k t}}\) with \(b\) and \(k\) substituted in.

Estimating Population at Given Time

In order to estimate when the population will reach 400 million, we set \(y=400\) in our logistic growth model equation and solve for \(t\). This will give us the estimated year when the population will reach 400 million.

Key concepts

Differential Equations

In mathematics, differential equations serve as a powerful tool to describe various phenomena. These equations involve functions and their derivatives, expressing how variables change with respect to each other. Logistic growth is a classic example that uses differential equations to model how populations grow over time.

For logistic growth, the equation is given by: \[\frac{dy}{dt} = ky\left(1 - \frac{y}{L}\right)\]Here:

• \(y\) represents the population at time \(t\).
• \(k\) denotes the relative growth rate of the population.
• \(L\) is the carrying capacity, which is the maximum population size that the environment can sustain.

This differential equation reflects that the population growth rate is proportional to both the existing population (\(y\)) and a factor that accounts for how close the population is to its carrying capacity (\(L\)). As the population nears \(L\), the growth rate slows down, preventing an indefinite increase.

Population Modeling

Population modeling is the study of mathematical models that describe how populations evolve over time. In this context, logistic growth models are particularly insightful. They provide a realistic depiction of population dynamics when resources are limited. An important logistic growth model is:\[y=\frac{L}{1+be^{-kt}}\]In this equation:

• \(L\) remains the carrying capacity.
• \(b\) and \(k\) are constants determined by initial population conditions.
• \(t\) denotes the time since the start of observation.

To find \(b\) and \(k\), we use known initial population data points, such as population at \(t=0\) and another time point. These constants shape how fast the population grows. Once determined, the model can be used to project future population sizes and examine potential impacts of different growth scenarios. Utilizing this model helps comprehend real-world changes and assists in planning and resource management.

Carrying Capacity

Carrying capacity is a fundamental concept in ecology and population studies. It refers to the maximum population size that a particular environment can support indefinitely. It considers factors like food availability, habitat space, water supply, and other resources.

In the logistic growth model, carrying capacity is signified as \(L\). It forms a crucial limitation to unlimited growth, representing a balancing point where the population stabilizes.

• Importance in Models: A logistic growth model predicts that as a population approaches its carrying capacity, growth decelerates.
• Real-world Implications: Understanding carrying capacity aids in resource management and sustainability efforts.
• Dynamic Nature: Carrying capacity is not static. Changes in environment or resources can alter \(L\).

When constructing population models, identifying the carrying capacity provides critical insights into potential growth limits and guides conservation and policy-making decisions.