Question

U.S. population projections According to the U.S. Census Bureau, the nation's population (to the nearest million) was 296 million in 2005 and 321 million in \(2015 .\) The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach: a. Assume \(t=0\) corresponds to 2005 and that the population growth is exponential for the first ten years; that is, between 2005 and \(2015,\) the population is given by \(P(t)=P(0) e^{\pi}\) Estimate the growth rate \(r\) using this assumption. b. Write the solution of the logistic equation with the value of \(r\) found in part (a). Use the projected value \(P(45)=398\) million to find a value of the carrying capacity \(K\) c. According to the logistic model determined in parts (a) and (b), when will the U.S. population reach \(95 \%\) of its carrying capacity? d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 410 million rather than 398 million. What is the value of the carrying capacity in this case? e. Repeat part (d) assuming the projected population for 2050 is 380 million rather than 398 million. What is the value of the carrying capacity in this case? E. Comment on the sensitivity of the carrying capacity to the 35-year population projection.

Short answer

Question: Analyze the sensitivity of the carrying capacity to the 35-year population projection based on the values of K found in parts b, d, and e. Answer: To analyze the sensitivity of the carrying capacity to the 35-year population projection, we can compare the K values obtained in parts (b), (d), and (e). By observing any significant differences or similarities in the K values, we can determine how sensitive the carrying capacity is to changes in the projected population for 2050. If the K values are significantly different, it would indicate that the carrying capacity is sensitive to changes in the projected population. On the other hand, if the K values are similar, it would indicate that the carrying capacity is not highly sensitive to changes in the projected population.

Step-by-step solution

a. Estimate the growth rate r

We are given that the population in 2005 (\(t=0\)) is \(P(0)=296\) million and the population in 2015 (\(t=10\)) is \(P(10)=321\) million. The exponential growth model is given by \(P(t)=P(0)e^{rt}\). Using the given information, we can find the growth rate \(r\): \(321 = 296e^{10r}\) Divide both sides by 296: \(321/296 = e^{10r}\) Take the natural logarithm of both sides: \(ln(321/296) = 10r\) Solve for r: \(r = \frac{ln(321/296)}{10}\)

b. Write the logistic equation and find carrying capacity K

The logistic equation is given by \(P(t) = \frac{K}{1+\frac{K-P(0)}{P(0)}e^{-rt}}\). We know the value of \(r\) from part (a) and the 2050 population is projected to be 398 million, which corresponds to \(t=45\) (since 2050 is 45 years from 2005). So, we have the equation: \(398 = \frac{K}{1+\frac{K-296}{296}e^{-45r}}\) Now, we have one equation with one unknown variable K, which we can solve for.

c. Calculate when the U.S. population will reach 95% of its carrying capacity

We want to find the time \(t\) when \(P(t) = 0.95K\). Using the logistic equation found in part (b), we can set up the following equation: \(0.95K = \frac{K}{1+\frac{K-296}{296}e^{-rt}}\) We have the values for \(K\) and \(r\), so we can solve for \(t\).

d. Find carrying capacity when projected population is 410 million

Similar to part (b), we will use the logistic equation with the new projected population for 2050 as 410 million: \(410 = \frac{K}{1+\frac{K-296}{296}e^{-45r}}\) Now, we have one equation with one unknown variable K, which we can solve for.

e. Find carrying capacity when projected population is 380 million

Similar to part (b), we will use the logistic equation with the new projected population for 2050 as 380 million: \(380 = \frac{K}{1+\frac{K-296}{296}e^{-45r}}\) Now, we have one equation with one unknown variable K, which we can solve for.

f. Comment on the sensitivity of the carrying capacity to the 35-year population projection

Compare the values of the carrying capacity K found in parts (b), (d), and (e) to discuss and analyze how the carrying capacity is affected by the projection of the population 35 years into the future.

Key concepts

Exponential Population Growth

Exponential population growth is crucial to understanding the early stages of a population's expansion when resources are abundant. In such a context, the growth rate of the population is proportional to the current size of the population. Visualize a scenario where each individual has more than enough resources to support offspring, leading to each generation being larger than the one before.

The mathematical model used to describe this type of growth is represented as:\[ P(t) = P(0)e^{rt} \]where \( P(t) \) is the population at time \( t \), \( P(0) \) is the initial population size, \( r \) is the intrinsic growth rate, and \( e \) is the base of the natural logarithm. When conditions are ideal, this model assumes that the population will grow indefinitely, doubling in size at a consistent rate.

However, this growth is not sustainable over the long term since it does not take into account environmental constraints such as limited food, space, or other resources. Thus, while exponential growth is a great starting point for understanding population dynamics, it is often a temporary phase in the life of a population.

Carrying Capacity

Carrying capacity, symbolized as \( K \), is a key concept in the logistic population model that represents the maximum number of individuals that a given environment can sustain indefinitely without being degraded. Once a population reaches its carrying capacity, the growth rate declines and the population size stabilizes.

Through the lens of the logistic model, carrying capacity is integrated into the equation as follows:\[ P(t) = \frac{K}{1+\left(\frac{K-P(0)}{P(0)}\right)e^{-rt}} \]This formula reflects a more realistic scenario where population growth slows as it approaches its carrying capacity, forming an S-shaped curve. Unlike the exponential model, the logistic model considers that for any given environment, there is a limit to the number of individuals that can be supported.

In practical applications like projecting the U.S. population, estimations of carrying capacity are sensitive to a variety of factors, such as technological advances, changes in policies, and unpredictable events that can affect resources and population growth. This inherent uncertainty means projections must be made with care and remain subject to revision as conditions change.

Population Projections

Population projections are estimates of future population size based on current demographic trends and assumptions about future changes in birth rates, death rates, and migration. These projections are vital for planning in areas such as healthcare, education, infrastructure, and environmental management.

Population projection models, including the logistic model, often require setting initial parameters like the growth rate and carrying capacity based on historical data and future expectations. Sensitivity analysis, like those seen in the provided exercise, can be used to assess the impact of changes in projected values on the final population estimates. It reveals that small variations in assumptions about future conditions can lead to significantly different estimates of the population and carrying capacity.

For instance, varying the 2050 population projection from 398 million to 410 million or 380 million leads to distinct carrying capacities, highlighting the need for careful analysis and consideration of uncertainties. Making informed projections means recognizing that they are educated guesses rather than certainties, requiring continuous monitoring and updating as new data emerges.