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:


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?

Verified step by step guidance

1. Understand that the logistic growth model is given by the differential equation: \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)\] where \(P\) is the population at time \(t\), \(r\) is the intrinsic growth rate, and \(K\) is the carrying capacity.
2. Identify the known data points: population in 2005 (\(P_0 = 296\) million), population in 2015 (\(P_{10} = 321\) million), and the projected population in 2050 (\(P_{45} = 410\) million). Here, \(t=0\) corresponds to the year 2005.
3. Use the logistic growth solution formula: \[P(t) = \frac{K}{1 + Ae^{-rt}}\] where \(A\) is a constant determined by initial conditions. At \(t=0\), \[P(0) = \frac{K}{1 + A} = 296\] which allows you to express \(A\) in terms of \(K\) and known initial population.
4. Set up two equations using the populations at \(t=10\) and \(t=45\): \[321 = \frac{K}{1 + Ae^{-r \cdot 10}}\] and \[410 = \frac{K}{1 + Ae^{-r \cdot 45}}.\] Substitute \(A\) from the initial condition into these equations to create a system with two unknowns, \(r\) and \(K\).
5. Solve the system of equations simultaneously to find the values of \(r\) and \(K\). The value of \(K\) you find will be the carrying capacity corresponding to the projected 2050 population of 410 million.

Key Concepts

Logistic Growth Model

The logistic growth model describes population growth that starts exponentially but slows as the population approaches a maximum limit called the carrying capacity. It is represented by a differential equation where growth rate decreases as population nears this limit, modeling realistic constraints like resources.

Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustain indefinitely given available resources. In logistic models, it acts as an upper bound, causing growth to slow and stabilize as the population approaches this value.

Parameter Estimation in Population Models

Estimating parameters like growth rate and carrying capacity involves using known data points to fit the logistic model. This often requires solving equations based on population values at different times to find the best values that describe the population trend.