Question

The human population (in thousands) of Nevada in 1950 was roughly 160 . If the carrying capacity is estimated at 10 million individuals, and assuming a growth rate of \(2 \%\) per year, develop a logistic growth model and solve for the population in Nevada at any time (use 1950 as time \(=0\) ). What population does your model predict for 2000 ? How close is your prediction to the true value of \(1,998,257\) ?

Short answer

Predicted: 423,360; Actual: 1,998,257. The prediction is an underestimate.

Step-by-step solution

Understand the Logistic Growth Model

The logistic growth model is expressed by the formula:\[ P(t) = \frac{K}{1 + \frac{K - P_0}{P_0} e^{-rt}} \]where \( P(t) \) is the population at time \( t \), \( K \) is the carrying capacity, \( P_0 \) is the initial population, and \( r \) is the growth rate. Here, we have \( K = 10,000 \) (since 10 million is 10,000 thousands), \( P_0 = 160 \) (since 160 is in thousands), and \( r = 0.02 \). The goal is to find \( P(50) \) for the year 2000.

Substitute Values into the Model

Substitute the known values into the logistic growth formula:\[ P(t) = \frac{10,000}{1 + \frac{10,000 - 160}{160} e^{-0.02t}} \]We need to calculate \( P(50) \) for the year 2000.

Calculate the Expression in the Denominator

Calculate the fraction in the denominator:\[ \frac{10,000 - 160}{160} = \frac{9840}{160} = 61.5 \]

Evaluate the Exponential Term

Calculate the exponential term for \( t = 50 \):\[ e^{-0.02 \times 50} = e^{-1} \]

Substitute and Solve for P(50)

Now simplify:\[ P(50) = \frac{10,000}{1 + 61.5 \times e^{-1}} \]Calculate \( e^{-1} \approx 0.3679 \) and substitute back:\[ P(50) = \frac{10,000}{1 + 61.5 \times 0.3679} \]\[ P(50) \approx \frac{10,000}{1 + 22.6275} \]\[ P(50) \approx \frac{10,000}{23.6275} \]\[ P(50) \approx 423.36 \] (thousand individuals)

Convert To Actual Number and Compare

The logistic model predicts a population in year 2000 as 423,360 individuals. Compared to the actual population of \(1,998,257\), this model significantly underestimates the real population.

Key concepts

Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustain indefinitely without being degraded. For Nevada, the carrying capacity is set at 10 million individuals or 10,000 in thousands.
This number represents the upper limit determined by resources such as food, space, and other environmental factors.
When a population reaches its carrying capacity, growth stops or stabilizes because the resources are insufficient to support more individuals. It’s like the maximum capacity of a theater – no more tickets can be sold when it is full.

• The environment provides a certain amount of resources.
• Beyond a point, resources cannot support additional population growth.
• This limit helps in understanding potential long-term sustainability.

Understanding this concept helps us predict when a population might stabilize or need to relocate. In the logistic growth model, carrying capacity plays a crucial role in addressing how population growth slows as it nears this limit.

Exponential Growth

Exponential growth occurs when a population size increases at a constant rate over time. It's characterized by rapid growth, where the population doubles over regular intervals.
In the Nevada example, an annual growth rate of 2% determined the initial phase of growth. This type of growth is common when there are plenty of resources, and fewer restrictions on population size.

• Starts with slow growth, but accelerates as the population increases.
• Requires ample resources to sustain the increasing numbers.
• Shows a J-shaped curve on a graph.

However, exponential growth is unsustainable over the long term as resources are finite. Once resources become scarce, growth slows down, shifting towards a logistic model where factors like carrying capacity come into play.

Population Dynamics

Population dynamics studies the size and age composition of populations over time, and the factors that affect them.
This can involve examining birth rates, death rates, immigration, and emigration.
In the context of Nevada's growth, considering population dynamics helps us understand why the logistic model may underestimate the population size.

• Includes studying how populations interact with the environment.
• Explores the impact of events like natural disasters or human activities.
• Looks at both short-term fluctuations and long-term trends.

This concept is significant for making informed decisions on managing natural resources and urban planning. While a logistic growth model uses a simplistic view, real-world population dynamics are often more complex and influenced by numerous unpredictable factors.