Problem 549 - Precalculus

Question

What is the carrying capacity for a population modeled by the logistic equation \(P(t)=\frac{250,000}{1+499 e^{-0.45 t}} ?\) What is the initial population for the model?

Step-by-Step Solution

Answer

The carrying capacity is 250,000, and the initial population is 500.

Step 1: Identify the Logistic Function

The logistic function is given in the form: \( P(t) = \frac{K}{1 + A e^{-rt}} \), where \( K \) is the carrying capacity, \( A \) is a constant related to the initial condition, and \( r \) is the growth rate. The provided logistic equation is \( P(t) = \frac{250,000}{1+499 e^{-0.45t}} \).

Step 2: Determine the Carrying Capacity \( K \)

Examine the logistic equation: \( P(t) = \frac{250,000}{1+499 e^{-0.45t}} \). The carrying capacity, \( K \), is the numerator of the fraction. Hence, \( K = 250,000 \).

Step 3: Determine the Initial Population

To find the initial population, evaluate \( P(t) \) at \( t = 0 \). Substitute \( t = 0 \) into the equation: \( P(0) = \frac{250,000}{1 + 499 e^{-0.45(0)}} = \frac{250,000}{1 + 499} \).

Step 4: Calculate the Initial Population

Solve the expression \( P(0) = \frac{250,000}{1 + 499} = \frac{250,000}{500} = 500 \). Thus, the initial population is 500 individuals.

Key Concepts

Carrying Capacity

The concept of carrying capacity is a key element in understanding logistic growth models. In ecology, carrying capacity refers to the maximum number of individuals in a population that an environment can support without leading to environmental degradation. This is a fundamental limit imposed by available resources like food, water, and space.

In the context of a logistic equation, the carrying capacity is represented by the constant \( K \). Looking at the logistic growth model provided in the exercise:

• The carrying capacity \( K \) appears as the numerator in the logistic equation \( P(t) = \frac{250,000}{1+499 e^{-0.45 t}} \).
• It indicates that this environment can support up to 250,000 individuals.

Understanding the carrying capacity helps predict the stabilization of population over time as it approaches this limit. This not only aids in planning and resource allocation but also in establishing sustainable practices for population management.

Initial Population

The initial population of a logistic model offers insight into the starting point or baseline of a population at time \( t = 0 \). For the logistic equation \( P(t) = \frac{250,000}{1+499 e^{-0.45 t}} \), determining the initial population involves substituting \( t = 0 \) into the equation.

Evaluating this provides:

• \( P(0) = \frac{250,000}{1+499} = \frac{250,000}{500} \)
• Which simplifies to an initial population of 500 individuals.

The initial population figure is crucial as it influences how quickly a population can reach its carrying capacity. A smaller initial population relative to the carrying capacity may result in a slower growth rate initially, while a larger initial population could indicate a faster approach towards reaching the environmental limits.

Population Growth Model

Population growth models are tools used to describe how populations change over time. The logistic growth model in particular is used to predict population dynamics in environments where resources are limited.

The logistic model is generally expressed as \( P(t) = \frac{K}{1+Ae^{-rt}} \), where:

• \( K \) is the carrying capacity.
• \( A \) is a constant that reflects initial conditions.
• \( r \) is the growth rate.

In the logistic equation provided, \( r = 0.45 \), demonstrating how quickly the population grows relative to resources available. Key distinctions of this model include the S-shaped curve, representing an initial growth phase, followed by a slowdown as resources become scarce, eventually stabilizing near the carrying capacity.

This model is especially helpful in understanding the balance between population increase and resource limitations, allowing for strategies that can sustain ecological and economic stability.