Problem 7 - Single Variable Calculus

Question

Use the given logistic equation to find (a) the growth constant, (b) the carrying capacity of the environment, and (c) the initial population. $$ P(t)=\frac{8000}{2+798 e^{-0.02 t}} $$

Step-by-Step Solution

Answer

The growth constant (a) is -0.02, the carrying capacity of the environment (b) is 4000, and the initial population (c) is 10.

Step 1: Identify the Carrying Capacity (K)

As seen from the equation, the carrying capacity \( K \) is the maximum population that the environment can support. In this logistic equation, this maximum value is reached when the denominator is at its minimum value, i.e. "\(1+\frac{K-P_0}{P_0} e^{-rt}\)" is maximized. In our case, we have: $$ P(t) = \frac{8000}{2 + 798e^{-0.02 t}} $$ The maximum population occurs when the term with the exponential in the denominator goes to zero, i.e., when "\(2 + 798e^{-0.02 t}\)" is at its minimum value of \(2\). So we have: $$ K = \frac{8000}{2} = 4000 $$ Thus, the carrying capacity of the environment is 4000.

Step 2: Identify the Growth Constant (r)

The growth constant, denoted as \(r\), is also given in the equation. Looking at the exponential term in the equation: $$ P(t) = \frac{8000}{2 + 798e^{-0.02 t}} $$ we can see that \(r = -0.02\). Thus, the growth constant is -0.02. Note that we are taking the negative sign because of the specific equation form provided. Sometimes growth constant can be represented as positive, meaning the equation will become: $$ P(t) = \frac{K}{1+\frac{K-P_0}{P_0} e^{(-(r) t)}} $$ However, it's not the case for this specific question.

Step 3: Identify the Initial Population (P_0)

To find the initial population, \(P_0\), we need to evaluate the population function at time \(t=0\). So we have: $$ P(0) = \frac{8000}{2 + 798e^{-0.02(0)}} = \frac{8000}{2 + 798} $$ Evaluating this value, we find: $$ P(0) = \frac{8000}{2 + 798} = 10 $$ Thus, the initial population \(P_0\) is 10. #Conclusion# The growth constant (a) is -0.02, the carrying capacity of the environment (b) is 4000, and the initial population (c) is 10.

Key Concepts

Growth Constant

In the logistic equation, the growth constant, often symbolized as \( r \), is a crucial parameter. It dictates how quickly the population grows or declines over time. In the given equation \( P(t) = \frac{8000}{2 + 798e^{-0.02 t}} \), you can find \( r \) in the exponent of the exponential term.
\( r \) is the rate at which population changes. Here, \( r = -0.02 \). This negative sign indicates a reduction rate, suggesting the growth slowing down as time progresses.

Understanding the growth constant helps in predicting population behavior over time:

• A positive \( r \) implies exponential growth.
• A negative \( r \) indicates a slowing down or stabilization.
• Value and sign changes influence how the equation models reality.

In practice, knowing \( r \) assists in managing and planning for resources aligned with population changes.

Carrying Capacity

Carrying capacity, denoted as \( K \), represents the maximum population size an environment can sustainably support. It’s a pivotal concept because it sets a cap on how large the population can grow, driven by resource limitations.
In the equation \( P(t) = \frac{8000}{2 + 798e^{-0.02 t}} \), \( K \) is determined when the denominator is minimized. This occurs when the exponential decays to zero. Hence, the maximum population is \( \frac{8000}{2} = 4000 \).

This means the environment can support up to 4000 individuals without undergoing degradation.

• Balanced resource use is assumed at capacity.
• Overcapacity may lead to resource depletion.

Understanding carrying capacity aids in ecological planning, ensuring long-term sustainability of the population within its environment.

Initial Population

The initial population, denoted as \( P_0 \), is the starting size of the population at time zero. This parameter is crucial for determining the initial conditions that drive the system's future behavior.
In this scenario, we calculate \( P_0 \) by setting \( t = 0 \) in the given logistic equation:
\[ P(0) = \frac{8000}{2 + 798e^{-0.02 \cdot 0}} = \frac{8000}{2 + 798} = 10 \]

Thus, the initial population is 10. This baseline provides insight into:

• Early growth patterns, dictated by the size of \( P_0 \).
• Foundation from which subsequent growth emerges, influenced by \( r \) and \( K \).

Understanding the initial population helps visualize initial growth dynamics and anticipate future population trends.