1. Identify L: The Carrying Capacity is the number in the denominator $(1 - y/L)$.
2. Identify k: The growth constant is the number in front of the $y$.
3. Fastest Growth: Happens at exactly $y = L/2$.
4. Equilibrium: The system is at rest when $y = 0$ or $y = L$.
For each logistic equation, identify the Carrying Capacity ($L$) and the Population of Maximum Speed ($L/2$).
$\frac{dy}{dt} = 0.2y(1 - \frac{y}{500})$
$\frac{dy}{dt} = 0.05y(1 - \frac{y}{10,000})$
$\frac{dy}{dt} = 3y - 0.01y^2$
Hint: Factor out the $3y$ first! $3y(1 - \frac{0.01}{3}y)$
The Village Growth: A village has a carrying capacity of 2,000 people. The current population is 1,800.
Is the growth rate ($y'$) increasing or decreasing? Why?
The Revival: A church has a capacity of 100 people ($L=100$). The current attendance is 10 people.
Is the growth rate likely to get faster or slower next week?
Look at the factor $(1 - y/L)$. What happens to the "Value" of this factor if $y = L$? What happens to the whole derivative ($dy/dt$)? Explain why this represents the "Peace of Fullness."
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On the grid below, sketch the solution curve for $\frac{dy}{dt} = 0.5y(1 - y/100)$ starting at $y(0) = 10$.
1. Draw a horizontal line at $y=100$.
2. Start at $y=10$. Curve upward steeply.
3. Slow down as you cross $y=50$.
4. Flatten out as you approach $y=100$.
A room has a capacity of 50 people. By mistake, 100 people are crammed inside ($y = 100$).
Using $\frac{dy}{dt} = 0.1y(1 - y/50)$:
Task: Calculate the sign of the growth rate. Are people entering or leaving? How fast is the "Correction" happening at this moment?
Objective: Explain the Logistic Curve to a younger student using a jar of candies.
The Activity:
1. Show an empty jar. "We can put candies in very fast!"
2. Fill it 95% full. "Try to put one more in. Is it easy or hard?"
3. "Now the jar is full. Can we add any more?"
The Lesson: "God made us to be like this jar. He fills us up with love until we are 'Full'. Once we are full, we stop 'Trying to get more' and we just 'Stay Full'."
Response: ___________________________________________________________