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Logistic Models

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Logistic Models - Complete Interactive Lesson

Part 1: The Logistic Differential Equation

Logistic Models

Part 1 of 7 — The Logistic Differential Equation

The Model

$\frac{dP}{dt} = kPleft(1 - \frac{P}{L}\right)$

where:

$P$ = population (or quantity)
$k$ = growth rate constant
$L$ = carrying capacity

Key Features

When $P ll L$ : nearly exponential growth ( $approx kP$ )
When $P = L/2$ : fastest growth rate
When $P = L$ : growth stops ()

Logistic Basics 🎯

Key Takeaways — Part 1

Logistic: $dP/dt = kP(1 - P/L)$ . Max growth at $P = L/2$ .

Part 2: The Solution

Logistic Models

Part 2 of 7 — The Solution

General Solution

$P(t) = \frac{L}{1 + Ae^{-kt}}$

Part 3: Inflection Point

Logistic Models

Part 3 of 7 — Inflection Point

The Inflection Point

The growth rate changes from increasing to decreasing at $P = L/2$ .

To verify: $\frac{d^2P}{dt^2} = 0$ when .

Part 4: Analyzing Logistic Problems

Logistic Models

Part 4 of 7 — Analyzing Logistic Problems

From the DE

$\frac{dP}{dt} = 0.1P(1 - P/2000)$ ,

Part 5: Logistic vs Exponential

Logistic Models

Part 5 of 7 — Logistic vs Exponential

Comparison

Feature	Exponential	Logistic
DE	$dP/dt = kP$	$dP/dt = kP(1-P/L)$

Part 6: Practice Workshop

Logistic Models

Part 6 of 7 — Practice Workshop

Workshop 🎯

Workshop Complete!

Part 7: Final Assessment

Logistic Models — Review

Part 7 of 7 — Final Assessment

Final 🎯

Logistic Models — Complete! ✅

where $A = \frac{L - P_0}{P_0}$ .

As $t \to infty$ : $P \to L$
$P(t)$ is always between $P_0$ and $L$ (assuming $0 < P_0 < L$ )
S-shaped (sigmoid) curve

Key Takeaways — Part 2

$P(t) = L/(1 + Ae^{-kt})$ where $A = (L-P_0)/P_0$ . Always approaches $L$ .

$\frac{dP}{dt} = kP - \frac{kP^2}{L}$

$\frac{d^2P}{dt^2} = k\frac{dP}{dt} - \frac{2kP}{L}\frac{dP}{dt} = \frac{dP}{dt}left(k - \frac{2kP}{L}\right)$

Set to zero (with $dP/dt \neq 0$ ): $k - 2kP/L = 0 implies P = L/2$ ✓

Key Takeaways — Part 3

Inflection at $P = L/2$ : growth changes from accelerating to decelerating.

Carrying capacity: $L = 2000$
Growth constant: $k = 0.1$
Max growth rate at $P = 1000$
Max $dP/dt = 0.1 cdot 1000 cdot (1 - 1000/2000) = 0.1 cdot 1000 cdot 0.5 = 50$

$kP(1 - P/L) = kP - kP^2/L$ : if given as $0.1P - 0.00005P^2$ , then $k = 0.1$ and $k/L = 0.00005$ so $L = 2000$ .

Key Takeaways — Part 4

Factor the DE to identify $k$ and $L$ . $kP - (k/L)P^2 = kP(1-P/L)$ .

Key Takeaways — Part 5

Logistic starts exponential, then levels off at $L$ . More realistic.

Logistic Models

Behavior

Key Takeaways — Part 2

Key Takeaways — Part 3

Reading the DE

Key Takeaways — Part 4

Key Takeaways — Part 5