AP Calc BC logistic growth cheat sheet

What is logistic growth?

Logistic growth models how a population increases rapidly at first and then slows as it approaches a carrying capacity.

True or False: Logistic growth has a constant growth rate.

False. Growth rate decreases as population approaches carrying capacity.

Define carrying capacity.

The maximum population size that an environment can sustain, denoted as $\displaystyle K$.

What is the logistic growth equation?

$\displaystyle P(t) = \\frac{K}{1 + \\frac{K - P_0}{P_0} e^{-rt}}$ where $\displaystyle P_0$ is initial population.

When does population growth begin to slow in logistic growth?

When the population size nears the carrying capacity $\displaystyle K$.

Fill in the blank: As $\displaystyle t \to \infty$, $\displaystyle P(t) \to$ _____

$\displaystyle K$ (the carrying capacity).

Example: Initial population $\displaystyle P_0 = 10$, $\displaystyle K = 100$, $\displaystyle r = 0.1$. Find $\displaystyle P(1)$.

Using $\displaystyle P(1) = \\frac{100}{1 + \\frac{100 - 10}{10} e^{-0.1}}$, $\displaystyle P(1) \approx 19.05$.

Cause → Effect: High initial growth rate leads to _____.

A rapid increase in population until resources become limited.

How does logistic growth differ from exponential growth?

Logistic growth has a limiting factor (carrying capacity), while exponential growth continues indefinitely.

What happens to growth rate at $\displaystyle P = K$?

The growth rate reaches zero as the population stabilizes.

True or False: The growth rate is highest at half of the carrying capacity.

True. Maximum growth occurs at $\displaystyle P = \\frac{K}{2}$.

Identify variables in $\displaystyle P(t) = \\frac{K}{1 + \\frac{K - P_0}{P_0} e^{-rt}}$.

$\displaystyle P(t)$: population at time $\displaystyle t$, $\displaystyle K$: carrying capacity, $\displaystyle P_0$: initial population, $\displaystyle r$: growth rate.

Example: If $\displaystyle K = 250$ and $\displaystyle P_0 = 50$, find $\displaystyle P(t)$ when $\displaystyle t = 2$.

Use the logistic model: $\displaystyle P(2) = \\frac{250}{1 + \\frac{250-50}{50} e^{-2r}}$. Solve for $\displaystyle r$ first.

What is the inflection point in logistic growth?

The point where the population growth rate transitions from increasing to decreasing, typically at $\displaystyle P = \\frac{K}{2}$.

Fill in the blank: The growth rate at any time $\displaystyle t$ is given by _____

$\displaystyle \\frac{dP}{dt} = rP(1 - \\frac{P}{K})$.

Cause → Effect: Limited resources lead to _____ in logistic growth.

A slowdown of population growth as it approaches carrying capacity.

True or False: Logistic growth can exceed carrying capacity.

False. Logistic growth levels off as the population approaches carrying capacity.

What does carrying capacity represent?

The maximum population size that an environment can sustain indefinitely.

Population of a store's customers over time?

Initially grows rapidly, then slows as it nears capacity, following a logistic curve.

If a population starts at 100 and has a carrying capacity of 1000, what happens?

The population will grow rapidly at first, then slow as it approaches 1000.

Calculate growth: Initial 50, capacity 500, rate 0.1.

Using the logistic growth model, the population will approach 500 over time.

Cause → Effect: High birth rate.

Leads to population growth until resources limit further increase.

Example of logistic growth in a restaurant.

Customer numbers rise quickly after opening, then plateau as seating capacity is reached.

What is the general logistic growth formula?

$\displaystyle P(t) = \\frac{K}{1 + \\frac{K-P_0}{P_0} e^{-rt}}$, where K = carrying capacity.

Fill in the blank: Logistic growth curves are_____

S-shaped, starting slowly, rising rapidly, and then leveling off.

True or False: Logistic growth is faster than exponential growth.

False. Exponential growth can initially be faster but does not level off.

Compare: Logistic vs. Exponential growth.

Logistic growth levels off; exponential growth continues to rise indefinitely.

Word problem: A population of 200 rabbits grows at 5% annually.

If carrying capacity is 800, expect rapid growth then stabilization around 800.

What happens when resources are abundant?

Logistic growth allows rapid population increases until limits are reached.

Example of logistic growth in technology adoption.

Initial rapid adoption of a new phone model, followed by slower growth as market saturates.

Calculate: 100 bacteria, capacity 1000, rate 0.2.

Population increases quickly and approaches 1000 over time, following a logistic curve.

Cause → Effect: Limited resources.

Causes a slowdown in population growth as it approaches carrying capacity.

Define the logistic growth equation.

$\displaystyle P(t) = \\frac{K}{1 + \\frac{K-P_0}{P_0}e^{-rt}}$ where: - $\displaystyle P(t)$ = population at time $\displaystyle t$ - $\displaystyle K$ = carrying capacity - $\displaystyle P_0$ = initial population - $\displaystyle r$ = growth rate.

Identify key features of logistic growth.

- Rapid initial growth - Slows as it approaches carrying capacity - Stabilizes at $\displaystyle K$ - S-shaped curve.

True or False: Logistic growth can exceed carrying capacity.

False: Logistic growth inherently includes a carrying capacity ($\displaystyle K$) that limits growth.

What happens when population is at carrying capacity $\displaystyle K$?

Population growth ceases. - $\displaystyle dP/dt = 0$ - Stable equilibrium.

Compare exponential growth and logistic growth.

Exponential: - Unlimited resources - Growth rate constant. Logistic: - Limited resources - Growth rate decreases as $\displaystyle P$ approaches $\displaystyle K$.

Fill in the blank: The inflection point of a logistic function occurs at ________.

$\displaystyle P = \\frac{K}{2}$, where growth rate is fastest.

Calculate population after 2 years: $\displaystyle K=100$, $\displaystyle P_0=10$, $\displaystyle r=0.3$.

Use: $\displaystyle P(t) = \\frac{100}{1 + \\frac{100-10}{10}e^{-0.6}}$ - Find $\displaystyle P(2) \approx 30.77$.

True or False: Logistic growth is only applicable to biological populations.

False: Applicable in various contexts, like technology adoption and resource consumption.

What does the term 'carrying capacity' mean?

Maximum population size that the environment can sustain indefinitely.

Cause → Effect: What causes the slow down in logistic growth?

Population approaching $\displaystyle K$ → Limited resources lead to reduced growth rate.

Identify the primary factor that influences growth rate $\displaystyle r$.

Birth rate minus death rate, and can be affected by environmental factors.

Evaluate: For $\displaystyle K=200$ and $\displaystyle P_0=50$, what is $\displaystyle P(t)$ at $\displaystyle t=1$ with $\displaystyle r=0.4$?

$\displaystyle P(1) = \\frac{200}{1 + \\frac{200-50}{50}e^{-0.4}} \approx 78.39$.