The logistic model is a mathematical representation of population growth limited by a carrying capacity.
What does K</latex> represent in the logistic differential equation?
Carrying capacity
In a logistic model, population growth slows as it approaches the carrying capacity.
What is the logistic differential equation for a bacterial colony with a growth rate of 0.1 per hour and a carrying capacity of 1000 cells?
dtdP=0.1P(1−1000P)
The logistic growth equation is \frac{dP}{dt} = rP \left(1 - \frac{P}{K}\right)</latex>, which shows population growth that plateaus at the carrying capacity.
Steps to solve the logistic differential equation
1️⃣ Separate variables
2️⃣ Use partial fraction decomposition
3️⃣ Integrate both sides
4️⃣ Solve for P
What is the solution to the logistic differential equation for P0=50, r = 0.2</latex>, and K=1000?
P(t)=1+19e−0.2t1000
The carrying capacity K in a logistic model represents the maximum population size the environment can sustain.
How does a higher growth rate r affect the initial growth of a population in a logistic model?
Leads to faster growth
Unlike exponential growth, a logistic model accounts for environmental constraints on population growth.
What is the logistic differential equation?
dtdP=rP(1−KP)
The carrying capacity in a logistic model represents the maximum population the environment can sustain
A logistic model accounts for environmental constraints on population growth.
Match the growth type with its equation:
Exponential ↔️ dtdP=rP
Logistic ↔️ dtdP=rP(1−KP)
What is the general solution to the logistic differential equation?
P(t) = \frac{K}{1 + Ae^{ - rKt}}</latex>
Steps to solve the logistic differential equation:
1️⃣ Separate variables
2️⃣ Apply partial fraction decomposition
3️⃣ Integrate both sides
4️⃣ Solve for P
The initial conditions are essential for determining the value of A
A higher growth rate in a logistic model leads to slower initial growth.
False
What does the carrying capacity define in a logistic model?
Maximum sustainable population
The initial population affects the shape of the growth
Match the real-world scenario with its application of logistic models:
Fish Populations ↔️ Growth in limited resources
Bacterial Growth ↔️ Colony size in a lab
Disease Spread ↔️ Infected individuals during an outbreak
The logistic equation can predict the growth of a fish population with limited resources.