7.9 Logistic Models with Differential Equations

Cards (22)

  • 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?
    dPdt=\frac{dP}{dt} =0.1P(1−P1000) 0.1P \left(1 - \frac{P}{1000}\right)
  • 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 PP
  • What is the solution to the logistic differential equation for P0=P_{0} =50 50, r = 0.2</latex>, and K=K =1000 1000?

    P(t)=P(t) =10001+19e−0.2t \frac{1000}{1 + 19e^{ - 0.2t}}
  • The carrying capacity KK in a logistic model represents the maximum population size the environment can sustain.
  • How does a higher growth rate rr 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?
    dPdt=\frac{dP}{dt} =rP(1−PK) rP \left(1 - \frac{P}{K}\right)
  • 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 ↔️ dPdt=\frac{dP}{dt} =rP rP
    Logistic ↔️ dPdt=\frac{dP}{dt} =rP(1−PK) rP \left(1 - \frac{P}{K}\right)
  • 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 PP
  • 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.