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(1P1000) 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+19e0.2t \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(1PK) 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(1PK) 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.
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