Logistic Growth Study Notes

Logistic Growth

  • Definition: Logistic growth refers to a type of limited growth characterized by a function that increases rapidly and then levels off over time, in contrast to exponential growth, which continues indefinitely without a cap.

    • Exponential Growth:
    • Explanation: In exponential models, growth is unbounded, meaning that populations can theoretically grow infinitely without limits.
    • Example: A situation where the population increases rapidly, for example, a bacterial culture in a nutrient-rich environment could exhibit exponential growth.
    • Logistic Growth Characteristics:
    • Growth Curve: The growth of a population starts rapidly and eventually stabilizes or plateaus as resources become limited.
    • Capacity Limit: Logistic growth takes into account the carrying capacity of the environment, meaning that there is a maximum population size that the environment can sustain.
    • Real-World Implication: For instance, the world population has a limit due to space and resources, demonstrating the concept of logistic growth.

  • Visual Representation:

    • Graph Description: The logistic growth function produces an S-shaped curve or sigmoidal curve, where the population initially grows exponentially before slowing down and approaching a maximum population level, known as the carrying capacity.
    • Key Points in the Graph:
    • Growth starts off steep, gradually slows, and finally reaches a plateau (the carrying capacity).

  • Key Components of Logistic Functions:

    • Formula: The standard formula for logistic growth can be represented as:
      N(t)=K1+K−N<em>0N</em>0e−rtN(t) = \frac{K}{1 + \frac{K - N<em>0}{N</em>0} e^{-rt}}

    • Where:

      • N(t)N(t) = population size at time tt
      • KK = carrying capacity
      • N0N_0 = initial population size
      • rr = intrinsic growth rate
      • tt = time

    • Asymptotes:

    • Horizontal Asymptotes: Each logistic function has two asymptotes:

      • Floor (Lower Asymptote): y = 0 (the population cannot go below zero).
      • Ceiling (Upper Asymptote): y = KK (the carrying capacity, or maximum population).

Examples of Logistic Growth

  • Example 1: Fast-Acting Flu Virus in Liberty School

    • Setting: The number of students and staff affected by a flu virus in a school.
    • Initial Population: 1,800 students and staff at Liberty.

  • Model Equation:

    • Write the equation to model the population:
      N(t)=18001+235e−0.38xN(t) = \frac{1800}{1 + 235 e^{-0.38x}}
    • Calculator Input: Ensure all parentheses are correct while entering the equation.

  • Graphing the Function:

    • Set the window for the graph:
    • xmin=−10x_{min} = -10
    • xmax=100x_{max} = 100
    • ymin=−10y_{min} = -10
    • ymax=1,900y_{max} = 1,900

  • Part A: Graph the function.

  • **Part B: Identify horizontal asymptotes:

    • y = 0
    • y = 1,800 (The maximum number of infected individuals possible).
    • Note that while 1,800 is an asymptote, instances may exist where the population can reach this value but ideally does not exceed it.

  • Part C: Determining Infected Population After Five Days:

    • Calculation Method: Use the table feature or hit "trace" on the calculator to find the value at x = 5 to find the number of infected individuals.
    • Result: 50 students infected on day five.

  • Part D: Time Taken for 150 Infected Individuals:

    • Set the equation to equal 150.
      150=18001+235e−0.38x150 = \frac{1800}{1 + 235 e^{-0.38x}}
    • Find the intersection of two equations set on the calculator by locating:
    • Graph the two functions and utilize the intersection function (trace -> #5).
    • Result: Approximately 8.0578.057 days.
    • Conversion into hours and minutes: Point 057 days is approximately 22 minutes, indicating that 150 individuals would be infected at around 1:22 AM.

  • Example 2: City Population Growth Modeling

    • Graphing Activity: Input the new population equation and derive its respective asymptotes, methods remain similar to example one.
    • Maximum Population Reached: Determine the numerical value indicating the city’s carrying capacity (approximately 1.432 million).

Summary

  • Understand the distinct differences between logistic and exponential growth.

  • Familiarize yourself with logistic growth formulas, graphing methods, and horizontal asymptotes.

  • Practice applying the concepts to real-world examples, highlighting the importance of carrying capacity and its effects on population dynamics.

  • Final Notes:

    • Make sure to practice with your calculator and graphing functions to become proficient in identifying and working with logistic growth models.