Logistic Models (DP IB Applications & Interpretation (AI)) : Revision Note

Dan Finlay

Last updated

Logistic Models

What are the parameters of logistic models?

  • A logistic model is of the form space f open parentheses x close parentheses equals fraction numerator L over denominator 1 plus C straight e to the power of negative k x end exponent end fraction

  • The L represents the limiting capacity

    • This is the value that the model tends to as x gets large

  • The C (along with the L) helps to determine the initial value of the model

    • The initial value is given by fraction numerator L over denominator 1 plus C end fraction

    • Once L has been determined you can then determine C

  • The k determines the rate of increase of the model

What can be modelled using a logistic model?

  • A logistic model can be used when the variable initially increases exponentially and then tends towards a limit

    • H(t) is the height of a giraffe t weeks after birth

    • P(t) is the number of bacteria on an apple t seconds after removing from protective packaging

    • P(t) is the population of rabbits in a woodlands area t weeks after releasing an initial amount into the area

What are possible limitations of a logistic model?

  • A logistic graph is bounded by the limit L

    • However in real-life the variable might be unbounded

      • For example: the cumulative total number of births in a town over time

  • A logistic graph is always increasing

    • However in real-life there could be periods where the variable decreased or fluctuates

Worked Example

The number of fish in a lake, F, can be modelled by the function

F open parentheses t close parentheses equals fraction numerator 800 over denominator 1 plus C straight e to the power of negative 0.6 t end exponent end fraction

where t is the number of months after fish were introduced to the lake.

a) Initially, 50 fish were introduced to the lake. Find the value of C.

2-6-3-ib-ai-hl-logistic-model-a-we-solution

b) Write down the limiting capacity for the number of fish in the lake.

2-6-3-ib-ai-hl-logistic-model-b-we-solution

c) Calculate the number of months it takes until there are 500 fish in the lake.

2-6-3-ib-ai-hl-logistic-model-c-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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