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IBMaths: AI HLDPRevision Notes2. Functions2.6 Further Modelling with Functions2.6.3 Logistic Models

Logistic Models (DP IB Maths: AI HL)

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Dan

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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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    Previous:2.6.2 Natural Logarithmic ModelsNext:2.6.4 Piecewise Models
    Dan

    Author: Dan

    Expertise: Maths

    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.