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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Logistic Models

What are the parameters of logistic models?

A logistic model is of the form $f (x) = \frac{L}{1 + C e^{- k x}}$
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 $\frac{L}{1 + C}$
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

$F (t) = \frac{800}{1 + C e^{- 0.6 t}}$

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

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

C

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

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

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

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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Author: Dan

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.