2.6.3 Logistic Models

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

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

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

DP IB Maths: AI HL

2.6.3 Logistic Models