What’s the population?

Question

The population (p) of a certain type of insect grows rapidly initially. Its growth rate decreases under constraints and reach the maximum population after a limit of time t (in days) by the equation of the form

P(t)=210/(1+(25 x e^-0.15t)). Its graph is called Logistic Growth curve.

a)What is the maximum population as t→infinity?

b) When will the population reach 116 ?

Answer 1

P(t) = 210 / (1 + 25e^-0.15t)

(a)

As t→∞, e^-0.15t → 0, so

P(t) → 210 / [1 + 25(0)] = 210

(b)

Set

P(t) = 116

and solve for t.

116 = 210 / (1 + 25e^-0.15t)

1 + 25e^-0.15t = 210/116

25e^-0.15t = 210/116 - 1

e^-0.15t = (210/116 - 1)/25

-0.15t = ln[(210/116 - 1)/25]

t = ln[(210/116 - 1)/25] / (-0.15) days = ? days

You should be able to finish from here.