Question

The population of walruses on an island satisfies a logistic model in which Po=800 in 2006. The carrying capacirty of the island is estimated at 5500, and P(1)=900.
a) Determin the logistic model for this population, where t is the number of years after 2006.
b) Use the logistic model from (a) to predict the year in which the population first exceed 2000.

Answer 1

Well, if I get it right, there is some multiplier you use to multiply each year's population to get the next one.
If that's true, then from the information you gave it has extended from 800 to 900 in one year.
Means, it grew up in factor of 900/800 = 9/8
So now we say the following:
\[ P(t) = Po \cdot (\frac{9}{8})_1 \cdot (\frac{9}{8})_2 \cdot .... \cdot (\frac{9}{8})_t \]
That's just like saying
\[ P(t) = Po \cdot (\frac{9}{8})^t \]

That should be for A.

For B, we wanna know in how much we have to multiply the Po (which is 800) to get 2000.
So 2000/800 = 2.5
Using P(t) we want to know what power of 9/8 would give us 2.5
\[ \log_{\frac{9}{8}}2.5 = 7.78 \]
Which basically means that after 7.78 years the population will grow to 2000.
But since the answer wanted is an integer number, and I know that somewhere along the 7th year it exceeded 2000, the answer should be 8, which is the first year to exceed 2000.

Answer 2

Thinking about it, I made a mistake here.
I mistakenly presumed there are measures that are taken in beginning of every year, and then the 8th year is the first one to exceed 2000. But just for an example it could be at the end of every year, and then the 7th year is the first one to exceed it.
This means I was thinking about it in a wrong way.

Instead, I should have thought: "In which year the occurrence of exceeding 2000 has occurred?" And that would be 7th. so B should be 7.
Very sorry =(

Answer 3

Thanks for explaining this for me.