Expert: Paul Klarreich - 1/31/2008

Question
Biologist stocked a lake with 400 fish and estimated the carrying capacity (the maximum population size for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year.

a, assuming that the size of the fish population satisfied the logistic equation, find an expression for the size of the population after t years.

For a, i i have P= 10,000/ 1+e^(-10,000*k*t)
but i couldn't find the value for k.
since it said "the number of fish tripled in the first year" would k = 3 ?

b, how long will it take for the opulation to increase to 5,000?

because i didn't have k, i couldn't figure this question either. When i find the k, would i make P =5,000 in the equation above then find the t?

c, What is the rate of grouth when the population is growing the fastest?

so far, i have  
k(P/2)(Max- P/2)
since P=10,000

= k(5,000)(10,000-5,000)
i tried plugging in 3 for k and got 75*10^6
but that number seems to big.

d, As an advisor to the biologists, what would be your estimate of when the lake would reach it's carrying capacity? Give your advice in the form of a memo, with complete sentences and explaining/ justifying your method for determining this number.

on this one, am i explaining how i'll find t when the lake reaches its capacity? if so, i should say, first plug in the 10,000 for P and then find t?

Thank you so much for your help.

Answer
Questioner:   ann
Category:  Calculus
Private:  No
 
Subject:  logistic growth
Question:  Biologist stocked a lake with 400 fish and estimated the carrying capacity (the maximum population size for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year.

a, assuming that the size of the fish population satisfied the logistic equation,
.........................
>> PLEASE -- NO SPECIALIZED VOCABULARY.  What is this equation??
.........................
find an expression for the size of the population after t years.

For a, i i have P= 10,000/ 1+e^(-10,000*k*t)
but i couldn't find the value for k.

since it said "the number of fish tripled in the first year" would k = 3 ?
.............................
>> No, but this equation is inconsistent with the information you have provided.  IF

P(t) = 10,000/ 1+e^(-10,000*k*t)

Then P(0) = 10000/(1 + 1) = 5000, not 400.
.....................................

b, how long will it take for the opulation to increase to 5,000?

because i didn't have k, i couldn't figure this question either. When i find the k, would i make P =5,000 in the equation above then find the t?

c, What is the rate of grouth when the population is growing the fastest?

so far, i have  
k(P/2)(Max- P/2)
since P=10,000

= k(5,000)(10,000-5,000)
i tried plugging in 3 for k and got 75*10^6
but that number seems to big.

d, As an advisor to the biologists, what would be your estimate of when the lake would reach it's carrying capacity? Give your advice in the form of a memo, with complete sentences and explaining/ justifying your method for determining this number.

on this one, am i explaining how i'll find t when the lake reaches its capacity? if so, i should say, first plug in the 10,000 for P and then find t?

Thank you so much for your help.
..........................................
Sorry I can't give you more help than this.  If you try resending the question, AND

1. Write it exactly as you see it.
2. Proofread it -- pretend it's going to the Dean of Admissions at the college you really want to get into. (Such as writing "I' for the first-person singular.)
3. Make sure you define your terms.