Thread: Population Models

Okay I have been doing this for a while now and I got t = 3, but clearly it was wrong because it actually equals 1.6 days.

I set up the model like this

dp/dt = k(p)^1/2


This is the problem

Let t denote time (in days) and let P denote the population of some species at time t. Suppose that the birth rate β (number of births per day per unit of population) is proportional to P^1/2 and that the death rate δ (number of deaths per day per unit of population) equals 0. Suppose the initial population is 4, and after one day the population is 16. When will the population equal 100?

Originally Posted by radioheadfan

Okay I have been doing this for a while now and I got t = 3, but clearly it was wrong because it actually equals 1.6 days.

I set up the model like this

dp/dt = k(p)^1/2


This is the problem

Let t denote time (in days) and let P denote the population of some species at time t. Suppose that the birth rate β (number of births per day per unit of population) is proportional to P^1/2 and that the death rate δ (number of deaths per day per unit of population) equals 0. Suppose the initial population is 4, and after one day the population is 16. When will the population equal 100?

Please show all your working that led to your answer of t = 3. Every step, please.

okay I'm guessing i setup the model correctly
So, I took the integrals of (1/(p)^(1/2)) dP = k dt

and I got 2p^(1/2) = kt + c

p = ((kt/2) + c)^2

c = Po^1/2 (initial population)

so i plugged in the initial values to find the constant k value

16 = ( k/2 + (4)^(1/2))^2 ... Took the square root of both sides

4 = (k/2 + 2)

2 = k/2

4 = k

then for the new value of P (100) i plugged in the values

100 = ( 4t/2 + (4)^1/2)^2

square root

10 = 2t +2

8 = 2t

4 = t

I actually got 4, but the Professor said it was equal to 1.6 days.

Originally Posted by radioheadfan

okay I'm guessing i setup the model correctly
So, I took the integrals of (1/(p)^(1/2)) dP = k dt

and I got 2p^(1/2) = kt + c

p = ((kt/2) + c)^2

c = Po^1/2 (initial population)

so i plugged in the initial values to find the constant k value

16 = ( k/2 + (4)^(1/2))^2 ... Took the square root of both sides

4 = (k/2 + 2)

2 = k/2

4 = k

then for the new value of P (100) i plugged in the values

100 = ( 4t/2 + (4)^1/2)^2

square root

10 = 2t +2

8 = 2t

4 = t

I actually got 4, but the Professor said it was equal to 1.6 days.

p = ((kt/2) + c)^2

Substitute t = 0, p = 4: c = 2.

Therefore p = ((kt/2) + 2)^2.

Substitute t = 1, p = 16. Get k = 4.

p = (2t + 2)^2.

Substitute p = 100: 100 = (2t + 2)^2 => t = 4.

Looks like you're correct.

are you sure the model is correctly written according to the word problem?

Originally Posted by radioheadfan

are you sure the model is correctly written according to the word problem?

Yes, it is correct according to the wording of the question you posted.