Expert: Paul Klarreich - 3/27/2011

Question
A troop of howler monkeys was introduced onto an island on 1 July 1965, and the approximate size of the population was estimated on 1 July in each subsequent year. The size of the initial population was 50, and it had grown to approximately 65 after one year. Some years later, the size of the population was found to be approximately 550, and the following year it had grown to approximately 660. Assume that the behaviour of this population satisfies the logistic model:

a)
Show that the annual proportionate growth rate for the population of size 50 was approximately 0.3, and that the annual proportionate growth rate for the population of size 550 was approximately 0.2.
(b)
Find the corresponding values of the annual proportionate growth rate for low population levels, r, and the equilibrium population size, E.
c)
Describe the long-term behaviour of the population as predicted by the model.

For part (a)  i did:
n=0, P(0)=50 n=1 P(1)=65
P(1)=(1+r)^1xP(0)
65=(1+r)^1x50
r=(65/50)^1-1=0.3  and for 0.2 r=(660/550)^1-1=0.2

Now im stuck in part b)

Many thanks for your time.

The logistic recurrence relation is Pn+1-Pn=rPn(1-Pn/E)

Answer
Questioner: elle
Category: Advanced Math
Private: No <<<<<<<<<<<<<<<<<<<< fixed.
Subject: maths
Question: A troop of howler monkeys was introduced onto an island on 1 July 1965, and the approximate size of the population was estimated on 1 July in each subsequent year. The size of the initial population was 50, and it had grown to approximately 65 after one year. Some years later, the size of the population was found to be approximately 550, and the following year it had grown to approximately 660. Assume that the behaviour of this population satisfies the logistic model:

a)
Show that the annual proportionate growth rate for the population of size 50 was approximately 0.3, and that the annual proportionate growth rate for the population of size 550 was approximately 0.2.
(b)
Find the corresponding values of the annual proportionate growth rate for low population levels, r, and the equilibrium population size, E.
c)
Describe the long-term behaviour of the population as predicted by the model.

For part (a)  i did:
n=0, P(0)=50 n=1 P(1)=65
P(1)=(1+r)^1xP(0)
65=(1+r)^1x50
r=(65/50)^1-1=0.3  and for 0.2 r=(660/550)^1-1=0.2

>>>>  yes, that looks clear.

Now im stuck in part b)

Many thanks for your time.

The logistic recurrence relation is Pn+1-Pn=rPn(1-Pn/E)
..................................
Ok, I think we can do something with it.

Pn+1 = Pn + rPn(1 - Pn/E)

Pn+1 = Pn( 1 + r(1 - Pn/E) )

Now that is simply an equation with two parameters, r, and E.  Therefore, you need two pairs of data.  Then you will get two simultaneous equations in r,E.

.................
You have P0 =  50  and P1 = 65, so:

65 = 50( 1 + r(1 - 50/E))

You also have

Pk = 550, for some k (we don't care which k), and Pk+1 = 660

660 = 550( 1 + r(1 - 550/E))

After some reducing, you have two equations:

13 = 10(1 + r(1 - 50/E))
6 = 5( 1 + r(1 - 550/E))

to solve for  r  and E.

13 = 10 + 10r(1 - 50/E)
3 = 10r - 500r/E


6 = 5 + 5r(1 - 550/E)

3 = 10r(1 - 50/E)
1 = 5r(1 - 550/E)

Dividing those equations:

      (1 - 50/E)
3 = 2 ------------
      (1 - 550/E)

3 - 1650/E = 2 - 100/E

1  =  1550/E

E = 1550

You can go back and finish up, to get  r.