[Guest]

Mr. Smith is a sheep breeder and he has a neat farm. He would like to know how affects the population of his animals if he sells each year h percent (same as every year) of the old sheeps (older than 1 year). The birth rate of the olds is 90%, the death rate is 20%. The young sheeps (younger than 1 year) are not able to reproduction and the farmer will not sell them, the death rate of this group is 10%. Mr. Smith wants to reach maximum profit and will not lose all of his animals. How to choose the value of h?

NOTE: you have to determine only the same constant value (h) for every year

I hope it's clear, if you need any further explanation, don’t hesitate to ask!

[Guest]

Do the births of the old sheep go into the young sheep category or do the births of the old sheep stay in the old sheep category?

[superprismatic]

Suppose the farmer has E old sheep and Y young ones.

The next year, if he sells none of them, he will have

.8E+Y old sheep and .9E young ones (the young ones

have become old). So, if he sells H of the old ones,

he will have (1-H)(.8E+Y) old ones and .9E young ones.

Assuming sheep are not discreet things, If H were

7/17, things will reach a steady state after a small

number of years no matter what reasonable number of

old and young sheep the farmer starts with. So, the

answer is H=7/17.

Edited March 20, 2010 by superprismatic

[Guest]

The births of the old sheep go into the young sheep category, of course.

[Guest]

Suppose the farmer has E old sheep and Y young ones.

The next year, if he sells none of them, he will have

.8E+Y old sheep and .9E young ones (the young ones

have become old). So, if he sells H of the old ones,

he will have (1-H)(.8E+Y) old ones and .9E young ones.

Assuming sheep are not discreet things, If H were

7/17, things will reach a steady state after a small

number of years no matter what reasonable number of

old and young sheep the farmer starts with. So, the

answer is H=7/17.

.8E+Y the young ones have also a death rate

[Guest]

Suppose the farmer has E old sheep and Y young ones.

The next year, if he sells none of them, he will have

.8E+Y old sheep and .9E young ones (the young ones

have become old). So, if he sells H of the old ones,

he will have (1-H)(.8E+Y) old ones and .9E young ones.

Assuming sheep are not discreet things, If H were

7/17, things will reach a steady state after a small

number of years no matter what reasonable number of

old and young sheep the farmer starts with. So, the

answer is H=7/17.

And another thing, “(1-H)(.8E+Y)”, in my solution the farmer sells only the h percent of the originally old ones, but I will accept your solution if correct. Would you mind showing how you got your answer, i mean the value of H?

Edited March 20, 2010 by det

[superprismatic]

Oops, I completely overlooked the 10% death rate for

the young ones! So, here's how I solve it. Let H

be the portion of elder sheep that the farmer sells,

let E be the original number of elder sheep, let Y

be the original number of young ones. Then, starting

with the ordered pair (old ones, young ones) of

(E,Y) for some year, the next year, the farmer will

have ([1-H]*[.8*E+.9*Y],[.9*E]). In the long run,

we want this to reach a steady state. So, we get two

equations:

(1) E=[1-H]*[.8*E+.9*Y] and

(2) Y=.9*E

Replacing the Y in (1) by .9*E from (2), we get

(3) E=[1-H]*[.8*E+.9*.9*E]

which simplifies to

(4) E=(1-H)*1.61*E

which makes

(5) (1-H)=1/1.61

and so,

(6) H=.61/1.61=61/161

which makes h to be approximately 37.9%. This amount

of selloff will make the number of (non-discrete) sheep

to reach a steady state.

[Guest]

Oops, I completely overlooked the 10% death rate for

the young ones! So, here's how I solve it. Let H

be the portion of elder sheep that the farmer sells,

let E be the original number of elder sheep, let Y

be the original number of young ones. Then, starting

with the ordered pair (old ones, young ones) of

(E,Y) for some year, the next year, the farmer will

have ([1-H]*[.8*E+.9*Y],[.9*E]). In the long run,

we want this to reach a steady state. So, we get two

equations:

(1) E=[1-H]*[.8*E+.9*Y] and

(2) Y=.9*E

Replacing the Y in (1) by .9*E from (2), we get

(3) E=[1-H]*[.8*E+.9*.9*E]

which simplifies to

(4) E=(1-H)*1.61*E

which makes

(5) (1-H)=1/1.61

and so,

(6) H=.61/1.61=61/161

which makes h to be approximately 37.9%. This amount

of selloff will make the number of (non-discrete) sheep

to reach a steady state.

Nice solution superprismatic! I've got the same result with your initial conditions, however I guess I overcomplicated a little bit

transformation matrix of the population change -> eigenvalues -> lambda1=1 -> h..

[superprismatic]

Nice solution superprismatic! I've got the same result with your initial conditions, however I guess I overcomplicated a little bit

transformation matrix of the population change -> eigenvalues -> lambda1=1 -> h..

I don't think your eigenvalue solution was overly complicated. It's just a very general solution. This problem is simple enough that you can do it more simply. I have to admit that my first thought was to bring out the big eigenvector guns! Thanks.