Dear readers,
The variables and constants are more or less based on the definitons in my previous thread: Rewrite variable in given constants and variables
There exists a stationary population , so:
, where and
( means the amount of people in age group at time . There are age groups, labeled from to )
Let with:
,
Now I have to show for a arbitrary population that is converges to the stationary distribution . To prove that, let and be 2 general populations.
And let
( . There are fertile age groups, labeled to )
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Show that by using the properties below:
with
(these also yield for )
and
is the probability of survival for people in age group . So the amount of people is equal to the amount of people multiplied by the probablity of survival . Hence,
This is different for the amount of people , because there are no things like and . So this is a special case. is the fertility factor of the people in age group . is between and , where , because people stop 'producing' children when they get past a certain age.
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My reasoning:
The survival probabilities of are in the numerator and denominator, so they cancel each other out. So the elements in the minimumfunction of and are mostly the same.
The only two differences are that has got the fraction and hasn't.
On the other hand has got and hasn't.
In short, when for all , then .
I thought that:
, but that doesn't help me either.
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Anyone who can lend me a hand?
- Fruitbowl.