Rewrite variable in given constants and variables

**Fruitbowl** · January 30th 2013, 05:00 AM

Hello everyone,

English isn't my native language, so I hope I can make myself clear. I'm new on this forum, because I have a question. I put this topic in this subforum, but I am not sure if this is the right one. Sorry if I'm wrong.

Given is some vector $A(t)$ and $t \in \{0, 1, ...\}$ with following conditions:

$A_{j+1}(t+1) = A_j(t)p_j$ with $j \in \{0, ..., l - 1\}$
$A_0(t+1) = \sum^r_{i=0} A_i(t)f_i$

In this case: $0 < r < l \in \mathbb{N}$ . $r$ and $l$ are constants.

$f_i$ with $i \in \{0, ..., r\}$ and $p_j$ with $j \in \{0, ..., l - 1\}$ are constants too.

Let $B$ be a vector with above mentioned conditions ánd:
$\frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_{j+1}(t)}{B_j(t)}$

Define $\alpha_j = \alpha_j(t) = \frac{B_{j+1}(t+1)}{B_j(t+1)}$ . Express $\alpha_j$ in $\alpha_0$ , $p_j$ and $p_0$ .

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I have a very difficult time doing this. This is how far I got:

$\alpha_0 = \alpha_0(t) = \frac{B_1(t+1)}{B_0(t+1)} = \frac{B_0(t)p_0}{\sum^r_{i=0} B_i(t)f_i}$ .

So for $j \in \{1, ..., l - 1\}$ :
$\alpha_j = \alpha_j(t) = \frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_j(t)p_j}{B_j(t+1)}$ .

I think I have to substitute $\alpha_0$ somewhere in $\alpha_j$ in a way that only $\alpha_0$ , $p_j$ and $p_0$ remain as constants/variables.

Later I found that:
$\alpha_0 = \alpha_0(t) = \frac{B_1(t+1)}{B_0(t+1)} = \frac{B_1(1)}{B_0(1)} = \frac{B_0(0)p_0}{\sum^r_{i=0} B_i(0)f_i} = \alpha_0(0)$

And for $j \in \{1, ..., l - 1\}$ :
$\alpha_j = \alpha_j(t) = \frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_{j+1}(1)}{B_j(1)} = \frac{B_j(0)p_j}{B_j(1)}$ .

But I don't know what I can do with that.

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Anyone who can help me with this?

Thanks in advance.

**chiro** · January 30th 2013, 05:57 PM

Hey Fruitbowl.

Can you outline broadly what you are trying to do? (What is the goal of your problem in non-mathematical terms)?

**Fruitbowl** · January 31st 2013, 01:24 AM

This task is about demography and making mathematical models. $A_j(t)$ (and $B_j(t)$ ) stands for the amount of people in age group $j$ at time $t$ . There are $l+1$ age groups labeled $0$ to $l$ . For example, in age group 0 the people are between 0 and 9 years old, in age group 1 between 10 and 19, etc.

$p_j$ is the probability of survival for people in age group $j$ . So the amount of people $A_{j+1}(t+1)$ is equal to the amount of people $A_j(t)$ multiplied by the probablity of survival $p_j$ . Hence, $A_{j+1}(t+1) = A_j(t)p_j$

This is different for the amount of people $A_0$ , because there are no things like $A_{-1}$ and $p_{-1}$ . So this is a special case. $f_i$ is the fertility factor of the people in age group $i$ . $i$ is between $0$ and $r$ , where $r < l$ , because people stop 'producing' children when they get past a certain age.

So the amount of people $A_0(t+1) = \sum^r_{i=0} A_i(t)f_i$ .

I hope this makes it more clear.

Edit:

And B is a stationary population in this case. So $\frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_{j+1}(t)}{B_j(t)}$ . So the proportion between $B_{j+1}$ and $B_j$ is always the same at a certain time $t \in \{0, 1, ...\}$ and for all $j \in \{0, ..., l -1\}$

**Fruitbowl** · January 31st 2013, 01:46 AM

< I'm sorry I posted above mentioned message twice >

**chiro** · January 31st 2013, 03:02 PM

I'm still a little lost: What exactly are you trying to answer? Do you want to find average survival? Maybe find some kind of optimal survival attribute? Maybe use this to design insurance premiums (like life insurance contracts)?

**Fruitbowl** · January 31st 2013, 03:14 PM

Later in this task I have to show that every population will converge to a stationary population (in this case $B$ ).

The thing I have to do here is to express $\alpha_j$ in $\alpha_0$ , $p_j$ and $p_0$ .

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