Rewrite variable in given constants and variables

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 $\displaystyle A(t)$ and $\displaystyle t \in \{0, 1, ...\}$ with following conditions:

$\displaystyle A_{j+1}(t+1) = A_j(t)p_j$ with $\displaystyle j \in \{0, ..., l - 1\}$

$\displaystyle A_0(t+1) = \sum^r_{i=0} A_i(t)f_i$

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

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

Let $\displaystyle B$ be a vector with above mentioned conditions ánd:

$\displaystyle \frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_{j+1}(t)}{B_j(t)}$

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

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

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

So for $\displaystyle j \in \{1, ..., l - 1\}$:

$\displaystyle \alpha_j = \alpha_j(t) = \frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_j(t)p_j}{B_j(t+1)}$.

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

Later I found that:

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

And for $\displaystyle j \in \{1, ..., l - 1\}$:

$\displaystyle \alpha_j = \alpha_j(t) = \frac{B_{j+1}(t+1)}{B_j(t+1)} = \frac{B_{j+1}(1)}{B_j(1)} = \frac{B_j(0)p_j}{B_j(1)}$.

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.

Re: Rewrite variable in given constants and variables

Hey Fruitbowl.

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

Re: Rewrite variable in given constants and variables

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

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

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

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

I hope this makes it more clear.

**Edit:**

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

Re: Rewrite variable in given constants and variables

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

Re: Rewrite variable in given constants and variables

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)?

Re: Rewrite variable in given constants and variables

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

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