# Math Help - differencial equation system

1. ## differencial equation system

Now ... I ve made it. Files are smaller.
I need help with question 2 part (b)
many thanks

2. Originally Posted by simfonija
Now ... I ve made it. Files are smaller.
I need help with question 2 part (b)
many thanks
I'm curious. If all you need is 2 (b) why did you post the first page?

Anyway, we have a matrix equation:
$x^{(k)} = Ax^{(k-1)}$
and an initial population distribution: $x^{(0)} = a_1 x_1 + ... + a_n x_n$
where $\{ x_n \}$ is the set of eigenvectors of A.

Let $\{ \lambda _n \}$ be the set of eigenvalues corresponding to the (respective) eigenvector.

Then
$x^{(1)} = Ax^{(0)} = A(a_1x_1 + ... + a_n x_n)$

$x^{(1)} = a_1 \lambda _1 x_1 + ... + a_n \lambda_n x_n$

Iterating this process gives:
$x^{(k)} = a_1 ( \lambda _1 )^k x_1 + ... + a_n ( \lambda_n )^k x_n$

Now, if $\lambda _1$ is the dominant eigenvalue (I think this said the first eigenvalue. It was hard to read.) then we may assume
$\lambda _1 >> \lambda _k$ for all $k \neq 1$. (Obviously only a positive eigenvalue can dominate the population.) This means that for large k there is effectively only one term in the $x^{(k)}$ that is significant: the first term. So
$x^{(k)} \approx a_1 ( \lambda_1 )^k x_1$ for large k.

This means the population is stable in the state represented by eigenvector $x_1$ and has a growth rate represented by $\lambda _1$.

If there is no dominant eigenvalue then the population will fluctuate in a series of eigenstates with no specific growth rate dominating. However, for large enough k the population should stabilize around a vector that is the sum of eigenvectors with the largest eigenvalues. For example, say that we know that $\lambda _1 = \lambda _2 > \lambda _k$ for $k \neq 1, 2$. Then, given a large enough k we would have:
$x^{(k)} \approx a_1 ( \lambda _1)^k x_1 + a_2 ( \lambda _2)^k x_2$
So the population would settle into a growth state vector $a_1 x_1 + a_2 x_2$ with (in this case) a growth of $\lambda _1 = \lambda _2$.

-Dan

3. Thank you so much for the detail answer. The first page was mistake, previously I saw there is a part of question 2 (b) on first page. Once again thank you, was great help!