1. ## matrix question

The population of rats in my farm moves constantly back and forth between the house and the field. Every week 40% of the rats in the field move to my house while 30% of those in my house move to the field. Suppose that, initially, the distribution of this population is: 70% in the field and 30% in my house.

(a) Find the migration matrix and set up a difference equation for this situation.
(b) What is the rats’ distribution after two week in my farm?

a) I get this migration matrix:

0.70 0.40
0.30 0.60

Is this correct? If so, how do I set up a difference equation?

b)

0.70 0.40
0.30 0.60

multiply this matrix by:

0.70
0.30

this will give the rats distribution after 1 week, and multiply by that again to get the distribution after two weeks. Is this correct?

Please can someone give me some guidance on this problem.

2. ## Re: matrix question

Just write what you are doing in matrix form

$\left[ \begin{array}{cc}H\\F \end{array} \right]_{n+1}=\left[ \begin{array}{cc}0.7 & 0.4\\0.3 & 0.6 \end{array} \right] \left[ \begin{array}{cc}H\\F \end{array} \right]_{n}.$

3. ## Re: matrix question

Yes I did write it in matrix form, which is why I said "I get this migration matrix" and then wrote it in matrix form.

Can someone help me with this problem please?

4. ## Re: matrix question

The "difference equations" are just $H_{n+1}= .7H_n+ .4Fn$ and $F_{n+1}= .3H_n+ .6F_n$ or, if you prefer the " $\Delta H$= H_{n+1}- H_n" and " $\Delta F= F_{n+1}- F_n$" form then $\Delta H= (.7- 1)H+ .4F= -.3H+ .4F$ and $\Delta F= .3H+ (.6- 1)F= .3H- .4F$, for each n.