# Char. Eq.

• Jun 27th 2007, 12:59 PM
flash101
Char. Eq.
Given:

M = ([0, f_2, f_3], [s_1, 0, 0], [0, s_2, 0])

(Note: [0, f_2, f_3] is the 1st row)

1.) Determine the characterist equation.

2.) Given f_2 = 2, f_3 = 5, s_2 = 0.8, how large must s_1 be to get L >= 1 (L is lambda)

3.) Given f_2 = 2, f_3 = 5, s_1 = 0.2, s_2 = 0.8, determine L with three-decimal-digit accuracy

4.) What's the stable population ratios for the preceding parameter values
• Jun 27th 2007, 01:30 PM
galactus
I hope I didn't make an error in my calculations, but I got a charpoly of:

$\displaystyle s_{1}s_{2}f_{3}+{\lambda}f_{2}s_{1}-{\lambda}^{3}$

If you set the parameters accordingly, when $\displaystyle s_{1}=\frac{1}{6}$, then $\displaystyle \lambda=1$
• Jun 27th 2007, 01:53 PM
flash101
Hmm, yes 1/6 is correct I think, but I don't see how you get it :confused:
• Jun 27th 2007, 05:11 PM
topsquark
Quote:

Originally Posted by flash101
Given:

M = ([0, f_2, f_3], [s_1, 0, 0], [0, s_2, 0])

(Note: [0, f_2, f_3] is the 1st row)

1.) Determine the characterist equation.

2.) Given f_2 = 2, f_3 = 5, s_2 = 0.8, how large must s_1 be to get L >= 1 (L is lambda)

3.) Given f_2 = 2, f_3 = 5, s_1 = 0.2, s_2 = 0.8, determine L with three-decimal-digit accuracy

4.) What's the stable population ratios for the preceding parameter values

The characteristic polynomial is given by the equation:
$\displaystyle det(A - \lambda I) = 0$

So we have the matrix
$\displaystyle A = \left ( \begin{array}{ccc} 0 & f_2 & f_3 \\ s_1 & 0 & 0 \\ 0 & s_2 & 0 \end{array} \right )$

Thus the characteristic polynomial will be given by:
$\displaystyle \left | \begin{array}{ccc} -\lambda & f_2 & f_3 \\ s_1 & -\lambda & 0 \\ 0 & s_2 & -\lambda \end{array} \right | = 0$

$\displaystyle -\lambda ^3 + f_2s_1 \lambda + f_3s_1s_2 = 0$

-Dan
• Jun 27th 2007, 05:29 PM
flash101
How do you get the s_1 = 1/6 though?
• Jun 27th 2007, 05:32 PM
Jhevon
Quote:

Originally Posted by flash101
How do you get the s_1 = 1/6 though?

plug in the given values and solve for s_1. you were given the values of f_2 ...etc in the questions