# Math Help - Char. Eq.

1. ## 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

2. I hope I didn't make an error in my calculations, but I got a charpoly of:

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

If you set the parameters accordingly, when $s_{1}=\frac{1}{6}$, then $\lambda=1$

3. Hmm, yes 1/6 is correct I think, but I don't see how you get it

4. 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:
$det(A - \lambda I) = 0$

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

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

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

-Dan

5. How do you get the s_1 = 1/6 though?

6. 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