# Predict without iterating?

• Apr 27th 2009, 03:45 PM
Mitchell
Predict without iterating?
When P0 is greater than .5Q0 “Q” will reach "0" first and become extinct. If P0 is less than .5Q0 “P” will reach "0" first and become extinct. However, if P0 = .5Q0, the population will eventually reach an equilibrium with 200 “P’s” and 400 “Q’s.” How could this be proven without using iteration?

Population model:

Pn+1 = Pn -.25Qn + 100

Qn+1 = -Pn + Q n+ 200
• Apr 27th 2009, 04:58 PM
HallsofIvy
Quote:

Originally Posted by Mitchell
When P0 is greater than .5Q0 “Q” will reach "0" first and become extinct. If P0 is less than .5Q0 “P” will reach "0" first and become extinct. However, if P0 = .5Q0, the population will eventually reach an equilibrium with 200 “P’s” and 400 “Q’s.” How could this be proven without using iteration?

Population model:

Pn+1 = Pn -.25Qn + 100

Qn+1 = -Pn + Q n+ 200

That can be written as the matrix equation $\displaystyle \begin{bmatrix} P_{n+1} \\ Q_{n+ 1}\end{bmatrix}= \begin{bmatrix}1 & -.25 \\ -1 & 1\end{bmatrix}\begin{bmatrix} P_n \\ Q_n\end{bmatrix}+ \begin{bmatrix}100 \\ 200\end{bmatrix}$

I would start by finding the eigenvalues and eigenvectors of that matrix.