# Math Help - Predict without iterating?

1. ## 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

2. 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 $\begin{bmatrix} P_{n+1} \\ Q_{n+ 1}\end{bmatrix}= \begin{bmatrix}1 & -.25 \\ -1 & 1\end{bmatrix}\begin{bmatrix} P_n \\ Q_n\end{bmatrix}+ \begin{bmatrix}100 \\ 200\end{bmatrix}$

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