I've just started learning Predator-Prey Models for the first time, though I find this question slightly strange.
Some animals have the ability to change sex if the population size gets too biased toward any one gender. It was found that in a particular group of fish the number of male and female fish follows the following equation:
\begin{eqnarray} M_{k+1}=0.8M_{k} + 0.1F_{k} \end{eqnarray} \begin{eqnarray} F_{k+1}=-pM_{k} + 1.1F_{k} \end{eqnarray}
$M_{k}$ and $F_{k}$ - number of male and female fish after k years respectively.
p is some positive constant.
I am required to find p so that the number of fish approaches a finite, non-zero limit.
I understand that normally you will have to find the eigenvalues and corresponding eigenvectors:
Starting off I would get matrix A: \begin{bmatrix} 0.8 & 0.1\\ -p & 1.1 \\ \end{bmatrix}
Finding eigenvalues by solving $det(A-\lambda I)$: \begin{bmatrix} 0.8-\lambda & 0.1\\ -p & 1.1-\lambda \\ \end{bmatrix}
I have: $det(A-\lambda I)=(0.8-\lambda)(1.1-\lambda)+0.1p$ but I can't help thinking that this step is unnecessary. Not too sure how this will help in finding p.
Many thanks in advance for any clarification/explanation.