I am currently working on a small project which involves mathematical biology. It's been a while since I've visited Linear Algebra so any recollection would be helpful.
Consider the Predator-Prey equation:
$\frac{dx}{dt}\equiv \dot{x}\left ( t \right )=\alpha x-\beta x y$
$\frac{dy}{dt}\equiv \dot{y}\left ( t \right )=\delta x y- \gamma y$
In this case, my physical constraints are
1) The predator do not die nor reproduce so their populations are constant with time
2) Standard assumption holds for Prey via the Predator-Prey model
Solving gives
$x\left ( t \right )=C_{2}e^{t\left ( \alpha-\beta C_{1} \right )}$
$y\left ( t \right )=C_{1}$
In matrix form:
$F=\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix}$
=$\begin{bmatrix} f_{1}\left ( x,y \right )\\ f_{2}\left ( x,y \right ) \end{bmatrix}$
$\begin{bmatrix} \alpha x -\beta x y\\ \delta x y-\alpha y \end{bmatrix}$
The fixed points are $\left ( 0,0 \right )$ and $\left ( \frac{\gamma}{\delta } ,\frac{\alpha}{\beta}\right )$.
But note that the system of equations are coupled so expressing them in matrix form may be impossible.
I'm thinking of taking the Jacobian J= $\begin{bmatrix} \alpha -\beta y &\beta x \\ \delta y & \delta x-\gamma \end{bmatrix}$
in the eigenvector equation $J\vec{v}=\lambda \vec{v}$ to determine the eigenvectors.
My second strategy is to ignore the non-linear terms and solve the differential equation.It would also enable me to express the system of equation sans the non-linear terms as a matrix/ vector.
Am I justified in doing so?