Consider the 2-dimensional system of non-linear ODEs, semplified instance of a predator-prey population model
$\dot x=\alpha x (1-x)-xy$
$\dot y = y(x-y)-\beta y$
with $\alpha = 1 ,\beta = 1/2$
Why does $x(0) > 0$ and $y(0) > 0$ imply that $x(t) > 0$ and $y(t) > 0$ for all times (populations remain positive)?
Could you give me at least a cue for this point? For all equilibrium points of saddle type, compute the invariant subspaces $E^s$ and $E^u$ of their linearization.