Prove for a coupled system of non-linear ODE the value of a variable is always positive

Question

There is a system of Non-Linear ODE of 3 variables: x,y,z defined as: $$\frac{dx}{dt} = -2kx^2 -kxy$$ $$\frac{dy}{dt} = kx^2 -kxy$$ $$\frac{dz}{dt} = 3kxy$$ with the initial conditions $x(0)=1$, $y(0)=0$ and $z(0)=0$ and k is any positive number. Show that the value of $y$ is always greater than 0.

Note: The equation for z is not of any direct use but it gives a conservation condition in tandem with the other ODEs ie. $x+2y+z=1$

Answer 1

Note that whenever $y = 0$, $dy/dt \ge 0$ (strictly except at the point $(0,0)$ which is an equilibrium).