fixed points of a system of ODEs

Question

I have two equations

$\frac{dx}{dt}=−x + a(y) + x^2$ and $\frac{dy}{dt}=b-a(y)-x^2(y)$

I solved the first equation for $y$ to be $y=\frac{x}{a+x^2}$ and plugged that in for my $y$ in the second equation and set that to 0. I am confused as to the next step.

Answer 1

You can also try setting

$$ \frac{dx}{dt} = 0, \quad \frac{dy}{dt} = 0 $$

and solve for corresponding values of $x$ and $y$ where they are $0$. These are the fixed points of the 2D dynamical system.

So one trivial fixed point of this $(x, y) = (0, 0)$ when $b = 0$. It may also help to plot this using pplane.