$2$-dimensional Runge-Kutta for system of polynomial ODEs I have just started getting into ODEs, and have come across the Runge-Kutta method for numerically solving them. However, in playing around with them to model hypothetical situations, I came across the equations:
$$\begin{aligned} \dot x &= x - x^2 - y\\ \dot y &= xy - y^2 \end{aligned}$$
I was trying to think of how one would use Runge Kutta methods to do this, and I couldn't figure it out. What is this type of differential equation called, and how can I simulate/solve it?

The Runge-Kutta formulas for a system of differential equations are really the same as for a single equation, it's just that your dependent variable is a vector rather than a scalar.  Write your system as 
$$\dfrac{dX}{dt} = F(t, X(t))$$
where $X = (x, y)$.  If you're using the classical fourth-order Runge-Kutta with step size $h$, your iteration is
$$ \eqalign{K_1 &= h F(t_n, X_n)\cr
            K_2 &= h F(t_n + h/2, X_n + K_1 /2)\cr
            K_3 &= h F(t_n + h/2, X_n + K_2 /2)\cr
            K_4 &= h F(t_n + h, X_n + K_3)\cr
            t_{n+1} &= t_n + h\cr
            X_{n+1} &= X_n + (K_1 + 2 K_2 + 2 K_3 + K_4)/6\cr}$$
