Find (a,b) Param Space for $\dot{x}, \dot{y}$

[Dustinsfl]

How do I find the (a,b) parameter space for

\begin{align}
\dot{x}=& -x+ay+x^2y\\
\dot{y}=& b-ay-x^2y
\end{align}

 

[gtoguy97]

The parameter space for this system of differential equations is the set of all (a,b) pairs for which the system is meaningful. To find this parameter space, we can use the characteristic equation of the system to determine for which values of a and b the system has a unique solution.The characteristic equation is given by$$\lambda^2+a\lambda+b-a=0$$This equation has two solutions given by$$\lambda_1=\frac{-a+\sqrt{a^2-4(b-a)}}{2} \quad \text{and} \quad \lambda_2=\frac{-a-\sqrt{a^2-4(b-a)}}{2} $$We can then determine the parameter space by setting the discriminant of the characteristic equation, $a^2-4(b-a)$, to be greater than or equal to zero. This gives us the following region in the (a,b) parameter space:$$\{(a,b) \mid a^2-4(b-a) \geq 0\}$$