Consider the system
$\frac{dx}{dt}$=$2-(b+1)x+ax^{2}y$
$\frac{dy}{dt}$=$bx-ax^{2}y$
For what values of a and b does a Hopf Bifurcation occure, given that a and b are positive, and in the region x,y ≥0
I found that the system had only one fixed point (2,$\frac{b}{2a}$) and computed the jacobian at that point. Should I be looking for eigenvalues here and figuring out for what values of a, b different fixed points occur?
You should find that the Jacobian of this system has two complex eigenvalues (which are complex conjugates of each other) at the fixed point which you have correctly found. A Hopf bifurcation occurs when a fixed point loses stability. That means it happens when the real part of the eigenvalues goes from negative (stable) to positive (unstable). Since you have the eigenvalues as a function of $a$ and $b$, you can calculate a relationship between $a$ and $b$ when that real part is zero, which is where the Hopf bifurcation occurs.