I have the following problem: Consider the predator-prey model: $\frac{du}{dt}=u(1-\alpha(u)-v), \frac{dv}{dt}=\rho(-1-\alpha(v)+u),$
where $\rho$ and $\alpha$ are positive parameters with $0\leq\alpha<0$.
The First bit asks that at the non-trivial fixed point, there is a centre at $\alpha=0$ which changes into a stable point for $0<\alpha<0$. I have shown this. However it then asks to find the range of values for the non-trivial fixed point to be a spiral?
I have that the det(A) at our non-trivial fixed point will be always greater than zero, hence I have to show that $\Delta<0$.
$\Rightarrow {(-\alpha u+\alpha\rho v)^2}-4({\alpha}^{2} \rho uv+u\rho v)<0$, then i tried using the our fixed point (u,v)= $(\frac{1+\alpha}{{\alpha}^{2}+1}, \frac{1-\alpha}{{\alpha}^{2}+1})$ implies u>1 and v<1. Eventually simplifying down i get an expression for $\rho$ and $\alpha$ but I am unable to solve for $\alpha$!
Is this the correct approach to take?
Many thanks in advance.