I have been asked to investigate the stability of a steady state to the following equation:
$\frac{\partial U}{\partial t} = D \frac{\partial^2 U}{\partial x^2} + rU (1- \frac{U}{K}) - EU$
with boundary conditions $U=0$ on $x=H$ and $\frac{\partial U}{\partial x}=0$ on $x=0$ where the constants are positive.
I'm told to perturb by setting $U(x,t)=\epsilon U_1(x)e^{\lambda t} + O(\epsilon ^2)$
But I cannot get to a solution for $U_1$ that is real, nor can I find a $\lambda$ that makes sense. I have reached the point of finding $U_1=0$, but I assume the question style suggests I should not be having to resort to higher order terms to determine stability, hence my confusion.
I would very much appreciate hints in the right direction!