Determine the conditions on $a$ and for which the nonzero equilibrium value of $b$ $x_{n+1}=ax_ne^{-bx_n}$ is stable, where $a$ and $b$ are positive constant.
I have learnt the condition of stability of system of difference equations, but I have no idea on this problem and I can not express $x_n$ in terms of n, could you please give me some hints? Thank you.
You've asked for some hints so I would start by considering different ranges for b (major clue: $b>0$ & $b<0$ due to the significance of the exponential sign in stability in infinite series)
Then consider what would happen if you kept calculating $x_n$ and what conditions you would have to apply for $a$ so that $|x_n|<\infty$