In order to sketch following question what I have done so far: $ lim_{Nt->0} F(N_{t})= 0 $ and $ lim_{Nt->\infty} F(N_{t})=r$ (For the case of s=0). Now since this a discrete time graph I am not sure if the derivative features can be used? My intuition is that you can't since the stationary points can't be find in that way but rather they are found by setting F(Nt)=0. Therefore, my question is what are the steps to go about sketching discrete time graphs?
It is just a qualitative behavior of $f$, to have an idea on how the population will behave in the next time step, so you can omit the details about the discrete nature of $t$, $N_t$ in itself is a continuous variable
In the limit $N_t \to 0$, $f(N_t)$ also goes to zero, independent of $s$ (apply L'Hopital's rule), and for large populations $f(N_t)\to r$