steady states and stability mathematical biology

Question

could anyone help on how to do this please?

find the steady states of $$F(N)=\frac{rN}{1+N}.$$

do you differentiate and set it equal to zero and find $N$?

many thanks in advance.

Answer 1

A steady state is one which stays the same when put into the function. So you want $N^*$ satisfying $$ \frac{rN^*}{1 + N^*} = F(N^*) = N^*,$$ or $(1+N^*)N^* = rN^*$. So $N^*$, your steady state, is a root of the quadratic $x^2 + (1-r)x = 0$. Then your two solutions are $x=0$ (which corresponds to a population of size $0$) and $x = r-1$.