Consider the model:
$y_{n+1}=ry_n(1-\frac{y_n}{k}); r>0$
a)Show that $y_{n+1}<0$ if and only if $y_n>k$.
b)Show that $y_{n+1}>k$ is possible with $0<y_n<k$ only for $r>4$.
c)What conditions on $y_0$ are necessary and sufficient to guarantee $y_n>0$ for n=1,2,3...
I block in question (c), I was unable to determine the condition.
I await your explanation.
Assuming a), b) are correct,
by b) if $r\le 4$ then given $y_0\in (0,k)$ we have $y_n\in (0,k)$ for all $n$.
If $r>4$ then it is possible that $y_{n+1}>k$ even if $y_n\in (0,k)$ for some $n$, and by a) it then implies that $y_{n+2} <0$. So $r>4$ then postivity is not guranteed.
Hence necessary and sufficient condition is $r\le 4$, and $y_0 \in (0,k)$
