The title is
Show that $n(t+1)=(1+r)e^{-\alpha n(t)}n(t)$ is equivalent
to $n(t+1)=(1+r)^{1-\frac{n(t)}{K}} n(t)$
where $K=\frac{\ln (1+r)}{\alpha}$
and the question body is
Show that $n(t+1)=(1+r)e^{-\alpha n(t)}n(t)$ is equivalent to
$n(t+1)=(1+r)^{1-\frac{n(t)}{K}} n(t)$ where
$K=\frac{\ln (1+r)}{\alpha}$
I also have to show that $n(t+1)$ will grow when $r<0$
and the initial population is greater than $n=k$.
I have also tried to show this but using both variations
of the equation I get an increasingly negative value.
its only tag is [mathematical-modeling]. As mentioned in comments, if you do plan to repost the question, make sure to improve the question. In addition to adding context, e.g. it might be useful to think of other tags; the problem statement could use a full-stop at the end; it seems that $k=K$; display math mode might look nicer; etc.
PS your edit description “got rid of the sad sub story” could have made it more likely that the post was deleted.