n[t]/.DSolve[{n'[t]==r(1-((n[t])/k))n[t],n[0]==n0},n,t]

Plot[(((E^(r t)) k n0)/(k-n0+(E^(r t)) n0))/.{r->2,k->10,n0->15},{t,0,5},PlotRange->{0,All}]

D[r n((1-(n/k) )/(1+c((n )/k))),n]

-((c n (1-(n/k)) r)/(k (((1+((c n)/k)))^2)))-((n r)/(k (1+((c n)/k))))+(((1-(n/k)) r)/(1+((c n)/k)))/.n->0

-((c n (1-(n/k)) r)/(k (((1+((c n)/k)))^2)))-((n r)/(k (1+((c n)/k))))+(((1-(n/k)) r)/(1+((c n)/k)))/.n->k

n+9 /.n->0

(* plot per capita of Smith equation *)
Plot[r ((1-(n/k) )/(1+c((n )/k)))/.{r->2,c->4,k->10},{n,0,12}]

(* use NDSolve[] for time solution, i.e. numerically solve *)
NDSolve[]