FP =FullSimplify[Solve[r*x*(1-x/k)==x^2/(1+x^2),x]]

Manipulate[
Plot[{r*(1-x/k),x/(1+x^2)},{x,0,20},PlotRange->{0,1}],
{{k,3},1,40},{{r,0.5},0,1}]

krofx=Solve[{r*(1-x/k)==x/(1+x^2),D[r*(1-x/k),x]==D[x/(1+x^2),x]},{k,r}]

kcVSrc =Flatten[{k/.krofx,r/.krofx}]

ParametricPlot[kcVSrc,{x,1.01,100},PlotRange->{{0,40},{0,1}},AspectRatio->1,Frame->True,FrameLabel->{"k","r"}]

Manipulate[
s=NDSolve[{x'[t]==r*x[t]*(1-(x[t]/k))-x[t]^2/(1+x[t]^2),x[0]==initial},x,{t,0,30}];
LogPlot[Evaluate[x[t]/.s],{t,0,30},PlotRange->{0.1,50},Frame->True,FrameLabel->{"Time","Population size x(t)"}],
{k,1,40},{r,0,1},{initial,0,50}]