DSolve[{y'[x]==k(1- y[x]/L)y[x],y[0]==y0},y[x],x]

y[x]->(((E^(k x)) L y0)/(L-y0+(E^(k x)) y0))/.{k-> 0.05,L->100,y0->5}

G1=Plot[(((E^(k x)) L y0)/(L-y0+(E^(k x)) y0))/.{k-> 0.05,L->100,y0->5},{x,-40,180},PlotRange->{-20,120}]

{x,D[(((E^(k x)) L y0)/(L-y0+(E^(k x)) y0)),x]}

G2=VectorPlot[{1,k(1- y/L)y/.{k-> 0.05,L->100,y0->5}},{x,-40,180},{y,-20,120},PlotLabel->Coarse,VectorPoints->Fine]

Show[{G1,G2}]

G3=Plot[(((E^(k x)) L y0)/(L-y0+(E^(k x)) y0))/.{k-> 0.05,L->100,y0->-5},{x,-40,180},PlotRange->{-20,120}];

G4=Plot[(((E^(k x)) L y0)/(L-y0+(E^(k x)) y0))/.{k-> 0.05,L->100,y0->15},{x,-40,180},PlotRange->{-20,120}];

G5=Plot[(((E^(k x)) L y0)/(L-y0+(E^(k x)) y0))/.{k-> 0.05,L->100,y0->115},{x,-40,180},PlotRange->{-20,120}];

Show[{G1,G2,G3,G4,G5}]