Manipulate[f[x_]:=c x (1-x);ListPlot[Table[{t,Nest[f,xo,t]},{t,0,time}],PlotRange->{{0,20},{0,1}},PlotStyle->PointSize[.02],Frame->True,FrameLabel->{"Time, t","(Subscript[x, t])"},LabelStyle->Directive[FontSize->14],PlotStyle->{Black}],{time,1,20,1},{{c,1.5,(Subscript[R, max])},{1.5,2,2.8,3,3.2}},{{xo,1/4,(Subscript[x, 0])},{1/8,1/4,1/2,5/8,7/8}}]