Manipulate[f[x_]:=(A^2)+2 A (1-k) Sqrt[x]+(((1-k))^2) x;ListPlot[Table[{t,Nest[f,2,t]},{t,0,time}],PlotRange->{{0,20},{0,20}},PlotStyle->PointSize[.02],Frame->True,FrameLabel->{"Time, t","P"},LabelStyle->Directive[FontSize->14],PlotStyle->{Black}],{time,1,20,1},{{A,0.5},{0.5,0.75,1.0}},{{k,1/4},{1/4,1/2,3/4}}]