Manipulate[h[x_]:=c x Exp[-x]-hh;ListPlot[Table[{t,Nest[h,1.0,t]},{t,0,time}],PlotRange->{{0,15},{.5,2.25}},PlotStyle->PointSize[.02],Frame->True,FrameLabel->{"Time, t","x"},LabelStyle->Directive[FontSize->14],PlotStyle->{Black}],{time,1,20,1},{{c,4,c},4,6,.01,Appearance->"Labeled"},{{hh,0,h},0,.72,.01,Appearance->"Labeled"}]

sol[c_,h_]:=x/.FindRoot[x==c x Exp[-x]-h,{x,2}]

Plot[{If[h<.72,sol[4,h],If[h>.725,0,Null]],Log[4]-(1/(Log[4])) h,Log[4]-(1/(Log[4])) h+((Log[4]-2)/(2 ((Log[4])^3))) (h^2)},{h,0,1},PlotStyle->{Black,Blue,Green},PlotRange->{-.1,1.5},Frame->True,FrameLabel->{"Harvesting rate, h","Limiting population size"},LabelStyle->Directive[FontSize->14]]

Export["1.eps",%129,"EPS"]