f[
        old_] := {old[[1]], (old[[1]] old[[2]] (1 - 
                old[[2]]) )};g[α_] := 
      Take[NestList[f, {α,  .1}, 1000], -500];l = 
      g[2.8];Do[
      l = Join[l, g[2.8 + 1.2/500 i]], {i, 
        500}];

ListPlot[l, Axes -> False,  Frame -> True,  
      PlotStyle -> {PointSize[ .001]}]