Manipulate[
Module[{T = 50, sol, sold},
sol = First[y /. NDSolve[{y'[t] == a y[t](1 - y[t]), y[0]== 0.1}, y,{t,0,T}]];sold = First[y /. NDSolve[{y'[t] == a y[t](1 - y[t - 1]), y[t /; t ≤ 0] == 0.1},y,{t,0,T}]];
 ParametricPlot[{{sol[t],sol[t - 1]},{sold[t],sold[t - 1]}},{t,1,T}, PlotRange->{{0,3},{0,3}}, PlotStyle->{Gray, Blue}]],{{a,1},0, 2}]