t0=3 ; t1=150 ;m=8;
a[0][t_]=0.8;
Do[ loesung=NDSolve[ {y'[t]==2 a[i][t]/(1+a[i][t]^m) - y[t]
                      , y[0]==a[i][t0]},
             y, {t,0,t0}
           ];
    a[i+1][t_]=y[t]/.loesung[[1]]
,  {i,0,Quotient[t1,t0]-1}
  ];
c[t_]:= a[ Quotient[t,t0] ] [Mod[t,t0]];
bild=ParametricPlot[{c[t],c[t-t0]},{t,t1/2,t1},
PlotRange->All,ImageSize->{400,300},
AxesLabel->{"c(t)","c(t-3)"}];