w[c_,t_]=a/(1+(a/w0 -1) E^(-c t));
a=100;
w0=10;
abb=Plot[Evaluate[Table[w[c,t],{c,0.2,1,0.2}]],{t,0,10},
PlotRange->{0,120},AxesLabel->{"t","x"},ImageSize->{400,300}];