Logistic Equation

x[t_,x0_,r_]:=1/(1+(1/x0-1) Exp[-r t])

Manipulate[If[fam,Plot[Evaluate@Table[x[t,x0,r],{x0,.01,1,.1}],{t,0,10},ImageSize->{450,300},PlotRange->{0,1},PlotStyle->Table[{Thickness[.007],ColorData["SunsetColors"][i]},{i,.01,1,.1}],AxesLabel->{Style["t",Italic,12],Style["x(t)",Italic,12]}],Plot[x[t,x0,r],{t,0,10},ImageSize->{450,300},PlotRange->{0,1},PlotStyle->{Thickness[.01]},AxesLabel->{Style["t",Italic,12],Style["x(t)",Italic,12]}]],{{x0,.25},0.01,1,Appearance->"Labeled",Enabled->!fam},{{r,1},-2,2,Appearance->"Labeled"},{{fam,False,"show family of curves"},{True,False}},SaveDefinitions->True]

Population growth can be modeled using the logistic equation, ((d x)/(d t))=r x (1-x).

Contributed by: Jeff Bryant