A Realistic Model for Population Dynamics
Author's Data: Housam Binous
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :
Varma and Morbidelli, Mathematical Methods in Chemical Engineering, Oxford University Press, 1997.

Bifurcation diagram

(Subscript[f, r])_[x_] := x Exp[r (1 - x)]

tbl[r_] := 
      Transpose[{Table[r, {102}], 
          Drop[NestList[(Subscript[f, r]), 0.1, 1001], 900]}]

plt1 = 
        ListPlot[Flatten[Table[tbl[r], {r, 0, 5,  .1}], 1], 
          PlotStyle -> {RGBColor[1, 0, 0], PointSize[0.02]}, 
          PlotRange ->  All];

A cycle of Period 3 appears at r≈3.1024

ListPlot[Drop[NestList[(Subscript[f, 3.1024]), 0.2, 200], 100], 
        PlotJoined -> False, 
        PlotStyle -> {RGBColor[1, 0, 0], PointSize[0.02]}];

Chaotic behavior appears at r>3.1024

ListPlot[Drop[NestList[(Subscript[f, 3.5]), 0.2, 200], 100], 
        PlotJoined -> False, 
        PlotStyle -> {RGBColor[1, 0, 0], PointSize[0.02]}];