The VRP-2 dynamics, with the parameter alpha value approaching the value equal to the value of the dimensionless fine-structure constant, features maximum variety with regard to the number of bifurcations with the minimum degree of chaosticity. 
D. B. Volov Specific behavior of one chaotic dynamics near the fine-structure constant http://arxiv.org/abs/1205.6091
http://chaosandcorrelation.org/Chaos/DV_1_5_2012.pdf
Volov_D__VTP_5_2011.pdf

http://www.sciteclibrary.ru/rus/catalog/pages/11612.html
This files contains an articles describing the Verhulst-Ricker-Planck dynamic and its relation to the fine structure constant.
The MathCAD's text program for the bifurcation diagram "four rats" (D.B.Volov, Russia, Samara).

Diagram "four rats" 
 
Source code in MATLAB to reproduce the bifurcation diagram "four rats" (A.P.Trounev, Toronto, Canada)
(smile must be replaced by a colon)
 

x(i+1)=-L(k)/(x(i)^2(exp(x(i))+alpha))

L=zeros(1,500);

y=zeros(1,128);
ly=zeros(1,128);
alpha=1/137.035999074;
dL=(3.7+exp(1.1989))/500;

for k=1:length(L)

    for i=1:length(y)

        if (k==1)

            L(k)=-3.8;

            y(k,=1;
            ly(k,=log(y(k,);

        else

            if (i==1)

                L(k)=L(k-1)+dL;
                V=y(k-1,128)*y(k-1,128)*(exp(-y(k-1,128))+alpha);

                y(k,i)=exp(L(k))/V;
               ly(k,i)=log(y(k,i));

            else
                V1=y(k,i-1)*y(k,i-1)*(exp(-y(k,i-1))+alpha);
                y(k,i)=exp(L(k))/V1;
                 ly(k,i)=log(y(k,i));

            end

        end

    end
end



for i=1:128

    plot(L,ly(:,i),'.k')

    hold on

    title('a=1/137');

    xlabel('lnK');ylabel('lnx');

end
The result: http://chaosandcorrelation.org/AS/4rat.gif
 The bifurcation diagram "four rats" one-dimensional dynamic. 
Code for the Wolfram Mathematica 8 & result (A.P.Trounev, Toronto, Canada):