system of iterated equations

daniel6874

The following came up in the context of a genetics question. With the convention that a-e on the LHS have subscript _n+1 and on the RHS a-e have subscript _n,

a = 2ad + bd
b = 2ae + bd + be + 2cd + ce
c = be + ce
d = 2ad + 2ae + bd + be + ce
e = bd + be + 2cd + ce

If the initial values for the variables is a=b=d=e=1, c=0 does the ratio
e/(a+b+c+d+e) converge as n gets large? Can the ratio be calculated without a computer?
Mathematica gives that e/(a+b+c+d+e) is about 0.109 for n=20, FWIW.