Convergence of Ratios in Genetic System of Iterated Equations

[daniel6874]

The following came up in the context of a genetics question.
\begin{align*}
a_{n+1} &= 2a_{n}d_{n} + b_{n}d_{n}\\
b_{n+1} &= 2a_{n}e_{n} + b_{n}d_{n} + b_{n}e_{n} + 2c_{n}d_{n} + c_{n}e_{n}\\
c_{n+1} &= b_{n}e_{n} + c_{n}e_{n}\\
d_{n+1} &= 2a_{n}d_{n} + 2a_{n}e_{n} + b_{n}d_{n} + b_{n}e_{n} + c_{n}e_{n}\\
e_{n+1} &= b_{n}d_{n} + b_{n}e_{n} + 2c_{n}d_{n} + c_{n}e_{n}
\end{align*}
If the initial values for the variables is $a_0=b_0=d_0=e_0=1, c_0=0$ does the ratio
$e_{n}/(a_{n}+b_{n}+c_{n}+d_{n}+e_{n})$ converge as $n$ gets large? Can the ratio be calculated without a computer?
Mathematica gives that $e_{n}/(a_{n}+b_{n}+c_{n}+d_{n}+e_{n})$ is about $0.109$ for $n=20$, FWIW.

 

[fresh_42]

I think the computer is the preferred choice here. If you plot the ratio versus $n$ you should see whether there is a monotone behavior or not. Handling it with algebra is a mess, except you have a better description for the genetic algebra this might define.