Recurrence relations with fractions I have the following two equations:
$$\alpha(t)=\frac{a}{b+\beta(t-1)}\\ 
\beta(t)=\frac{c}{d+\alpha(t-1)}$$
where $a,b,c,d$ are constants.
Question is, is there an analytical form for $\alpha$ as $t\to\infty$.
Furthermore, what are the conditions on $a,b,c,d$ for convergence?

The behaviour is easy: $\alpha(t)$ and $\beta(t)$ are both constant (more properly, this is true if you require them to be continuous; without this requirement, each $\alpha(t)$ and $\beta(t)$ can take two different values and convergence can fail). 
If you substitute the second equation into the first one, you get 
$$
\alpha(t)=\frac{a}{b+\frac{c}{d+\alpha(t)}}=\frac{ad+a\alpha(t)}{bd+b\alpha(t)+c}.
$$
This is a degree two equation on $\alpha(t)$, namely 
$$
b\alpha(t)^2+(bd+c-a)\alpha(t)-ad=0.
$$
