Consider two sequences described below: $$\alpha_{t+1} = (1-\beta_t^2)\alpha_t,$$ $$\beta_{t+1} = (1-C \alpha_t\alpha_{t+1})\beta_t,$$ where $C>0$ and we know $0 <\beta_1<1$ and $0<\alpha_1<1$. Determine $\alpha_t$ and $\beta_t$ as a function of $C$ and $\beta_1$ and $\alpha_1$ for all $t \geq 2$ (if doing so is possible).
The answer to the following simpler task is sufficient in my case: what value $\alpha$ and $\beta$ converge to and with what rate?
Intuitively, I think $\alpha$ should converge (nearly) linearly to $0$ but $\beta$ should converge sublinearly to a nonzero quantity. On the other hand, this is also reminiscent of the fixed-point iteration which may help with showing the convergence.