Given: $$ k_{r+1} = k_r \exp(-k_r\tau_r) \\ \tau_{r+1} = -\frac1{k_r}\ln\left(1-\frac{b}{k_r \sum_{i=0}^{r}\tau_i}\right) \\ $$ And that $k_0, \tau_0, b$ are constants, is there a closed form representation of $k_r, \tau_r, \forall r \in \mathbb{Z}^{+}$? And can $\tau_0$ be represented in terms of $k_0$?
Edit: after calculating the sequence values computationally it appears that as $r\to\infty$, $k_r \to \infty, \tau_r\to\infty$, but the value of $e^{-k_r\tau_r} $ actually converges towards a limit. Why is this? Can this value be expressed in terms of $b, k_0, \tau_0$?