The steady-state equilibrium is $ d^* = \frac{1-p_1-\sqrt{(p_1-1)^2-4p_0p_2}}{2p_2} $. Based on a plot, I guessed the solution $ d_m = d^*(1-e^{-\alpha m}) $, which is pretty close but not correct. What other method can I use to solve this?
Context: This is to calculate the extinction probability of a branching process at each step, as in http://en.wikipedia.org/wiki/Branching_process#Example_of_extinction_problem.