A 2 To solve this recurrence relation, it is convenient to write the following expansion: F n ~ ( s ) ≃ f n − s b n. A 3 Then F n ~ ( s = 0 ) = f n yields the extinction probability.
To proceed further, we substitute (A 3) into (A 2): ( s n + λ + ϕ ) ( f n − s b n ) = ϕ ( f n − 1 − s b n − 1 ) + λ ( f n + 1 − s b n + 1 ).