Branching processes with two types of individuals. Assume that each individual can have descendants of either kind; the numbers of descendants of the two types are regulated by two bivariate generating functions P(S1, S2) and P2(S1, S2). We have now two extinction probabilities x, y depending on the type of the ancestor. Show that the pair (x, y) satisfies the equations (6.1) x = P(x, y), y = P2(x, y). Prove that these equations have at most one solution 0 x 1, 0 y I different from (1, 1). The solution (1, 1) is unique if, and only if, "11 1, - 22 1 and (1 11)(122) 1221 where is = ap,(1, 1) as