  This paper develops ideas from a previous paper described as `an appetizer
for non-linear Wiener--Hopf theory', but is completely independent of that
paper. It again considers only the simplest possible case in which the
underlying motion of the branching particles is described by a two-state Markov
chain. Key generating functions provide solutions of a simple two-dimensional
dynamical system, and the main interest is in the way in which Probability
Theory and ODE theory complement each other. ODE pictures convey rather
strikingly a simple phase transition. No knowledge of either ODE theory or
Wiener--Hopf theory is assumed.
