  We consider a broad class of continuous-time two-type population
size-dependent Markov Branching Processes. The offspring distribution can
depend on the current (alive) and total (dead and alive) populations. Using
stochastic approximation techniques, we show that the time-asymptotic
proportion of the populations either converges to the equilibrium points or
infinitely often enters every neighbourhood and exits some neighbourhood of a
saddle point of an appropriate ordinary differential equation with a certain
probability (almost surely for the process with attack and proportion-dependent
branching process). The result holds under finite second-moment conditions. We
also show that certain normalized trajectories of the embedded chain almost
surely converge to the solution of the ordinary differential equation uniformly
over any finite time window as time progresses. In addition to extending the
analysis of several existing BPs, we analyze two new variants: BP with attack
and acquisition, and BP with proportion-dependent offspring. Using these
results, we study competition in viral markets and fake news control on online
social networks.
