continuous-time Markov branching
The Galton-Watson process advances in discrete generations, but real populations reproduce in continuous time: individuals live exponentially-distributed lifetimes and, at death, are replaced by a random number of offspring. The continuous-time Markov branching process is the natural continuous-time analogue, and the simplest and most important special case is the linear birth-death process, where each individual independently gives birth at rate lambda and dies at rate mu.
Each individual lives for an Exponential(a) time, independent of all others, and at the end of its life produces k offspring with probability p_k (so a is the total event rate and the offspring law (p_k) governs what happens at each branching event). Because lifetimes are memoryless, the total population Z_t is a continuous-time Markov chain on the nonnegative integers, with absorbing state 0. Its generating function F(t, s) = E[s^(Z_t)] solves the nonlinear backward Kolmogorov equation dF/dt = u(F), where u(s) = a (f(s) - s) is the branching mechanism and f is the offspring pgf; this is the continuous-time shadow of the discrete composition rule. The mean grows exponentially, E[Z_t] = e^(alpha t) with Malthusian parameter alpha = a (m - 1) = u'(1), and the trichotomy reads alpha < 0 (subcritical), alpha = 0 (critical), alpha > 0 (supercritical). For the pure linear birth-death process (birth rate lambda, death rate mu per individual) one can solve explicitly: extinction probability is min(1, mu/lambda), E[Z_t] = e^((lambda - mu) t), and conditioned on survival the population grows geometrically with an explicit limit law.
Why it matters: continuous-time branching is the backbone of stochastic models in epidemiology (the early phase of an outbreak is a birth-death branching process, with R_0 = lambda/mu), population genetics, queueing (the M/M/infinity and related systems), and the dynamics of clonal cell populations and tumours. It is also the bridge to continuous-state branching via a scaling limit. The honest cautions: exponential lifetimes are an idealization — general lifetime distributions give the more flexible but harder Crump-Mode-Jagers (general branching) processes, which are no longer Markov in the population count alone; and explosion (infinitely many births in finite time) can occur if birth rates grow too fast with population, which the standard linear model excludes by keeping per-capita rates constant.
A linear birth-death process with birth rate lambda = 2 and death rate mu = 1 per individual is supercritical (alpha = lambda - mu = 1 > 0). Its extinction probability is mu/lambda = 1/2, and on the survival event the population grows like e^t times a random factor. This is exactly the textbook model for the early, before-saturation phase of an epidemic with R_0 = 2.
Memoryless lifetimes make the population count a Markov chain; the Malthusian rate alpha = a(m-1) sets criticality.
Exponential lifetimes are essential for the Markov property of the population count; general lifetimes give non-Markov Crump-Mode-Jagers processes. With population-dependent rates that grow too fast, the process can explode (infinitely many events in finite time).