  Recently, a first step was made by the authors towards a systematic
investigation of the effect of reaction-step-size noise - uncertainty in the
step size of the reaction - on the dynamics of stochastic populations. This was
done by investigating the effect of bursty influx on the switching dynamics of
stochastic populations. Here we extend this formalism to account for bursty
reproduction processes, and improve the accuracy of the formalism to include
subleading-order corrections. Bursty reproduction appears in various contexts,
where notable examples include bursty viral production from infected cells, and
reproduction of mammals involving varying number of offspring. The main
question we quantitatively address is how bursty reproduction affects the
overall fate of the population. We consider two complementary scenarios:
population extinction and population survival; in the former a population gets
extinct after maintaining a long-lived metastable state, whereas in the latter
a population proliferates despite undergoing a deterministic drift towards
extinction. In both models reproduction occurs in bursts, sampled from an
arbitrary distribution. In the extinction problem, we show that bursty
reproduction broadens the quasi-stationary distribution of population sizes in
the metastable state, which results in an exponential decrease of the mean time
to extinction. In the survival problem, bursty reproduction yields an
exponential increase in survival probability of the population. Close to the
bifurcation limit our analytical results simplify considerably and are shown to
depend solely on the mean and variance of the burst-size distribution. Our
formalism is demonstrated on several realistic distributions which all compare
well with numerical Monte-Carlo simulations.
