The Markov branching process with density-independent catastrophes I. Behaviour of extinction probabilities

Source abstract

Let (X t : t ≥ 0) be the Markov branching process (MBP) with a density independent catastrophe component. It is denned to be the Feller process on the non-negative integers having the generator Here {p j } is the offspring distribution which satisfies p 1 = 0 and p 0 < 1, ρ is the per capita birth rate, κ is the rate of occurrence of catastrophe events, {δ j : j ≥ 0} is the decrement distribution and . Thus X t can be interpreted as the size of a population in which individuals reproduce according to the rules of a MBP – see Athreya and Ney[1], chap, III – and where there is an external and independent Poisson process of catastrophe events, κ per unit time, and if j < i each such event reduces the population size by j with probability δ j . Usually we assume that δ 0 = 0 on the basis that a catastrophe always reduces the population size. Let f(s) = Σp j s j and assume that . This ensures that the MBP obtained by setting κ = 0 is regular ([1], p. 105) and hence (X t ) is the unique Markov process corresponding to the above generator when κ > 0.