Hard harvesting of a stochastically changing population

We consider a population whose size changes stochastically under a branching process, with the added modification that each generation a fixed number of individuals are removed, irrespective of the size of the population. We call removal that is independent of population size 'hard harvesting'. A key feature of hard harvesting occurs if the size of the population is smaller than the fixed number that are harvested. In such a case, the dynamics cannot continue and must terminate. We find that even for populations with a tendency to grow, there is a finite probability of termination. We determine the probability of termination, and given that termination occurs, we characterise the statistical properties of the random time to termination. We determine the impact of hard harvesting on the size of the population, in populations where termination has not occurred.