Markovian Trees Subject to Catastrophes: Would They Survive Forever?
We consider multitype Markovian branching processes subject to catastrophes which kill random numbers of living individuals at random epochs. It is well known that the criteria for the extinction of such a process is related to the conditional growth rate of the population, given the history of the process of catastrophes, and that it is usually hard to evaluate. We give a simple characterization in the case where all individuals have the same probability of surviving a catastrophe, and we determine upper and lower bounds in the case where survival depends on the type of individual. The upper bound appears to be often much tighter than the lower bound.
Key wordsbranching processes matrix-analytic methods catastrophes extinction criteria Lyapunov exponent
This work was subsidized by the ARC Grant AUWB-08/13–ULB 5 financed by the Ministère de la Communauté française de Belgique. The first author also gratefully acknowledges the support from the Fonds de la Recherche Scientifique (FRS-FNRS) and from the Australian Research Council, Grant No. DP110101663.
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