Finding the Probability of Extinction: Error Analysis and Optimal Use of Markovian Population Models

The use of Markovian transition matrices has proved to be a flexible way of finding the probability of extinction for various discrete-time models of population growth. However, it requires handling of large matrices and is thus time-consuming. Two approaches are available to keep the computing effort limited: the necessary introduction of an upper limit to the population, and the optimal reduction of the transition matrix by grouping several states of the population together. Both methods bring about errors to the solution of the extinction probability. Here, two methods are presented, by which the error associated to a chosen matrix can be assessed without knowledge of the true solution. The results indicate that while grouping allows the solution of a much wider range of problems with an equal amount of computing effort, it should not be used without assessing the possibility of an erroneous approximation with applying both error analysis methods presented here.