Reduced Dimensional Population Transition Matrices: Extinction Distributions from Markovian Dynamics

Population transition matrices yield a full distribution of extinction times, which is valuable in decision-making for the conservation of small populations. However, matrices of even moderate dimension have proved computationally impractical. A scheme based on a generalization of the Fibonacci series is developed whereby integer population numbers (0, 1, 2, 3,...) can be collapsed in a many-to-one fashion onto an integer series of fewer terms. This homomorphic mapping permits a reduction in dimensionality of matrix models of stochastic population processes, thereby greatly lowering associated computations. Error contributed by reduction in dimensionality is investigated with general models of how growth rate and variance in growth rate change with population size.