The use of Markovian metapopulation models: Reducing the dimensionality of transition matrices by self-organizing Kohonen networks

Abstract

Markovian population models are used in conservation biology to find an accurate estimate of a population's extinction probability. Such models require handling of large transition matrices and calculations are thus extremely time-consuming when large populations have to be studied. To accomplish these problems, some authors have suggested to group together several states/sizes of the population. Unfortunately, this so-called binning frequently results in errors in estimates obtained. The main problem with binning is that it assumes that grouped states behave nearly identical with respect to the underlying stochastic population process and that so far binning methods implicitly violate this assumption. In this paper, we present a new binning method based on self-organizing Kohonen neural networks for time-homogeneous Markovian metapopulation models. The neural networks are used to analyse one-step transitions of the Markov chain in order to group only nearly identical states. We show that the new method is more qualified for the use in conservation than deterministic methods that we had discussed in a previous paper (first order Fibonacci binning, pairs binning). It reveals more accurate and more reliable estimates than these methods. Errors in estimated extinction probabilities that were introduced by binning did not exceed one order of magnitude and errors in the global population size did not exceed 30%. These errors in estimates correspond to a low inaccuracy in model parameter values of population growth and migration ranging from ±1 to ±10%. The reduction in the state space of the studied metapopulations ranged from 21 to 33% per subpopulation. The resulting decrease in computing time caused by our binning method is substantial particularly with regard to simulations tasks such as comparing the extinction risk of several populations or performing a detailed sensitivity analysis for model parameters assumed. The successful estimation of the extinction risk of eight natural butterfly populations demonstrates the applicability of our new binning method in conservation biology. A comparison of extinction probabilities and mean population sizes estimated by Monte Carlo simulations [Griebeler, E.M., Seitz, A., 2002. An individual based model for the conservation of the endangered Large Blue butterfly, Maculinea arion (Lepidoptera: Lycaenidae). Ecol. Model. 134, 343–356] with those obtained from the respective Markov chains for these butterfly populations revealed similar results on the accuracy of estimates and the reduction in transition matrices that were predicted by the comparative error analysis for binning methods.