The mathematics of Markov models: what Markov chains can really predict in forest successions

Abstract

The formalism of Markov chains is presented as a mathematical description of a succession process with a known scheme of successional transitions and their time characteristics. The basic hypotheses and assumptions behind the formalism are formulated and discussed from the viewpoint of the prognostic potential in the resulting models. Fundamental mathematical results from the theory of stochastic processes guarantee the existence of a stationary probability distribution of states in any finite, regular, time-homogeneous Markov chain, and any initial distribution of chain states does converge to some stationary distribution, matching the paradigm of classical succession theory. We propose a general method for estimation of time-homogeneous transition probabilities applicable for any kind of successional scheme, yet with strong requirements to the expert data: average duration times should be known for each specified stage of succession as well as the likelihood proportions among the transitions from the ramifying states of the scheme. Constructed by this method on a single set of data, the discrete- and continuous-time models are proved to converge to the identical distributions, while the average sojourn times in each state are shown to be also the same in the two models. Succession through forest types in a mixed (coniferous–deciduous) forest in Central Russia is considered as an example.