Logofet D. O.

Laboratory of Mathematical Ecology, IFARAN, Pyzhevsky Per. 3, Moscow, 109017, Russia logofet@omega.ifaran.ru

If the stages of succession are defined and a scheme is known of transitions
among them in the course of succession, then a simple direct way to formalize
the geobotany knowledge available is provided by a Markov chain whose states
are identified with the stages of succession, while the transition probabilities
are estimated from empirical evidence. As in physics, a Markov description
of the process does not reveal any internal cause-effect mechanisms that
each transition is determined by, but gives rather an external description
for the course of the process in terms of probabilities of its states.
In this sense, a Markov description represents the “phenology” of succession.

When the transition probabilities are estimated as constant numbers (and the chain is thereafter called homogeneous), it means that the invariance hypothesis is adopted, either implicitly or explicitly, concerning the terms of environment which determine the course of succession. Then the property, in common of homogeneous models, to converge to a stable limit distribution conforms the main paradigm of succession theory, namely, a regular movement from pioneer stages to the (poly)climax one. The time scale of succession is however tens and hundreds of years (for terrestrial ecosystems), during which certain changes do happen in the environment, i.e. the invariance hypothesis is implausible. This is perhaps a reason why the homogeneous Markov chains, while being an excellent didactic means, have not become nevertheless into a tool for reliable ecological predictions.

A move “towards causation” is made in a new generation of Markov-chain models, i.e. inhomogeneous chains where transition probabilities are designed as functions of key environmental factors that determine the course of succession. A technique has been developed in the case study on overgrowing of spoil-banks formed by an open ore-minery in the forest steppe zone (Kursk Magnetic Anomaly). The notion of adaptation range is defined for vegetation of the given zone, in terms of the temperature (T) and moistening (H) factors and the range is found by statistical routines from time series of weather data. Constructing the matrix of transition probabilities P(T, H) as a function defined over the adaptation range, has required one to adopt certain empirical generalization concerning the effects the factors render upon the course of succession, as well as to prove that the backward problem of recovering the transition matrix from observed values of stages' duration is mathematically correct.

The inhomogeneous model loses the algebraic elegance of homogeneous chain analysis, but its predictions now gain sensitivity to climate indices. Given scenarios of how T and H vary with time, the inhomogeneous model is now able to answer questions of practical importance concerning the fate of a particular territory. Its inner parameter, the probability of “steppe or forest” bifurcation, can be used, for instance, in global biosphere models to describe shifts in the bioclimatic zones in response to climate changes.