Using long-term direct observations in a Polytrichum-Myrtillus pine forest, we have constructed and verified a homogeneous Markov chain model for two dominant species (Vaccinium myrtillus and V. vitis-idaed) at the late stages of succession. The sampling design features a large sample size (2000 quadrats) on permanent transects, several re-examinations with the interval of 5 years, and the use of species rooted frequency. As a model of the process under concern, the discrete Markov chain accounts for the following four states: both species being absent on the quadrat, one of them being present alone, and the joint presence of the both; the model time step coincides with the time interval between observations. The model is calibrated on the data of two successive examinations and verified on that of one more examination. 

All possible transitions between the states are revealed to realize in quadrats for one time interval, as well as the absence of transitions at each state, which results in the complete digraph (directed graph) of transitions. Major model results are obtained by the formulae of finite Markov chain theory: the steady-state square distribution, cyclicity characteristics, and the mean durations of stages in the fine-scale dynamics. As a steady-state (stable) outcome of succession, the distribution among quadrats is expected where 30% of quadrats are occupied by V. myrtillus alone, 11% by V. vitis-idaea alone, both species are present on 18% of quadrats, and 41% of quadrats are 'empty'. This demonstrates a possibility for V. myrtillus and V. vitis-idaea to coexist stably at the latest stages of succession, with the clear predominance of V. myrtillus, yet without competitive exclusion. The quantitative characteristics of cyclicity and the durations of stages in the fine-scale dynamics enable us to estimate the total duration of secondary post-fire succession as about 45 years (to reach a distribution of states that differs less than 5% from the steady-state one). Out of the four states specified, the quadrats with V. vitis-idaea alone persist for the least time (8 years) on the average, while 'empty' ones persist for the greatest time (18 years). Forecasting the dynamics for one model time step forward and comparing the forecast with the real square distribution have revealed the measure of difference to be 5.4%. This illustrates the efficiency of the (time-)homogeneous Markov chain as a short-term forecast tool, yet leaves open the question whether the homogeneity hypothesis be true in the longer term.
Forests Markov Chains