Analyzing the Fine-Scale Dynamics of Two Dominant Species in a Polytrichum–Myrtillus Pine Forest. I. A Homogeneous Markov Chain and Cyclicity Indices

Abstract

Long-term direct observations in a Polytrichum–Myrtillus pine forest are used to construct and verify a homogeneous Markov chain model of fine-scale dynamics of two dominant species (Vaccinium myrtillus and V. vitis-idaea) at the late stages of succession. The sampling design featured a large sample size (2000 quadrats) on permanent transects, re-examinations with an interval of 5 years, and the use of species rooted frequency. The discrete Markov chain used to model the process under consideration consists of four states: an absence of both species from the quadrat, the presence of one species of the two, and the joint presence of both species. The time step of the model coincides with the interval between the observations. The data of two successive examinations are used to calibrate the model, and the data of one additional examination are used to verify the model. All possible transitions between states, as well as the absence of transitions from each specific state, were revealed in quadrats within one time interval, and this resulted in a complete digraph (directed graph) of transitions. The main results of the model obtained from the formulas of finite Markov chain theory included steady-state square distribution, cyclicity characteristics, and the mean durations of stages in the fine-scale dynamics. The distribution among quadrats expected for the steady-state (stable) outcome of succession was the following: 30% of quadrats were occupied by V. myrtillus alone, 11% by V. vitis-idaea alone, 18% by both species, and 41% were “empty.” This demonstrates the possibility of stable coexistence of V. myrtillus and V. vitis-idaea at the latest stages of succession with a clear predominance of V. myrtillus, yet without competitive exclusion. The quantitative characteristics of cyclicity and the durations of stages in finescale dynamics enabled us to estimate the total duration of secondary postfire succession (until the attainment of a state distribution that differed from the steady-state one by less than 5%) at about 45 years. Of the four specific states, the average time of persistence was the smallest (8 years) for quadrats with V. vitis-idaea alone, while the “empty” ones persisted for the longest time (18 years). Comparison of the real square distribution and the forecast of the dynamics for one model time step (5 years) showed that the measure of difference was 5.4%. This illustrates the efficiency of the (time-)homogeneous Markov chain as a short-term forecast tool, but the question of validity of the homogeneity hypothesis in the longer term is left open.