Bilberry vs. cowberry in a Scots pine boreal forest: III. Another forest, another method, and similar conclusions

Abstract

Markov chain formalism developed formerly for the joint dynamics of cowberry (Vaccinium vitis-idaea) and bilberry (V. myrtillus) in the course of post-fire succession (Logofet, Maslov, 2019) is applied here to a similar set of empirical data, namely, the rooted presence/absence of these species on a great number of small (20×20 cm) quadrats laid down in a protected sub-xeric Cladina-Vaccinium pine forest (boreo-nemoral zone) and re-examined each 5 years during 25. As before, a discrete Markov chain describes the 5-year transitions between the four quadrate states: species-free, V. myrtillus alone, V. vitis-idaea alone, and both species present. In contrast to the former study, we summarize the total observation period by weighted averaging the frequencies of each kind of transition, thus getting a single mean-frequency transition matrix, Pmf, rather than the pattern-multiplicative average, G, of the 5 one-step transition matrices. We have verified the correctness of the mean frequencies (i.e., the strata homogeneity) by a statistical (Mantel–Haenszel) test and obtain the standard spatial and temporal characteristics of the observed dynamics from the fundamental matrix of the Markov chain. The conclusion for ecology is that there is a unidirectional trend revealed in temporal changes for three of the four chain states: the share of V. myrtillus quadrats and that of both species quadrats are permanently increasing, while the share of species-free quadrats decreases. The fourth (V. vitis-idaea alone) state has a local maximum at the middle stages of succession, after which its share decreases. The terminal stable outcome of species dynamics is expected to be a distribution where 29.1% of quadrats are occupied by the Vaccinium myrtillus alone, 15.2% by V. vitis-idaea alone, 44.4% by the both species, and 11.3% be species-free. The results confirm that the coexistence of V. myrtillus and V. vitis-idaea can be stable at the final stages of succession. The total time of post-fire succession is estimated at approximately 140 years. From the forest typology point of view, we have a gradual transformation of Cladina-Vaccinium pine forest into Vaccinium and further into Vaccinium-Myrtillus pine forest.The challenge for modeling was to compare the former method of averaging transition matrices vs. the current one of the mean transition frequencies (forming matrix Pmf) and find the limits to their applications. The homogeneity of samples (e.g., a positive outcome of the Mantel– Haenszel test) is an obvious limitation of our current method, and there are published instances where it fails. Former averaging reduced to taking the 5th matrix root from the product of 5 stochastic matrices as an exact solution, G, to the averaging equation, and whether the root did exist and was stochastic too represented the basic limitation of the method. Our set of data does luckily fall within the limitation, though just a random perturbation of these data leads to the matrix root that is no longer stochastic. However, beyond the limitation, we could always find an approximate solution via minimizing the approximation error under the constraint of stochasticity, and this is the final advantage of the matrix average vs. Pmf.