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.

中文翻译：

---

苏格兰松树北方森林中的越橘与越橘：III。另一种森林，另一种方法和类似的结论

摘要 先前为越橘 (Vaccinium v​​itis-idaea) 和越橘 (V. myrtillus) 在火灾后演替过程中 (Logofet, Maslov, 2019) 的联合动力学而开发的马尔可夫链形式主义在这里应用于一组类似的经验数据，即这些物种在大量小型（20×20 厘米）样方上的根存在/不存在，这些样方位于受保护的亚旱赤松林（boreo-nemoral 区）中，每 5 年重新检查一次在 25. 和以前一样，离散马尔可夫链描述了四个方形状态之间的 5 年过渡：无物种、单独的桃金娘、单独的 V. vitis-idaea 以及两个物种都存在。与之前的研究相比，我们通过对每种转变的频率进行加权平均来总结总观察期，从而得到一个单一的平均频率转变矩阵，Pmf，而不是 5 个一步转换矩阵的模式乘法平均值 G。我们已经通过统计（Mantel-Haenszel）测试验证了平均频率（即地层同质性）的正确性，并从马尔可夫链的基本矩阵中获得了观测动力学的标准空间和时间特征。生态学的结论是，四个链状态中的三个在时间变化上显示出单向趋势：桃金娘象限和两个物种象限的份额都在不断增加，而无物种象限的份额下降。第四个（仅 V. vitis-idaea）状态在演替的中期阶段具有局部最大值，之后其份额减少。物种动力学的最终稳定结果预计为 29. 1% 的样方被单独的 Vaccinium myrtillus 占据，15.2% 被 V. vitis-idaea 占据，44.4% 被两个物种占据，11.3% 是无物种的。结果证实，桃金娘和葡萄藤在演替的最后阶段可以稳定共存。火灾后演替的总时间估计约为 140 年。从森林类型学的角度来看，我们已经将克拉迪纳-越桔松林逐渐转变为越桔，并进一步转变为越桔-桃金娘松林。建模的挑战是比较前一种平均转移矩阵的方法与当前的平均转移频率之一（形成矩阵 Pmf），并找到它们的应用限制。样品的均匀性（例如，Mantel-Haenszel 测试的积极结果）是我们当前方法的一个明显限制，并且有已发布的实例失败。以前的平均化简化为取5个随机矩阵的乘积的第5个矩阵根作为平均方程的精确解G，根是否存在并且是否随机也代表了该方法的基本局限性。幸运的是，我们的数据集确实在限制范围内，尽管这些数据的随机扰动会导致矩阵根不再随机。然而，超越限制，我们总是可以通过在随机性约束下最小化近似误差来找到近似解，这就是矩阵平均相对于 Pmf 的最终优势。以前的平均化简化为取5个随机矩阵的乘积的第5个矩阵根作为平均方程的精确解G，根是否存在并且是否随机也代表了该方法的基本局限性。幸运的是，我们的数据集确实在限制范围内，尽管这些数据的随机扰动会导致矩阵根不再随机。然而，超越限制，我们总是可以通过在随机性约束下最小化近似误差来找到近似解，这就是矩阵平均相对于 Pmf 的最终优势。以前的平均化简化为取5个随机矩阵的乘积的第5个矩阵根作为平均方程的精确解G，根是否存在并且是否随机也代表了该方法的基本局限性。幸运的是，我们的数据集确实在限制范围内，尽管这些数据的随机扰动会导致矩阵根不再随机。然而，超越限制，我们总是可以通过在随机性约束下最小化近似误差来找到近似解，这就是矩阵平均相对于 Pmf 的最终优势。幸运的是，我们的数据集确实在限制范围内，尽管这些数据的随机扰动会导致矩阵根不再随机。然而，超越限制，我们总是可以通过在随机性约束下最小化近似误差来找到近似解，这就是矩阵平均相对于 Pmf 的最终优势。幸运的是，我们的数据集确实在限制范围内，尽管这些数据的随机扰动会导致矩阵根不再随机。然而，超越限制，我们总是可以通过在随机性约束下最小化近似误差来找到近似解，这就是矩阵平均相对于 Pmf 的最终优势。