[Local population of Eritrichium caucasicum as an object of mathematical modelling. I. Life cycle graph and a nonautonomous matrix model].

For the plant species, which is considered a short-lived perennial, we have composed a scale of ontogenetic stages and the life cycle graph (LCG) according to annual observations on permanent sample plots in an Alpine lichen heath during the 2009-2014 period. The LCG that reflects seed reproduction has been reduced to the one that avoids the stage of soil seed bank, yet preserves the arcs of annual recruitment. The corresponding matrix model of stage-structured population dynamics has four stages: juvenile plants (including seedlings), virginal, generative, and 'terminally generative' (the plants die after seed production). Model calibration reduces to directly calculating the rates of transition between stages and those of delays within stages from the data of only one time step, while keeping the two reproduction rates uncertain, yet confined to the quantitative bounds of observed recruitment. This has enabled us to determine a feasible range for the dominant eigenvalue of the model matrix, i.e., the quantitative bounds for the measure of how the local population adapts to its environment, at each of the five time steps, resulting in aformally nonautonomous model. To obtain 'age-specific parameters' from a stage-classified model, we have applied the technique that constructs a virtual absorbing Markov chain and calculates its fundamental matrix. In a nonautonomous model, the estimates of life expectancy also depend on the time of observation (that fixes certain environmental conditions), and vary from two to nearly seven years. The estimates reveal how specifically short lives the short-lived perennial, while their range motivates the task to average the model matrices over the whole period of observation. The model indicates that Eritrichium caucasicum plants spend the most part of their life span in the virginal stage under each of the environment conditions observed, thus revealing the place retention strategy by C. K6rner (2003), or the delayed-development strategy by L.A. Zhukova (1995). We discuss the prospects of model experiments with a logically nonautonomous model to forecast the long-term dynamics of E. caucasicum under a scenario of climate changes.

[From uncertainty to an exact number: Developing a method to estimate the fitness of a clonal species with poly variant ontogeny].

The task to estimate the fitness of a clonal plant with polyvariant ontogeny reduces to compiling a life cycle graph, constructing and calibrating the corresponding matrix model of the discrete-structured population, and calculating the dominant eigenvalue (λ1) of the model matrix. We demonstrate a solution to this task with a sample of Calamagrostis epigeios , a perennial long-rhizome grass propagating vegetatively, and data on the age-stage structure of its local population. A traditional technique of successive censuses fixing the age-stage status of all individuals on a permanent sample plot (SP) provides for calculating frequencies of ontogenetic transitions directly from the data, but leaves uncertain the status-specific reproduction rates as the recruit parents are unknown ("reproductive uncertainty"). Uncertainty in data was leading to that in the estimation and dictating a need to change the method of field study: the description of above-ground parts of plants has been completed with the analyses of rhizome parent-daughter links revealed after digging SPs out. However, the traditional, fixed area of SPs (1 m 2) forced cutting the links off along its perimeter, while those within the SP turned out quite entangled already in a 4-year-old colony. A result, the reproductive uncertainty were not eliminated completely, and the next step in the method development has become to determine the contour of the entire woodreed colony and to carefully dig it out. Analysing both the above- and below-ground spheres of the colony has enabled us to calculate uniquely all the elements of the matrix model, hence the value of λ1, while accounting for the actual area of the contour in the current and previous years amends the value of λ1 needed for comparison with the results of previous studies.

[Analyzing the fine-scale dynamics of two dominant species in a Polytrichum­Myrtillus pine forest. I. A homogeneous Markov chain and cyclicity indices].

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.

[Polyvariant ontogeny in woodreeds: Novel models and new discoveries].

Polyvariant ontogeny (PVO) is expressed visually in the life cycle graphs (LCGs) for Calamagrostis woodreeds as a variety of pathways for individual plants to develop through many of their states distinguishable by the ontogenetic stage and chronological age (in years). PVO is recognized as the basic mechanism of adaptation in local plant populations to their environments, while they find a quantitative measure of the adaptation via developing a matrix model of double-structured population, calibrating the matrix of vital rates from empirical data, and calculating its dominant eigenvalue λ1. This approach encounters an obstacle typical for rhizome grasses: while the rates of aging and ontogenetic transitions can be deterinined from field data mainly by the morphology of aboveground parts of the plant, the rates of vegetative propagation can be reliably determined only from digging up the belowground rhizome system, i.e., by destroying the sample plot ('reproductive uncertainty'). Therefore, the former (non-destroying) calibrations of matrix models were subjective to an extent, resulting in correspondingly subjective estimations. A novel method to overcome the 'reproductive uncertainty' makes use of the maximation hypothesis: the uncertain rates are such that λ1 attains its maximal possible value under the given conditions. To test the hypothesis, we have conducted a field experiment by a new technique with a model species, the woodreed Calamagrostis epigeios (L.). Roth, that reproduces vegetatively in a meadow phytocoenosis and a spruce forest clearance. Excavating the whole system of ramets with their rhizomes and analyzing the maternal-child links in the laboratory have provided for (in addition to the former data on the local population structures and ontogenetic transitions) a new kind of data to calculate the status-specific rates of reproduction. The novel method of calibration has enabled us to find an exact range of λ1 values, i.e., the true quantitative bounds of adaptation for a given local population. Obtained under the reproductive uncertainty and maximation hypothesis, the values of λ1 have turned out close to the upper bounds of the ranges, thus verifying the hypothesis. The experiment has discovered generative subsidiary plants sprouting from the rhizomes of maternal ramets without entering the virginal stage. As a result, the LCG enriches with new reproductive pathways, and the new (not yet published) situations emerge where λ1 fails in its accuracy as a measuring tool of comparative plant demography. We propose a general method to adjust the adaptation measure in this kind of situation.

[Succession caused by beaver (Castor fiber L.) life activity: I. What is learnt from the calibration of a simple Markov model].

A homogeneous Markov chain of three aggregated states "pond--swamp--wood" is proposed as a model of cyclic zoogenic successions caused by beaver (Castor fiber L.) life activity in a forest biogeocoenosis. To calibrate the chain transition matrix, the data have appeared sufficient that were gained from field studies undertaken in "Bryanskii Les" Reserve in the years of 2002-2008. Major outcomes of the calibrated model ensue from the formulae of finite homogeneous Markov chain theory: the stationary probability distribution of states, thematrix (T) of mean first passage times, and the mean durations (M(j)) of succession stages. The former illustrates the distribution of relative areas under succession stages if the current trends and transition rates of succession are conserved in the long-term--it has appeared close to the observed distribution. Matrix T provides for quantitative characteristics of the cyclic process, specifying the ranges the experts proposed for the duration of stages in the conceptual scheme of succession. The calculated values of M(j) detect potential discrepancies between empirical data, the expert knowledge that summarizes the data, and the postulates accepted in the mathematical model. The calculated M2 value falls outside the expert range, which gives a reason to doubt the validity of expert estimation proposed, the aggregation mode chosen for chain states, or/and the accuracy-of data available, i.e., to draw certain "lessons" from partially successful calibration. Refusal to postulate the time homogeneity or the Markov property of the chain is also discussed among possible ways to improve the model.

[Two paradigms of mathematical population biology. Attempting to synthesise].

Whatever popular the slogan of nonlinear ecological interactions has been in theory, practical ecology professes the projection matrix paradigm, which is essentially linear, i.e., the linear matrix model paradigm for discrete-structured population dynamics. The dominant eigenvalue lamda1, of the projection matrix L is considered as a growth potential of the population. It provides for a quantitative measure of the fitness at which the species is adapted to the given environment, the measure being adequate and accurate, given the data of"identified individuals" type. The case of"identified individuals with unknown parents" bears uncertainty in the status-specific reproduction rates, which eliminates in a unique way (for a broad class of structures and life cycle graphs) by maximizing lamda1(L) under the constraints ensuing from the data and knowledge of species biology. The paradigm of linearity gives way to nonlinear models when modeled are species interactions, such as competition for shared resources, and where the outcome of interaction depends on the population structure of the competitors. This circumstance dictates a need for the synthesis of two paradigms, which is achieved in nonlinear matrix operators as models of interaction between the species whose populations are discrete-structured.

[Svirezhev's substitution principle and matrix models for dynamics of populations with complex structures].

Matrix models of discrete-structured population dynamics became a traditional tool in plant and animal demography, aided with developments in the proper mathematics and wide spread of software products, which greatly facilitate creating models in the man-machine dialogue mode, but leave behind the scenes the issue of whether the methods applied are adequate to the problem posed. A principal discrepancy of this kind does appear in a problem to calibrate the projection matrix on observation data of the "identified individuals with uncertain parents" type: simplifying recipes from an ecological software package contradict the idea of polyvariant ontogeny as an adaptation mechanism. The problem becomes solvable if we substitute an extremal adaptation principle for the uncertainty in data as follows: the unknown reproduction rates are assumed to distribute among the reproductive groups in such a way that maximizes the potential growth rate of the model population under the current conditions. In combination with findings from mathematical analysis of a wide class of matrix models, this principle turns the model into a reliable tool to test research hypotheses. Considered as an example is a matrix model (published elsewhere) for the population dynamics of Calamagrostis canescens woodreed, a perennial clonal plant species with a complex, age-stage-based population structure.

[On the competition among discrete-structured populations: a matrix model for population dynamics of woodreed and birch growing together].

Presented is a synthesis of field, theoretical and modelling studies on joint dynamics of two species--common birch (Betula pendula Roth) and wood small reed (Calamagrostis epigeios (L.) Roth)--overgrowing a spruce forest clear-cut. A nonlinear matrix model for population dynamics of two species, which both possess non-trivial population structures and compete for a resource in common was developed as an expansion of the linear models for single-species, age-stage-structured population dynamics. Constant values of the age-stage-specific survival and reproduction rates have been modified with some decreasing functions of the (competitive group) abundances in the competitor species or/and the species itself. Special aggregation of the age-stage structure for each of the competitor species has reduced the dimension of the nonlinear matrix operator down to the level that admits accurate calibration of the model parameters on the observation data, as well as the search for an equilibrium and its stability analysis. When calibrated, the nonlinear model exhibits convergence to the steady equilibrium--a state of the phytocoenosis that is interpreted as young, closed-canopy, birch forest with suppressed woodreed population. The model illustrates the observed course of forest renewal: the appearance of birch germs and the growth of birch population overpass the woodreed competitive resistance and result in formation of young birch forest, where the birch exerts a strong suppressive impact on both the woodreed growth and the own young growth. Remarked is a potential of the model as an object of more general mathematical study and a tool to predict the course of forest renewal.

[Successions in the forest-steppe under climate changes: a modelling approach].

The paper represents an attempt to apply the general principles of modelling vegetation dynamics under climate changes to a study of the long-term vegetation dynamics in the forest-steppe zone of the European territory of Russia, with a purpose to forecast under special climatic scenarios. An original technique is used to construct a Markov chain as a model of vegetation succession. The technique emanated from gebotanic knowledge generalized as a scheme of successional transitions with estimates of the average duration for certain stages of succession. Whenever the knowledge related the stage duration to certain (climate-sensitive) factors of the environment, the fundamental potentiality arises to model the temporal course of succession as a function of a given scenario for how the key factors change. In the formal terms, the model represents a random chain of the Markov kind with a finite number of states and discrete time of transitions by the given scheme. Relative square distributions of succession stages under concern at any time moment (within an adopted scenario) appear as the model outcome (forecasts), as well as estimates of the attainment time for certain states of the vegetation in the territory under study. A method is proposed to describe dynamics of the phytomass production and stores (and the corresponding model trajectories are obtained) for a given scenario.

[The structure and dynamics of woodreed Calamagrostis canescens population: a modelling approach]. / Struktura i dinamika populiatsii veinika sedeiushchego Calamagrostis canescens: model'nyi podkhod.

A scale of ontogenetic states has been developed for woodreed Calamagrostis canescens, a perennial species dominating the grass layer of fell forest areas. The population structure is considered as a set of age-stage groups of individuals differing both in the ontogenetic stage and the chronological age measured in years. to describe the dynamics through years a special kind of matrix formalism has been proposed which is reducible neither to the classic Leslie matrix for an age-structured population, nor to the well-known Lefkovitch matrix for a stage-structured one, and which does not suffer from excessiveness of the "two-dimensional" representation for the structure implying the projection matrix of a block pattern. It has been shown however that the protection matrix corresponding to C. canescens life-history graph embodies the canonical features of matrix formalism for structured population dynamics, such as the exponential population growth or decline, the convergence to a stable equilibrium structure, the calculable indicator of growth/decline/equilibrium (i.e., a measure of the population reproductive potential) as well as possibility to determine the relative reproductive value of each group. On the other hand, "left-sidedness of the age spectrum", a property that is often observed in real populations and is inherent in Leslie models of growing populations, may fail in the age-stage-structured model. The aggregation of age-stage groups into the age classes is possible only under special strict relationship among the age-stage-specific vital rates of the population. The both circumstances serve a methodical indication that an additional dimension such as the stages, for example, ought to be introduced into the age structure of the model population.

The model for human population dynamics as a part of the global biosphere model: some aspects of modeling in a dialogue regime.

"A matrix model for an age structured population with 4 groups is presented.... The demography matrix is identified with data from global demographic statistics for the 1970s. When calibrating the matrix elements, a semi-formal procedure was used to calculate the dominant eigenvalue and corresponding eigenvector. This procedure was based essentially on the dialogue mode of computation provided by the programing language APL." The advantages of using APL are discussed