[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.

[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.

[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.

[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.