Modelling of matter cycle in a mesotrophic bog ecosystem I. Linear analysis of carbon environs

By means of environ analysis the four-compartment model of matter cycle through the ecosystem of a mesotrophic bog is investigated. The aggregated compartments are: plants, animals, fungi combined with bacteria, and litter. The model is based on data obtained by observations and estimations made on the experimental area of “Tajozhny Log” ocated in the southern taiga sub-zone. Environ analysis enables one to set up, in the quantitative form, certain conclusions on the relative importance of individual system components and diverse processes in the matter cycle, as well as to obtain a number of quantitative characteristics of the turnover. Environ analysis is discussed as a preliminary stage in developing more complex, dynamic models of the matter cycle.     

Modelling of matter cycle in a mesotrophic bog ecosystem II. Dynamic model and ecological succession

Based upon the the results of static environ analysis of the organic matter cycle through the ecosystem, a number of dynamic models are developed, in this second part, for the matter and nitrogen cycles. Probable outcomes of ecological succession are obtained within the framework of the dynamic model without using the ergodic hypothesis that is implicitly adopted in ecological studies. The dynamics of both stocks in the ecosytem components and flows among them can be observed within the model in order to compare some turnover characteristics of the succession series with those of the ecological one.     

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     

Convexity in projection matrices: Projection to a calibration problem

Convexity, as a fundamental property of sets and functions defined on convex sets, plays an important role in many mathematical and applied disciplines, including extremal and optimal-control problems. We prove the set of all feasible projection matrices in a general class of matrix models for stage-structured population dynamics to be convex and the dominant eigenvalue (λ₁) of any projection 2 × 2 matrix to be either a convex, or a concave function on a simplex of the matrix first-row entries (i.e., stage-specific reproduction rates). The latter is also conjectured for the general n × n case. Though looking far from practical needs of matrix population models, this mathematical result has appeared to be quite useful in solving a practical problem to calibrate the projection matrix, i.e., to estimate all the stage-specific vital rates, from empirical data. The data from monitoring of individual life histories of marked plants on permanent sample plots during successive years enable direct calculation of the stage-specific survival and ontogenetic transition rates, but the rates of reproduction do remain uncertain as far as the parent plants can hardly be determined for the (not yet marked!) recruitment.     

The monoculture vs. rotation strategies in forestry: formalization and prediction by means of Markov-chain modelling

The monoculture strategy of forest management, where the same tree species (e.g., Picea abies) is cultivated in a number of successive planting-growing-felling cycles, is generally considered to be economically efficient, yet not sustainable as it reduces biodiversity in the forest. The sound alternative suggests a long-term strategy of forest management in which different forest types rotate either with planting after clear cutting, or by natural forest succession, yet the commercial output remains dubious. We suggest an approach to formalization and modelling forest dynamics in the long-term by means of Markov chains, the monoculture strategy resulting in an absorbing chain and the rotation one in a regular chain. The approach is illustrated with a case study of Russkii Les, a managed forest located in the Moscow Region, Russia, and the nearby forest reserve having been used as a data source for undisturbed forest dynamics. Starting with conceptual schemes of transitions among certain forest types (states of the chain) in the monoculture and rotation cases, we estimated the transition probabilities by an original method based on average duration of the corresponding states and on the likelihood of alternative transitions from a state into the next one. Formal analysis of the regular chain reveals an opportunity to achieve particular management objectives within the rotation strategy, in particular, to get the distribution of forest types in accordance with an adopted hierarchy of their commercial values, i.e. more valuable types have greater shares.