Yu.M. Svirezhev

Aggregation of variables in simulation models of water ecosystems

Abstract The problem of variable aggregation in simulation model is considered. By aggregation is understood the unification of some variables of the system which are related to one another, into some sort of blocks each of which is a new variable with the properties defined by the aggregation laws. Let us take such a model, described by a system of differential equations with the set of variables {Xi} (i = 1,…,N). The problem is tofind a set of variables {Yj} (j=1,…,M; M⩽N) that is related to the set {Xi} by equations Y=φj(X) such that the new system of differential equations can be described by variables {Yi} only. it brings us to the problem of the representability of a function of many variables {Xi} in the aggregated variables {Yj}. Two criteria of representability are given. The withthe help of these criteria, the criterion of aggregation for some general model of a two-level ecological system is deduced.

MATHEMATICAL MODELLING OF A FISH POND ECOSYSTEM

Abstract A mathematical model is constructed for a fish breeding pond for carp, silver carp and bighead. The model is a system of ordinary differential equations describing the material transformations in the ecosystem. It allows a choice of optimal regimes of the aeration, feeding and fertilization of a pond for different climatic conditions in order to maximize the yield.

Optimisation of global CO2 emission based on a simple model of the carbon cycle

A simple model has been designed to describe the interaction of climate and biosphere. Carbon dioxide, understood as a major emitted gas, leads to a change of global climate. Economic interpretation of the model is based on the maximisation of the global CO2 cumulative emissions. The two most important profiles of emission have been obtained: optimal and multi-exponential suboptimal profiles, each displaying different characteristics.

A minimal model of eutrophication in freshwater ecosystems

Abstract A very simple model of eutrophication is presented. It includes several quite aggregated parameters and can be studied by analytical methods. Nevertheless, its dynamic behaviour reflects well real reservoir evolution as observed under the impact of increasing biogenic pollution. Such models, simple in construction but sensitive to the main trends of an ecosystem, are termed ‘minimal’. The state variables in the model are the concentrations of: (1) phytoplankton; (2) biogenic elements (nutrients); (3) detritus; and (4) dissolved oxygen. Transformations among the substances are described by a system of four ordinary differential equations. Steady-state dynamics are studied (the so-called quasi-stationary process). The qualitative analysis undertaken shows that the total amount of substances in a reservoir (phytoplankton + detritus + nutrients) is a very important ecosystem control parameter. In fact, it is this parameter that determines the rate and the degree of eutrophication. It also turns out that the relations between some observed characteristics of the ecosystem may define beforehand the future behaviour of the reservoir.

“Emission game”: some applications of the theory of games to the problem of CO2 emission

If there are no doubts that we must reduce the total emission of carbon dioxide, then the problem of how much different countries should be allowed to contribute to this amount remains a serious one. We suggest this problem to be considered as a non-antagonistic game (in Germeiers sense). A game of this kind is called an “emission game”. Suppose that there are n independent actors (countries or regions), each of them releasing a certain amount of CO2 per year (in carbon units) into the atmosphere, and that the emission would be reduced by each actor. Each actor has his own aim: to minimise the loss in the Gross Domestic Product (GDP) caused by the reduction of emissions. On the other hand, taking into account that it is impossible to estimate more or less precisely the impact of the climate change on GDP for each country today, a common strategy will be to reduce the climate change. Since one of the main leading factors in global warming is the greenhouse effect, then the common aim will be to reduce the sum of emissions. This is a typical conflict situation. How to resolve it? We can weigh the “egoistic” and “altruistic” criteria for each actor introducing the so-called “coefficients of egoism”. This coefficient is very large, if the actor uses a very egoistic strategy, and conversely, if the actor is a “super-altruist”, then the corresponding coefficient is very small. Using these coefficients we get the general solution of the game in a form of some Paretos equilibrium. The solution is stable and efficient.

On some general properties of trophic networks

Abstract A new description in the class of non-linear Volterra models for such structures as trophic networks is suggested, and some fully plausible assumptions are formulated when it is possible. These assumptions allow us to construct a system of non-linear differential equations corresponding to the graphic representation of trophic networks in the form of a flow diagram. It is possible to calculate the Jacobi matrix for network equations, using data on the matter inside compartments and flows between them. It is very interesting that the Jacobi matrix does not depend on the type of interaction between compartments. With the help of the Jacobi matrix, some general properties of trophic networks, such as stability, sensitivity, etc. can be estimated. The sensitivity matrix and entropy function, as well as the other scalar measures, are calculated using the implicit function theorem and the Jacobi matrix, and then interpreted from the ecological point of view. A property of the Jacobi matrices such as stability, can be used as a criterion for selection of observed data in real trophic networks.

Climate impact on social systems: the risk assessment approach

A novel approach to the problem of estimating climate impact on social systems is suggested. This approach is based on a risk concept, where the notion of critical events is introduced and the probability of such events is estimated. The estimation considers both the inherent stochasticity of climatic processes and the artificial stochasticity of climate predictions due to scientific uncertainties. The method is worked out in some detail for the regional problem of crop production and the risks associated with global climate change, and illustrated by a case study (Kursk region of the FSU). In order to get local climatic characteristics (weather), a so-called “statistical weather generator” is used. One interesting finding is that the 3% risk level remains constant up to 1.0–1.1°C rise of mean seasonal temperature, if the variance does not change. On the other hand, the risk grows rapidly with increasing variance (even if the mean temperature rises very slowly). The risk approach is able to separate two problems: (i) assessment of global change impact, and (ii) decision making. The main task for the scientific community is to provide the politicians with different options; the choice of admissible (from the social point of view) critical events and the corresponding risk levels is the business of decision makers.

Constraints on information measures of embodied solar energy fluxes in the biosphere

The structure of the input-output (I-O) model of the Biosphere is investigated from the viewpoint of the information theory. The model was used to estimate the direct and indirect solar energy cost of different natural and artificial commodities. We showed that the distribution of aggregated fluxes, different types among compartments, can reflect the physical and functional ordering and natural hierarchy in the Biosphere to be found in the values of embodied energy. So, if we divide all the sets of commodities into three groups according to their order of magnitude of energy intensity, and then calculate the information measures of fluxes due to proposed subdivisions, we get the following: (1) that each distribution of aggregated fluxes among the compartments in chosen groups is increasingly far from equi-distribution; and (2) information of one distribution in respect to another is increasingly consequently.