in biology are used to simulate biological structures, functions, and processes at different levels in the organization of living matter: molecular, subcellular, cellular, organosystemic, organismic, and population-biocenotic. It is also possible to simulate various biological phenomena and conditions of the vital activity of individuals, populations, and ecosystems. Basically, three types of models are used in biology: the biological, the physicochemical, and the mathematical (logico-mathematical).

Biological. Biological models reproduce in laboratory animals certain conditions or diseases that occur in man or animals. These models permit the experimental study of the pathogenesis and the course and outcome of a given condition or disease; they also make it possible to influence the course of a pathological condition. Examples of such models include artificially induced genetic disturbances, infectious processes, and poisonings; anH the simulation of hypertonic and hypoxic conditions, malignant tumors, hyperfunction or hypofunction of certain organs, and neuroses and emotional conditions.

Biological models make use of various methods, including altering the genetic structure of an organism, infecting an organism with microbes, introducing toxins, removing certain organs or introducing the products of their vital activity (for example, hormones), altering the action of the central and peripheral nervous systems, excluding certain substances from food, and placing of an organism in an artificially created habitat. Biological models are widely used in genetics, physiology, and pharmacology.

Physicochemical. Physicochemical models make use of physical or chemical means to simulate biological structures, functions, or processes; as a rule, these models are only very approximate likenesses of the biological phenomenon being simulated. Attempts at creating physicochemical models of the structure and some functions of cells were first made in the 1860’s. For example, in 1867 the German scientist M. Traube imitated the growth of a living cell by growing CuSO₄ crystals in an aqueous solution of K₄ [Fe(CN)₆]. In 1907 the French physicist S. Leduc immersed molten CaCl₂ in a saturated solution of K₃PO₄ and, as a result of surface tension and osmosis, obtained structures that outwardly resembled algae and fungi. By mixing olive oil with various water-soluble substances and putting the mixture in a drop of water, O. Biitschli obtained, in 1892, a frothy microscopic substance that bore a superficial resemblance to protoplasm; the same model even duplicated amoeboid movement.

Beginning in the 1860’s, various physical models of nerve excitation were also proposed. In a model created by the Italian scientist C. Matteucci and the German scientist L. Hermann, a nerve was represented by a wire sheathed in an electrolytic conductor. When the wire and its sheath were connected to a galvanometer, the scientists observed a potential difference that changed when electrical “stimulation” was applied to a part of the “nerve.” This model simulated several bioelectrical phenomena that occur during nerve excitation. In his model of a wave of excitation spreading along a nerve, the French scientist R. Lillie simulated a number of phenomena observed in nerve fibers (refractory period, all-or-none law, bilateral conduction). The model consisted of a steel wire that was placed first in concentrated and then in weak nitric acid. The wire was covered with an oxide that was reduced as a result of various factors; the process of reduction that occurred in one part of the wire spread along the wire. These models, which demonstrated the possibility of simulating the appearance and some properties of living matter by physicochemical means, were based on an external, qualitative resemblance. Such models are now only of historical interest.

More complex models based on a much greater quantitative similarity were later constructed by applying the principles of electrical engineering and electronics. For example, electronic circuits modeling bioelectric potentials in the nerve cell, process, and synapse were based on data from electrophysiological research. Electronically controlled mechanical devices modeling complex behavioral acts (formation of conditioned reflexes; central inhibition) have also been constructed. These models are usually given the shape of a mouse, turtle, or dog. Such models oversimplify phenomena observed in the living organism and are more important for bionics than for biology.

Significantly greater progress has been made in modeling the physicochemical conditions in which living organisms or their organs and cells exist. For example, solutions of inorganic and organic substances, such as Ringer’s, Locke’s, and Tyrode’s solutions, are used to simulate the internal environment of the organism and to prolong the functioning of isolated organs or of cells cultivated outside the organism.

Models of biological membranes (a film consisting of natural phospholipids separates an electrolyte solution) are used to study the physicochemical principles of ion transport and the various factors that affect ion transport. Oscillatory processes characteristic of many biological phenomena (differentiation, morphogenesis, phenomena in complex neuronal networks) are modeled by means of chemical reactions occurring in solutions under self-oscillating conditions.

Mathematical. Mathematical models deal with the mathematical and logico-mathematical description of the structure, relationships, and lawlike regularities of functioning of living systems. These models are constructed from experimental data; they may also be abstract, formalized descriptions of a hypothesis, theory, or a discovered lawlike regularity of some biological phenomenon. Mathematical models require further experimental verification. Different versions of such experiments reveal the limited applicability of mathematical models and provide data for their subsequent correction.

However, by “feeding” a mathematical model to a computer it is often possible to predict the nature of the changes that may occur in a biological process under conditions that are difficult to reproduce in an experiment. In some cases, mathematical models help to predict phenomena not previously known to the researcher. For example, the model of cardiac activity proposed by the Dutch scientists van der Pol and van der Mark based on relaxation oscillations theory demonstrated the possibility of an unusual cardiac arrhythmia that was later discovered in man.

Also important among the mathematical models of physiological phenomena is the model of nerve fiber excitation developed by the English scientists A. Hodgkin and A. Huxley. The theory of neural networks advanced by the American scientists W. McCulloch and W. Pitts formed the basis for the logico-mathematical model of the interaction of neurons. Systems of differential and integral equations underlie the modeling of biocenoses (V. Volterra, A. N. Kolmogorov). O. S. Kulagina and A. A. Liapunov constructed a Markov mathematical model of the evolutionary process. I. M. Gel’fand and M. L. Tsetlin used game theory and finite automata device theory to develop conceptual models of the organization of complex forms of behavior. In particular, they showed that the many muscles in the body are controlled by the formation of several functional blocks, or synergies, in the nervous system and not by the independent control of each muscle. The creation, use, and improvement of mathematical and logico-mathematical models are important for the advancement of mathematical and theoretical biology.

(in economics). Models have been used in economics since the 18th century. The ideas contained in F. Quesnay’s Tableau économique (The Economical Table), which K. Marx called “incontestably the most brilliant for which political economy had up to then been responsible” (K. Marx and F. Engels, Soch., 2nd ed., vol. 26, part 1, p. 345), essentially constituted the first attempt to formalize the entire process of social reproduction. The models of reproduction created by Marx and developed by V. I. Lenin had an immense influence on the science of economics. The theory of intersector balance was a direct outgrowth of this approach.

Models came into especially wide use in economic research during the mid-20th century, when a number of new fields of mathematics emerged and electronic computers were introduced. Mathematical models in economics have been used by such foreign scholars as L. Walras, J. von Neumann (the creator of the first electronic digital computer and one of the founders of games theory and mathematical economics in general), J. M. Keynes, R. Frisch, J. Tinbergen, P. Samuelson, K. Arrow, W. Leontief, G. B. Dantzig, J. Debré, T. Koopmans, H. Nikaido, M. Morishima, R. Harrod, and J. Hicks.

In the USSR, the development of economic modeling has been linked primarily with the names of L. V. Kantorovich (first among the world’s economists to formulate a model of the socialist economy as a mathematical problem in linear programming), A. L. Lur’e, V. S. Nemchinov, V. V. Novozhilov, A. G. Aganbegian, A. L. Vainshtein, V. A. Volkonskii, L. M. Dudkin, A. A. Makarov, V. L. Makarov, S. M. Movshovich, lu. A. Oleinik, V. F. Pugachev, E. lu. Faerman, N. P. Fedorenko, and S. S. Shatalin.

Economic research through modeling is typically divided into a number of phases. In the first phase the general problem is formulated and the object of investigation is fixed appropriately. Such an object of study might be the world economy in its entirety, the economy of the world capitalist and socialist systems, or an individual country, economic sector, enterprise, or component; or it might be a particular aspect of the functioning of an economic system such as supply and demand, income distribution, or price formation. Requirements are then imposed on the nature of the initial information, which may be statistical (obtained through observation of actual economic processes) or normative, such as input-output coefficients or rational consumption requirements. Next the most basic initial characteristics of the object being modeled are studied and hypotheses about its development are advanced. In resolving problems in efficient economic management in this fashion, fundamental importance must be given to such characteristics as the limitations on material, labor, and natural resources that prevail at any given time, the level of scientific and technical knowledge achieved by society, and the determining assortment of technological procedures available for obtaining needed products from actual resources. Equally important is the multivariant nature of possible curves of economic development, which in turn dictates the problem of working out the criteria for selecting the most efficient curve.

The information obtained in this first phase is necessary for the creation of a model of the economic system, which constitutes the second phase of this research. Various models are used to study different aspects of the functioning of economic systems. The most general regular patterns of economic development are investigated through macroeconomic models, including balance models, optimization models, equilibrium models, and gaming models. Macroeconomic models are used to analyze and forecast the movement and interrelationship of large economic aggregates such as national income, employment, rate of return on assets, consumption, savings, and investment; microeconomic models of production, transportation, trade, supply, and marketing are employed in research into specific economic situations. For the study of complex economic systems mathematical models are generally used, inasmuch as they are best suited for analyzing most basic economic processes, as for example in transportation; these are known as analog models, and include electrical, mechanical, and hydraulic models. Early in the 1960’s simulation models became widely known, and are now used to study the real processes of the functioning of economic systems in cases where mathematical analysis is complicated or impossible; to a certain degree, such models have come to substitute for experimental study of economic systems. In the form of business games, such models are also used to teach managers the techniques of efficient economic management.

Economic models are classified according to the following basic criteria: goals and basic problems, object of investigation, research apparatus employed, and nature of initial information. Based on this last criterion a distinction is made between statistical and normative models. Clearly such classifications are largely arbitrary, inasmuch as actual models may occupy an intermediate position—for example, one portion of the information may be given normatively, while another portion is derived from statistical analysis of the behavior of the economic system. In addition, the more general models may subsume partial models. For example, models of sectors or enterprises, designated as submodels, may be elements of a model of the national economy; on the other hand, requirements that grow out of such analysis of an entire economy may be introduced into more confined models.

In this phase of the construction of mathematical models, the results of empirical research are translated from the specific language of the object under study to the universal language of mathematics; a schematic design for the model is selected; and the principal variables, parameters, and functional relationships are introduced. The model thus obtained is then compared with already available models. If it emerges that models of this particular type have been sufficiently studied and that appropriate analytic methods are readily available, the corresponding mathematical problem can be solved. If such is not the case, the question becomes whether it is possible to simplify the premises of the model in such a way that it retains the significant specific characteristics of the object under study, yet at the same time is brought under a class of structures that has already been studied in mathematics. In turn, the construction of models with characteristics that have not yet been studied stimulates the development of new areas in mathematics.

The third phase is mathematical analysis of the model, by which means not only quantitative but also qualitative conclusions are obtained. Here it is important to determine which questions can be answered by the model and which cannot; a typical error is to attempt to use analysis of a model to explain phenomena that extend beyond the bounds of the model. The qualitative conclusions obtained from analysis of economic models make it possible to discover characteristics of the economic system that were not known before, pertaining to such problems as systemic structure, dynamics of development, systemic stability, relationship among macroeconomic parameters, and the characteristics of cost factors. Marx, for example, obtained a relationship between the constant capital of the first subdivision and the variable capital and surplus value of the second subdivision from his schematic presentations of reproduction. Lenin’s presentations of reproduction have made it possible to determine the kind of technological progress required for the operation of the law of preferential growth in production of the means of production. On the basis of what is known as the model of equilibrium growth, it has become possible to identify the asymptotic characteristics of efficient economic curves, that is, the tendency toward continuous growth at a maximum rate. Models of optimal planning are also used to study theoretical problems of price formation.

The quantitative conclusions arising from economic models are relevant for optimal planning for the development of particular economic units, to forecasts of economic dynamics, and to price calculation; in this fashion they are already producing considerable economic benefit. The relevant economic models constitute an important element in automated control systems.

Different models have differing requirements. Theoretical or abstract models are required to reflect only the most general characteristics of economic systems. Mathematical models are employed here to prove the existence within the system of an efficient state—consisting of either an equilibrium or an optimal curve—after which the characteristics of this state are studied. If possible, the algorithm yielding this efficient state is also determined; the representation of processes actually taking place in the object being modeled often serves as the algorithm for solving economic problems.

Models used for concrete calculations have as their theoretical basis abstract models, fully analyzed. Concrete models give a sufficiently complete reflection of the specific characteristics of the object under investigation; if they did not, calculations made on the basis of them could not be introduced in practice. This phase concludes with the economic interpretation of the results obtained: mathematical concepts are translated into the language of the object under study. Qualitative results are interpreted as characteristics and patterns of development within the economic system, the derived algorithm is interpreted as the mechanism of the system’s planning and functioning, and numerical results are then taken as plans or forecasts.

Prior to utilization of the conclusions thus obtained in either theory or practice, a fourth phase in research related to modeling must be carried out. This consists of confirming the results obtained. The researcher faces enormous difficulties here. The usual methods of the natural sciences, such as experimentation and comparison of results obtained with the known characteristics of real processes, are by no means always applicable. For example, when a program for the development of an economic unit obtained through use of a model demonstrates that there are possibilities for improving that unit’s efficiency, it is still not clear whether this reflects genuine flaws in existing methods of planning, management, and administration of incentives, or results instead from the failure of the initial model to take into account conditions actually prevailing, thus making the improvements outlined infeasible. Thus theoretical testing of the initial premises of the model becomes especially important while still in the first phase of investigation. Experimentation with the object under study, or with an analog device that simulates it, which would make possible a check of the results obtained in modeling, is rarely done because of major costs involved. Actual physical experimentation also entails numerous socioeconomic difficulties.

The fifth and last phase, that of actual introduction, in the event of a positive outcome in the preceding phase should lead to the improvement of theory and methods in management, price formation, and planned economic development. If such does not take place, the initial premises of the model must be refined and each phase previously outlined must be carried through again. Thus research into economic systems through use of models is constructive in nature. In capitalist society models offer specific benefits, primarily within a single company. But the practical application of models on a national scale is substantially limited as a result of the antagonistic contradictions inherent in capitalism. Under socialist conditions, however, fundamentally new opportunities emerge for the use of models in solving problems of planning and managing the entire national economy.

The use of models in economics has certain limits. Not all information about economic processes can be completely formalized, nor is all the needed information accessible; not every model is subject to theoretical analysis. In addition, even the most advanced computers cannot handle the enormous volume of computations that must be performed to solve certain concrete economic problems. Therefore the use of models must be supplemented by other methods, including utilization of the experience acquired by economic managers. In turn, the results of calculations made on the basis of models can be of great assistance to economic managers in their work.