FISHERIES RESEARCH BOARDOF CANADA rteJq Translation SeÈies No. 2348 A method of mathematical modelling of complex ecological systems by A. G. Ivakhnenko, Yu. V. Koppa N. N. •Todua, and G. Petrake Original title: Metod matematychnoho modelyuvannya sladnykh ekologichnykh system From: Avtomatykâ, Instytut kibernetyky AN URSR•(Automatic control, Institute of Cybernetics of the Academy of Sciences of the Ukrainian Soviet Socialist Republic), (4) : 20-34, 1971 Translated by the Translation Bureau(JS/TTH) Foreign Languages Division Department of the Secretary of State of Canada Department of the .Environnent Fisheries Research Board of Canada Marine Ecology Laboratory Dartmouth, N. S. 1973 33 pages typescript 7 4 e t It F PJ:3 • DEPAiTMEIT OF THE SECRETARY OF STATE r- SECRÉTARIAT D'ÉTAT 4 TRANSLATION BUREAU BUREAU DES TRADUCTIONS MULTILINGUAL SERVICES DIVISION DES SERVICES CANADA DIVISION MULTILINGUES TRANSLATED FROM - TRADUCTION DE INTO - EN Ukrainian English AUTHOR - AUTEUR A. G. Ivakhnenko, Yu. V. Koppa, N. N. Todua and H. Petrake TITLE IN ENGLISH - TITRE ANGLAIS A method of mathematical modelling of complex ecological systems. TITLE irt , OREIGN LAN ( UAGE (TRANSLITERATE FOREIGN CHARACTERS) TITRE EN LANGUE ÉTRANGÉ'RE (TRANSCRIRE EN CARACTkRES ROMAINS) Metod matematychnoho modelyuvannya skladnykh ekologichnykh system. REFERENCE IN Ï-- OREIGN LANGUAGE (NAME OF BOOK OR PUBLICATION) IN FULL. *TRANSLITERATE FOREIGN CHARACTERS. RÉFÉRENCE EN LANGUE ÉTRANGÉRE (NOM DU LIVRE OU PUBLICATION), AU COMPLET, TRANSCRIRE EN CARACTIRES ROMAINS. Avtomatyka, Instytut kibernetyky AN URSR, No. 4, 1971. REFERENCE IN ENGLISH - RÉFÉRENCE EN ANGLAIS Automatic control, Institute of Cybernetics of the Academy of Sciences of_Ihe Ukrainian Soviet Socialist Republic, No. 4, 1971. PUBLISHER- ÉDITEUR PAGE NUMBERS IN ORIGINAL DATE OF - PUBLICATION NUMÉROS DES PAGES DANS Institute of Cybernetics of • DATE DE PUBLICATION L'ORIGINAL the Acad. of Sci. of the Ukrainian SSR. 20-34 YEAR ISSUE NO. VOLUME PLACE OF PUBLICATION ANNÉE NUMÉRO NUMBER OF TYPED PAGES LIEU DE PUBLICATION NOMBRE DE PAGES Kyiv, UKR.SSR DACTYLOGRAPHIÉES 1971 4 33 REQUESTING DEPARTMENT TRANSLATION BUREAU NO. MINISTÈRE-CLIENT Environment NOTRE DOSSIER NCI 183074 Fisheries Res. Board BRANCH OR DIVISION Marine Ecology Lab. TRANSLATOR (INITIA LS) J.S. erlIr DIRECTION OU DIVISION TRADUCTEUR (INITIALES) uo ea, e Bedford Inst. of Oceanography 1 PERSON REQUESTING Dr. K. H. Mann CD UNEDITED TRANSL/4TI011 DEMANDÉ PAR ni or "4,4tormation.only YOUR NUMBER TRADUCT,ION NON REVISPI: VOTRE DOSSIER NCI M.f...p.rnution souleir DATE OF REQUEST DATE DE LA DEMANDE July 25, 1972 SOS.200-10-5 (REV. 2/58) 75'30.21.029.5333 Tr DEPARTMENT OF THE SECRETARY OF STATE SECRETARIAT D'ÉTAT TRANSLATION BUREAU BUREAU DES TRADUCTIONS MULTILINGUAL SERVICES DIVISION DES SERVICES DIVISION MULTILINGUES CL I ENT'S NO, DEPARTMENT DI VISI ON/BRANCH CITY N° DU CLIENT MINIS Ti.RE DIVISION/DIRECTION VILLE F.R.B. Environment Marine Ecology Lab. Dartmouth, N.S. BUREAU NO. LANGUAGE TRANsLAT0R(INITIALs) N ° DU BUREAU LANGUE TRADUCTEUR (INITIALES) DEC 1 4 972 183074 Ukrainian J.S. •or inliormar:on Automatic control. TRADUCTION Institute of Cybernetics of the Academy of Sciences of the Ukrainian Soviet Socialist Republic. • A method of mathematical modelling of complex ecological systems. By: A. G. Ivakhnenkô, Yu. V. Koppa, N. N. Todua and G. Petrake (Kyiv.) • Summary . The method of data handling by group (M1.)1IG) is applied to synthesize an analog predicting the quantity of bacteria in the I"Zybinslt reservoir with extrapolation time for a year. The method is based on the principle of self-organization at which it is enough to observe only a small part of the characteristic vector components, as a result of which a complex problem of simulation turns into a comparatively simple one. SOS.-2.00-10-31 7530.21-029.5332 ■ 7 2 Statement of the problem of modelling aquatic ecological systems. In the coming years, automatic computerized control centres will be created. These centres will be connected by means of telemetered systems to the transducers, which will apt upon the active elements controlling the ecological conditions in reservoirs. A reservoir will thus become an object of automatic control, and, because of this, mathematical modelling of ecological systems in the reservoirs will be more and more necessary. Below, an attempt is made to adopt a new approach for the simulation of aquatic ecological systems by introducing heuristic self-organization in which, among others, nonlinear high-degree finite-difference equations ("polvnnmial descriptions") are used instead of differential equations. This method is more adequate for the problems involved in the simulation of complex Systems, and maY provide not only qualitative, but also quantitative estimations of the variables. The models available so far are useful only for a qualitative analysis of various processes, a fact admitted even by the authors of these models. For example, in (2), where one of the best deterministic models is described we read the following: "The results of the analysis of an aquatic ecosystem provided by this model can only be treated as purely qualitative; in order to obtain well-founded quantitative data, a considerable amount of further work is required". 3 The authors of the present paper claim, however, to have developed a mathematical model, which provides not only qualitative, but also quantitative estimates. Accuracy in the simulation of complex systems requires an increase in the complexity of mathematical descriptions. There exists a certain discrepancy between the complexity of the objects of mathematical modelling and the simplicity of the means employed for this purpose. Until recently, modelling was done either by means of deterministic methods (based on the study of simple differential equations, e.g. of the linear equations of convective . diffusion), or by statistical methods of simpie regression analysis (which do not reach beyond the scope of the linear or quadratic regression). A simple . substitution of . finite differences for deriv- atives suffices to show that the complexity of the mathematical description is extremely low in all cases and that, in principle, it cannot ensure accuracy in modelling complex systems. Example. The equation of convective diffusion in the diffetential form is - OS ô OS) - -1- - I\ •• Ot `Oxi where S stands for concentration of matter; t for time; -v. for stream /p.21 velocity;xi for coordinates; for the coefficient of turbulent Kij diffusion; K for the coefficient of non-conservation. 4 Let us approximate the derivatives by finite differences DS-• A S S. — S S • — , i i•1 DS AS S 1 - • , . OXe b à By substituting the expressions obtained in the original equation, we obtain the algebraic equation Sivi+ ao It may thus be seen that the above differential equation corresponds, from the poinE of view of its complexity, to the linear regression equation. The fact that the striking discrepancy between the complexity of the mathematical apparatus on the one hand, and the complexity of the object on the other, is not even noticed (the reasons for the inaccuracy are often explained by the fact that some other factors should also be considered) is due to the deterministic way of thinking of the researchers, which has become deeply rooted and represents the - main shortcoming of contemporary mathematical modelling. This deficiency is eliminated by means of the MDHG. Method of data handling by group ( ,IDHG). The method of data handling by group (MDHG) is similar to the methods of mass selection of plants or animais (7). A certain number of input data (which are called factors or arguments) is used 5 for the construction of all their possible combinations by pairs. Each pair of arguments provides a "partial description", the coefficients of which are determined by the solution of a small system of normal equations (on the basis of the minimum mean square deviation). In the above procedure use is made of a certain experimental selection of data referred to as the learning sequence. The complete description is obtained by excluding intermediate variables from the set of partial descriptions. 0.1it «MI 0.10 Choice according to "the rule of the left angle". 005“ 0.04 Oubip ia070 rigria" 0.02 0015 _ 4 8 quCel pöô Ceti Number of selecti.on steps Fig. 1. Use .of the index of regularity (deviations in control sequence) in increasing the number of selection steps (the problem of forecasting the number of bacteria at the number of nodes in the learning sequence = 9, in the control sequence = 5). The essential difference between the MDHG arid other mathematical methods (recurrent methods, decomposition methods, et al.) is that both initial sequences of experimental data (the learning and the control sequence) increase in each consecutive selection step: the best results of a preceding step are used as the initial data in the next selection step, and so on. 6 The structure of the MDHG algorithm is multiple-stage: the intermediate variables, obtained in the first step, are used to form combinations by pairs in the next step of selection etc. At the end of each step a threshold self-selection takes place, similar to that observed in the mass selection of plants or animais: only a certain percentage of the most regular intermediate variables is admitted into the next step. The regularity of the variables is defined either by their correlation coefficient, or by the value inverse to the mean square deviation, determined from a separate control sequence. The rule for stopping the increase in the number of selection steps is as follows: as soon as the regularity criterion rises to the permissible level or begins to decrease systematically, the increase in the complexity of the full description discontinues ("the rule of the left angle"). We may choose the first or the second local minimum of deviation (Fig. 1). Over 20 different modifications of the MDHG algorithm have been proposed so far.