(1967), and Gushing (1959a, 1969). In the Soviet Union, the works of G. 

 G. Vinberg and S. A. Anisimov (1966) and of A. A. Lyapunov (1971), have 

 been significant in the development of mathematical modelling of aquatic 

 ecosystems. 



The approach of A. A. Lyapunov to the modelling of complicated 

 systems (Lyapunov, Yablenskiy, 1963) is similar in its general features 

 to the macroscopic method of H. T. Odum (1971). The system is presented 

 as an assemblage of relatively independently functioning elements. 

 These elements are interconnected by various communication channels. 

 The role of signals passing through these channels may be placed by 

 portions of matter (energy) or information. Correspondingly, we can 

 distinguish material and information connections between elements of the 

 system. 



One of the basic principles involved in the construction of 

 simulations of the functioning of ecologic systems is the principle of 

 conservation of matter and energy, which is interpreted in the form of 

 balance relationships for each (animate or inanimate) element of the 

 ecosystem. The mathematical model of balance relationships in an 

 ecologic system can be constructed only if a certain degree of 

 completeness has been reached in the study of the object being 

 modelled. In other words, we must have an idea of the distribution of 

 matter and energy among the corresponding elements, the regularities 

 which define the intensity of flows among the elements, what is included 

 in the ecosystem and what leaves or is removed from the ecosystem, and 

 in what quantities. 



A. A. Lyapunov, in composing his model, assumed that all processes 

 occur without delay. This assumption, generally speaking, is not always 

 correct; however, upon transition to a discrete time step, one day or 

 more in length, the assumption of non-inertial elementary processes, for 

 example as concerns phy topi ank ton, is quite justified. 



The model of A. A. Lyapunov contained merely six elements: light 

 (I), the concentration of nitrogen in assimilable ionic form (n^^), the 

 concentration of assimilable phosphorus (n ), the biomass of 

 phytoplankton (p), the biomass of zooplankton (f) and the concentration 

 of detritus (d). The following assumptions were made concerning each of 

 these elements: light is absorbed by the water (a), phytoplankton (a^), 

 zooplankton (a2) and detritus (a3). Nitrogen and phosphorus are 

 expended in the formation of primary production, in the process of 

 photosynthesis (coefficients h^ and hp) and liberated as a result of 

 decomposition of detritus (v^ and Vp). The intensity of photosynthesis 

 is limited by the light conditions and the concentration of nutrients. 



Pp = mindl, gi^n^, QpHp), (1.1) 



where z, g^ and gp are coefficients. 



The consumption of phytoplankton by zooplankton is assumed to 

 follow Vol terra (coefficient e). The effects of multiplication of 

 zooplankton, consumption of detritus by zooplankton and the process of 



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