Physical Oceanography, Vol.
6, March, 2012 (Ukrainian Original No.
6, November–December, 2011)
NUMERICAL EXPERIMENTS WITH AN ADAPTIVE MODEL OF A MARINE
ECOSYSTEM REPRESENTED BY THE REACTION-DIFFUSION EQUATIONS
E. V. Romanovskii
and I. E. Timchenko
We consider the role of diffusion in the formation of space-time processes in an adaptive model of a ma-
rine ecosystem. It is shown that the consideration of the diffusion in the ecosystem model constructed
by the method of adaptive balance of causes results in a system of Kolmogorov–Petrovsky–Piskunov re-
action-diffusion equations. We propose a model of an ecosystem including seven Kolmogorov–
Petrovsky–Piskunov equations for the concentrations of phytoplankton and zooplankton, biological re-
sources, oxygen, nutrients, detritus, and carbon dioxide. In numerical experiments performed using the
model, we obtained one-dimensional and two-dimensional distributions of parameters of the ecosystem
for various coefficients of diffusion. The analysis of results shows that the model of ecosystem based on
the Kolmogorov–Petrovsky–Piskunov equations can generate dissipative structures, i.e., distributions of
concentrations of interacting substances that are stationary in time and inhomogeneous in space.
Keywords: model of a marine ecosystem, diffusion, ABC-method, dissipative structures.
Mathematical models of marine ecosystems are developed for the solution of various problems. One of the
most important problems is the study of the dynamics of populations of marine organisms. This trend gains spe-
cial importance due to the increase in the anthropogenic loading on marine ecosystems. Most models of marine
ecosystems contain an integral description of processes, which does not include spatial dynamics . This situa-
tion is explained by serious difficulties in the modeling of distributed systems because the space–time models of
such systems involve the solution of the problems of transfer of nonconservative admixtures that interact with
one another and are subject to the influence of the environment.
At the same time, exactly these problems are of high scientific and practical interest, and they constitute a
branch of the extensively developing theory of adaptive complex systems with self-organization properties,
whose models are generally called the “reaction-diffusion” equations. This trend was started by Turing in his
work The Chemical Basis of Morphogenesis published in 1952. His investigations were devoted to the mathe-
matical theory of formation of structures in an initially homogeneous system in which some chemical reactions
run simultaneously with energy consumption and in which the dynamics of a medium in the form of transfer
(diffusion) is present. Turing’s ideas were substantially developed in the works of the group guided by Prigog-
ine  devoted to the study of the phenomenon of self-organization in complex systems on the basis of funda-
mental laws of thermodynamics and chemical kinetics. It was established that special types of structures capable
of self-organization, i.e., of passing from thermal chaos to ordered states, can spontaneously appear in open sys-
tems under nonequilibrium conditions. Prigogine called these structures dissipative, emphasizing that the proc-
ess of irreversible energy loss plays a constructive role in their appearance. Of special importance for systems
composed of a large number of objects that interact with one another are random fluctuations of parameters of
Marine Hydrophysical Institute, Ukrainian National Academy of Sciences, Sevastopol, Ukraine.
Corresponding author; e-mail: firstname.lastname@example.org.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
59–72, November–December, 2011. Original article submitted June 23,
2010; revision submitted December 1, 2010.
424 0928–5105/12/2106–0424 © 2012 Springer Science+Business Media, Inc.