The Model of the Pollution Spread in the Cascades of Ponds Within the Protected Areas Vitaliy I. Zatserkovnyi , Kateryna A. Kazantseva (&) , and Ludmila V. Plichko Institute of Geology, Taras Shevchenko National University of Kyiv, 90 Vasilkivska Str., Kyiv 03022, Ukraine vitallii.zatsekovnyi@gmail.com, djanaia@ukr.net Abstract. The paper presents the developed model of the pollution spread in the cascades of ponds in the Holosiivskyi National Natural Park. The mathe- matical model is based on graph theory and dynamic equations. Lake cascades are shown in the form of oriented graphs which form a single hydrological network of a complex hierarchy. Therefore, from the point of view of mathe- matics, they form a tree that is a strictly hierarchical orgraph whose vertices are loaded using a dynamic equation. As a result, the pollution of ponds is modeled as a dynamic not statistical system and is not empirically defined as a constant (Lotka-Volterra model). In this paper, pollution is a dynamic that occurs in a geo-ecosystem and fluctuates between pollution and recovery approaching equilibrium. It means that the dynamic equation tries to achieve equilibrium. The task of loading graph vertices is based on the Lotka-Volterra equations with constraints that enable to assess the behavior of the environment, which is in constant progress according to the pollution. Ponds in a cascade are either being polluted or self-healing after pollution. The authors propose to consider pollu- tion as a dynamic process that consists of pollution and recovery, unlike whose who consider this indicator as an empirically defined constant. Keywords: Mathematical modeling  Pollution  Graph theory  Dynamic equations  Geo-ecology 1 Introduction Although a number of issues on modeling the spread of pollution in water bodies have been investigated [1–5], they are mainly represented by probabilistic or statistical models that inadequately describe the processes occurring in them. Much has been done in the field of researching the pollution spread in the cascades of ponds within protected areas, but undoubtedly much remains to be done. Since natural systems are stable in time, throughout their existence, when a polluter is introduced, they seek to counteract this process by self-purification. Section 2, There is defined and mathematized a task of modeling the process of polluting and restoring the environment. This dynamic model seeks to achieve equi- librium. In Sect. 3 there are shown the main results of two permissible lim - 1 - when © Springer Nature Switzerland AG 2020 A. Palagin et al. (Eds.): MODS 2019, AISC 1019, pp. 29–36, 2020. https://doi.org/10.1007/978-3-030-25741-5_4