FUZZY DIFFERENTIAL MODELS AND THEIR APPLICATION FOR MATHEMATICAL MODELING OF COMPLEX ECOLOGICAL SYSTEMS 

Butusov Oleg Borisovich, doctor of physical and mathematical sciences, professor, sub-department of mathematics, Moscow Polytechnic University (107023, 38 B. Semenovskaya street, Moscow, Russia), E-mail: butusov-1@mail.ru
Dikusar Vasily Vasilevich, doсtor of physical and mathematical sciences, professor, sub-department of higher mathematics, Moscow Institute of Physics and Technology (141701, 9 Institutsky lane, Dolgoprudny, Russia), E-mail: dikussar@yandex.ru
Redikul'tseva Nina Ivanovna, candidate of technical sciences, associate professor, sub-department of applied mathematics, Moscow Humanitarian University (111395, 5 Yunosty street, Moscow, Russia), E-mail: redik_ni@mail.ru 

519.8 004.3 

10.21685/2307-4205-2019-4-2 

For mathematical modeling of complex ecological systems and processes, the classical Verhulst model is proposed with the following modifications: 1) it was proposed to model complex processes of forest ecosystem degradation under the influence of industrial emissions using positive and negative virtual biomass flows; 2) it was proposed to model the influence of unknown or littleknown parameters using fuzzy numbers (fuzzy initial conditions and fuzzy virtual biomass flows); 3) it was proposed to use vegetation periods rather than months, as a unit of time for mathematical modelling of forest degradation in northern latitudes. Mathematical modeling shows the presence of a threshold in polluted ecosystem dynamics, near which evolutionary processes change to catastrophic ones. For computer simulation, three models are proposed. The first model of virtual fuzzy dynamics modeling using the separately upper and separately lower boundaries of the virtual fuzziness interval (the model is designated as “OA” (Ordinary Arithmetic)). The second model of fuzzy dynamics using a right-hand derivative (the model is designated as “IA1” (Interval Arithmetic-1)). The third model of fuzzy dynamics with using the leftside derivative (the model is designated as “IA2” (Interval Arithmetic-2)). The “OA” model shows asymptotic dynamics. However, fuzziness boundaries can converge to different limits. The “IA1” model has no asymptotes and the fuzziness interval catastrophically diverge. The “IA2” model has asymptotes, the fuzziness interval quickly approaches the stationary limit and after that remains constant. 

fuzzy derivative, fuzzy differential equations, modified Verhulst equation, virtual negative biomass flow, interval arithmetic, environmental systems quality, environmental monitoring