maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2020-3-344-355mais-1353Research ArticleComputer System OrganizationОсобенности алгоритмической реализации разностных аналогов логистического уравнения с запаздываниемFeatures of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delayhttps://orcid.org/0000-0002-6403-4061ГлызинСергей ДмитриевичGlyzinSergey D.

Заведующий кафедрой компьютерных сетей, доктор физико-математических наук, профессор.

150003, Ярославль, ул. Советская, 14

Head of the Department of Computer Networks, professor.

Sovetskaya str., 14, Yaroslavl, 150003

glyzin@uniyar.ac.ruhttps://orcid.org/0000-0002-4846-6040КащенкоСергей АлександровичKashchenkoSergey A.

Первый проректор, доктор физико-математических наук, профессор.

150003, Ярославль, ул. Советская, 14

First Vice-Rector, professor.

Sovetskaya str., 14, Yaroslavl, 150003

kasch@uniyar.ac.ruhttps://orcid.org/0000-0001-5668-3929ТолбейАнна ОлеговнаTolbeyAnna O.

Доцент кафедры компьютерных сетей, кандидат физико-математических наук.

150003, Ярославль, ул. Советская, 14

Associate Professor, PhD.

Sovetskaya str., 14, Yaroslavl, 150003

bekva@yandex.ruЦентр интегрируемых систем, Ярославский государственный университет им. П.Г. ДемидоваРоссияCentre of Integrable Systems, P. G. Demidov Yaroslavl State UniversityRussian Federation202020092020273344355Copyright © Глызин С.Д., Кащенко С.А., Толбей А.О., 20202020Глызин С.Д., Кащенко С.А., Толбей А.О.Glyzin S.D., Kashchenko S.A., Tolbey A.O.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/1353

Логистическое уравнение с запаздыванием или уравнение Хатчинсона представляет собой одно из фундаментальных уравнений популяционной динамики и находит широкое применение в задачах математической экологии. В работе рассматривается семейство отображений, построеннное для этого уравнения на основе центральных разделенных разностей. Такие разностные схемы обычно используются при численном моделировании данной задачи. Представленные отображения сами по себе могут служить моделями динамики популяций, поэтому их изучение представляет значительный интерес. В работе сопоставляются свойства траекторий данных отображений и исходного уравнения с запаздыванием. Показано, что поведение решений отображений, построенных на основе центральных разделенных разностей, не сохраняет, даже при достаточно малой величине шага по времени, основных динамических свойств логистического уравнения с запаздыванием. В частности, у этого отображения при колебательной потере устойчивости ненулевого состояния равновесия не бифурцирует устойчивая инвариантная кривая. Эта кривая соответствует в таких отображениях устойчивому предельному циклу исходного непрерывного уравнения. Тем самым показано, что такая разностная схема не может быть использована для численного моделирования логистического уравнения с запаздыванием.

The logistic equation with delay or Hutchinson’s equation is one of the fundamental equations of population dynamics and is widely used in problems of mathematical ecology. We consider a family of mappings built for this equation based on central separated differences. Such difference schemes are usually used in the numerical simulation of this problem. The presented mappings themselves can serve as models of population dynamics; therefore, their study is of considerable interest. We compare the properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of the mappings constructed on the basis of the central separated differences does not preserve, even with a sufficiently small value of the time step, the basic dynamic properties of the logistic equation with delay. In particular, this map does not have a stable invariant curve bifurcating under the oscillatory loss of stability of a nonzero equilibrium state. This curve corresponds in such mappings to the stable limit cycle of the original continuous equation. Thus, it is shown that such a difference scheme cannot be used for numerical modeling of the logistic equation with delay.

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The authors declare that there are no conflicts of interest present.