Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay
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 us...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2020-09-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/1353 |
| _version_ | 1797877839252422656 |
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| author | Sergey D. Glyzin Sergey A. Kashchenko Anna O. Tolbey |
| author_facet | Sergey D. Glyzin Sergey A. Kashchenko Anna O. Tolbey |
| author_sort | Sergey D. Glyzin |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T02:24:25Z |
| format | Article |
| id | doaj.art-e59c6e45f7f2471cb0276c29659ad388 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:24:25Z |
| publishDate | 2020-09-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-e59c6e45f7f2471cb0276c29659ad3882023-03-13T08:07:35ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172020-09-0127334435510.18255/1818-1015-2020-3-344-3551012Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with DelaySergey D. Glyzin0Sergey A. Kashchenko1Anna O. Tolbey2Центр интегрируемых систем, Ярославский государственный университет им. П.Г. ДемидоваЦентр интегрируемых систем, Ярославский государственный университет им. П.Г. ДемидоваЦентр интегрируемых систем, Ярославский государственный университет им. П.Г. ДемидоваThe 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.https://www.mais-journal.ru/jour/article/view/1353логистическое уравнение с запаздываниемотображениебифуркации |
| spellingShingle | Sergey D. Glyzin Sergey A. Kashchenko Anna O. Tolbey Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay Моделирование и анализ информационных систем логистическое уравнение с запаздыванием отображение бифуркации |
| title | Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay |
| title_full | Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay |
| title_fullStr | Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay |
| title_full_unstemmed | Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay |
| title_short | Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay |
| title_sort | features of the algorithmic implementation of difference analogues of the logistic equation with delay |
| topic | логистическое уравнение с запаздыванием отображение бифуркации |
| url | https://www.mais-journal.ru/jour/article/view/1353 |
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