maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2025-3-242-251mais-1961Research ArticleComputing Methodologies and ApplicationsАлгоритм исследования динамики пространственно-распределенного логистического уравнения с запаздыванием и учетом миграцииAlgorithm for studying the dynamics of a spatially distributed logistic equation with delay and taking into account migrationhttps://orcid.org/0009-0006-1946-6806КащенкоДмитрий СергеевичKashchenkoDmitry S.d.kashchenko@uniyar.ac.ruhttps://orcid.org/0000-0001-9500-8280ЛогиновДмитрий ОлеговичLoginovDmitry O.dimonl@inbox.ruhttps://orcid.org/0000-0001-5668-3929ТолбейАнна ОлеговнаTolbeyAnna O.a.tolbey@uniyar.ac.ruЯрославский государственный университет им. П.Г. ДемидоваРоссияP.G. Demidov Yaroslavl State UniversityRussian Federation202524092025323242251Copyright © Кащенко Д.С., Логинов Д.О., Толбей А.О., 20252025Кащенко Д.С., Логинов Д.О., Толбей А.О.Kashchenko D.S., Loginov D.O., 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/1961

Рассматривается важное в математической экологии логистическое уравнение с запаздыванием и диффузией. Предполагается, что граничные условия на одном из концов отрезка [0,1] содержат параметр. Исследован вопрос о локальной — в окрестности состояния равновесия — динамике соответствующей краевой задачи при всех значениях параметров граничных условий. Выделены критические случаи в задаче об устойчивости состояния равновесия и построены нормальные формы — скалярные комплексные обыкновенные дифференциальные уравнения первого порядка. Их нелокальная динамика определят поведение решений исходной задачи в малой окрестности состояния равновесия.

The logistic equation with delay and diffusion, which is important in mathematical ecology, is considered. It is assumed that the boundary conditions at one end of the interval [0,1] contain a parameter. The question of local — in the neighborhood of the equilibrium state — dynamics of the corresponding boundary value problem for all values of the boundary condition parameters is investigated. Critical cases in the problem of stability of the equilibrium state are identified and normal forms — scalar complex ordinary differential equations of the first order — are constructed. Their nonlocal dynamics determine the behavior of solutions of the original problem in a small neighborhood of the equilibrium state.

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