Stability of solutions to the logistic equation with delay, diffusion and nonclassical boundary conditions

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Resumo

The work is devoted to the logistic equation with delay and diffusion with non-classical boundary conditions. The stability of a nontrivial equilibrium state is investigated, and the resulting bifurcations are studied numerically.

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Sobre autores

I. Kashchenko

P. G. Demidov Yaroslavl State University

Autor responsável pela correspondência
Email: iliyask@uniyar.ac.ru

Regional Scientific and Educational Mathematical Center of Yaroslavl State University

Rússia, Yaroslavl

S. Kashchenko

P. G. Demidov Yaroslavl State University

Email: kasch@uniyar.ac.ru

Regional Scientific and Educational Mathematical Center of Yaroslavl State University

Rússia, Yaroslavl

I. Maslenikov

P. G. Demidov Yaroslavl State University

Email: igor.maslenikov16@yandex.ru

Regional Scientific and Educational Mathematical Center of Yaroslavl State University

Rússia, Yaroslavl

Bibliografia

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  3. Kuang Y. Delay differential equations: with applications in population dynamics. Academic Press, 1993.
  4. Murray J.D. Mathematical biology II: Spatial models and biomedical applications. New York : Springer, 2001. V. 3.
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  6. Кащенко С.А., Логинов Д.О. Бифуркации при варьировании граничных условий в логистическом уравнении с запаздыванием и диффузией // Математические заметки. 2019. Т. 106. № 1. С. 138–143.
  7. Wright E.M. A non-linear difference-differential equation // J. fur die reine und angewandte Math. (Crelles Journal). 1955. V. 194. P. 66–87.
  8. Кащенко С.А. Динамика моделей на основе логистического уравнения с запаздыванием. М.: КРАСАНД, 2020.
  9. Кащенко С.А. , Толбей А.О. Бифуркации в логистическом уравнении с диффузией и запаздыванием в граничном условии // Матем. заметки. 2023. Т. 113. № 6. С. 940–944.
  10. Rudyi A.S. Theoretical fundamentals of the method for thermal diffusivity measurements from auto-oscillation parameters in a system with a thermal feedback // International J. of Thermophysics. 1993. V. 14. P. 159–172.

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2. Fig. 1. Images of curves (11). The stability region of the zero solution (3), (7) is highlighted in gray. Parameter values: T = 1, r = 1, d = 10−1 and a) h = 10−1, b) h = 10−2, c) h = 10−3, d) h = 0.

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3. Fig. 2. The domain Ω for parameters T = 1, r = 1 and a) d = 0.1, b) d = 0.2, c) d = 0.5, d) d = 1.

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4. Fig. 3. Graphs of the solution amplitude of solution (3), (6) for d = 0.1, T = 1, r = 1, x0 = 0.5, a) α = −26.5, b) α = −26.9, c) α = −27.

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5. Fig. 4. Graphs of the dependence u(t,1) (left) and u(t*, x) (right) of the solution (3), (6) for d = 0.1, T = 1, r = 1, x0 = 0.55 α = −26.9.

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6. Fig. 5. Graphs of the solution amplitude of solution (3), (6) for d = 0.1, T = 1, r = 1 , x0 = 0.55, a) α = −38, b) α = −100.5, c) α = −101, d) α = −118.

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7. Fig. 6. Graphs of the dependence u(t,1) (left) and u(t*, x) (right) of the solution (3), (6) for d = 0.1, T = 1, r = 1 , x0 = 0.55, a) α = −100.5, b) α = −118.

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