EXISTENCE OF MAXIMUM OF TIME AVERAGED HARVESTING IN THE KPP-MODEL ON SPHERE WITH PERMANENT AND IMPULSE COLLECTION

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

On a two-dimensional sphere, a distributed renewable resource is considered, the dynamics of which is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource, carried out by constant or periodic impulse harvesting. It is shown that after choosing an admissible exploitation strategy, the dynamics of the resource tend to the limiting dynamics corresponding to this strategy, and that there is an admissible harvesting strategy that maximizes the time averaged harvesting of the resource.

About the authors

E. V. Vinnikov

Lomonosov Moscow State University; NUST MISIS

Author for correspondence.
Email: evinnikov@gmail.com
Russian Federation, Moscow; Russian Federation, Moscow

A. A. Davydov

Lomonosov Moscow State University; NUST MISIS

Author for correspondence.
Email: davydov@mi-ras.ru
Russian Federation, Moscow; Russian Federation, Moscow

D. V. Tunitsky

Institute of Control Sciences RAS

Author for correspondence.
Email: dtunitsky@yahoo.com
Russian Federation, Moscow

References

  1. Verhulst P.F. Notice sur la loi que la population poursuit dans son accroissement // Correspondance mathematique et physique. 1838. V. 10. P. 113–121.
  2. Арнольд В.И. Теория катастроф. М.: Наука, 1990. 128 с.
  3. Арнольд В.И. “Жесткие” и “мягкие” математические модели Электронное издание М.: МЦНМО, 2014. 32 с. ISBN 978-5-4439-2008-5.
  4. Колмогоров А.Н., Петровский И.Г., Пискунов Н.С. Исследование уравнения диффузии, соединенной с возрастанием вещества, и его применение к одной биологической проблеме // Бюллетень МГУ. Сер. А. Математика и Механика. 1937. Т. 1. № 6. С. 1–26.
  5. Fisher R.A. The Wave of Advance of Advantageous Genes//Annals of Eugenics, 1937. 7 (4), pp. 353–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
  6. Fourier J.B.J. Theorie Analytique de la Chaleur. Paris: F. Didot, 1822.
  7. Berestycki H., Francois H., Roques L. Analysis of the periodically fragmented environment model: I Species persistence// J. Math. Biol. 2005. V. 51. P. 75–113. https://doi.org/10.1007/s00285-004-0313-3
  8. Berestycki H., Francois H., Roques L. Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts. J. Math. Pures Appl. 2005. V. 84. P. 1101–1146. https://doi.org/10.1016/j.matpur.2004.10.006
  9. Pethame B. Parabolic equations in biology: Growth, reaction, movement and diffusion. Springer, 2015, Lecture Notes on Mathematical Modelling in the Life Sciences, 978-3-319-19499-8; 978-3-319-19500-1.
  10. Давыдов А.А. Существование оптимальных стационарных состояний эксплуатируемых популяций с диффузией//Избранные вопросы математики и механики, Сборник статей. К 70-летию со дня рождения академика Валерия Васильевича Козлова, Труды МИАН, 310, МИАН, М., 2020. P. 135–142; Proc. Steklov Inst. Math. 2020. V. 310. P. 124–130. https://doi.org/10.1134/S0081543820050090
  11. Davydov A.A. Optimal steady state of distributed population in periodic environment// AIP Conf. Proc. 2021. V. 2333. P. 120007. https://doi.org/10.1063/5.0041960
  12. Давыдов А.А., Мельник Д.А. Оптимальные состояния распределенных эксплуатируемых популяций с периодическим импульсным отбором// Тр. ИММ УрО РАН. 2021. Т. 27. № 2. С. 99–107. Optimal States of Distributed Exploited Populations with Periodic Impulse Harvesting // Proc. Steklov Inst. Math. 2021. V. 315 (Suppl. 1). P. S1–S8. https://doi.org/10.1134/S008154382106007910.1134/S0081543821060079 https://doi.org/10.21538/0134-4889-2021-27-2-99-107
  13. Davydov A.A., Vinnikov E.V. Optimal cyclic dynamic of distributed population under permanent and impulse harvesting// Dynamic Control and Optimization. DCO 2021. Springer Proceedings in Mathematics & Statistics. 2023. V. 407. P. 101–112. https://doi.org/10.1007/978-3-031-17558-9_5
  14. Туницкий Д.В. О разрешимости полулинейных эллиптических уравнений второго порядка на замкнутых многообразиях// Изв. РАН. Сер. матем. 2022. V. 86. № 5. P. 97–115. https://doi.org/10.4213/im9261
  15. Tunitsky D.V. On Initial Value Problem for Semilinear Second Order Parabolic Equations on Spheres// Proceedings of the 15th International Conference “Management of large-scale system development” (MLSD), 26–28 September, 2022, Moscow, Russia. IEEE Explore, 9 November 2022. P. 1–4. https://ieeexplore.ieee.org/document/9934193 https://doi.org/10.1109/MLSD55143.2022.9934193
  16. Nicolaescu L.I. Lectures on the Geometry of Manifolds. New Jersey: World Scientific, 2021.
  17. Tunitsky D.V. On Solvability of Second-Order Semilinear Elliptic Equations on Spheres / Proceedings of the 14th International Conference “Management of large-scale system development” (MLSD), 27–29 September, 2021, Moscow, Russia. IEEE Explore, 22 November 2021. P. 1–4. https://ieeexplore.ieee.org/document/9600203. ISBN 978-1-6654-1230-8 https://doi.org/10.1109/MLSD52249.2021.9600203
  18. Showalter R.E. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. AMS, Providence, RI, 1997.
  19. Lions J.L. Equations differentielles operationnelles et problemes aux limites, Springer-Verlag, Berlin, 1961.
  20. Пале Р. Семинар по теореме Атьи–Зингера об индексе. М.: Мир, 1970.
  21. Уэллс Р. Дифференциальное исчисление на комплексных многообразиях. М.: Мир, 1976.
  22. Koopman B.O. The theory of search. III. The optimum distribution of search effort // Operations Res. 1957. V. 5. № 5. P. 613–626.
  23. Жиков В.В. Математические проблемы теории поиска// Тр. Владимир. политех. ин-та. 1968. С. 263–270.
  24. Lieberman G.M. Second Order Parabolic Differential Equations. New Jersey: World Scientific, 2005.

Copyright (c) 2023 Е.В. Винников, А.А. Давыдов, Д.В. Туницкий

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies