Abstract
A distributed renewable resource of any nature is considered on a two-dimensional sphere. Its dynamics is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource is 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 limiting dynamics corresponding to this strategy and there is an admissible harvesting strategy that maximizes the time-averaged harvesting of the resource.
REFERENCES
P. F. Verhulst, “Notice sur la loi que la population poursuit dans son accroissement,” Corresp. Math. Phys. 10, 113–121 (1838).
V. I. Arnold, Catastrophe Theory (Nauka, Moscow, 1990) [in Russian].
V. I. Arnold, “Hard” and “Soft” Mathematical Models (Mosk. Tsentr Neprer. Mat. Obrazovan., Moscow, 2014) [in Russian].
A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Byull. Mosk. Gos. Univ. Mat. Mekh. 1 (6), 1–26 (1937).
R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7 (4), 353–369 (1937). https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
J. B. J. Fourier, Theorie Analytique de la Chaleur (Didot, Paris, 1822).
H. Berestycki, H. Francois, and L. Roques, “Analysis of the periodically fragmented environment model: I. Species persistence,” J. Math. Biol. 51, 75–113 (2005). https://doi.org/10.1007/s00285-004-0313-3
H. Berestycki, H. Francois, and L. Roques, “Analysis of the periodically fragmented environment model: II. Biological invasions and pulsating travelling fronts,” J. Math. Pures Appl. 84, 1101–1146 (2005). https://doi.org/10.1016/j.matpur.2004.10.006
B. Pethame, Parabolic Equations in Biology: Growth, Reaction, Movement and Diffusion (Springer, Cham, 2015).
A. A. Davydov, “Existence of optimal stationary states of exploited populations with diffusion,” Proc. Steklov Inst. Math. 310, 124–130 (2020). https://doi.org/10.1134/S0081543820050090
A. A. Davydov, “Optimal steady state of distributed population in periodic environment,” AIP Conf. Proc. 2333, 120007 (2021). https://doi.org/10.1063/5.0041960
A. A. Davydov and D. A. Melnik, “Optimal states of distributed exploited populations with periodic impulse harvesting,” Proc. Steklov Inst. Math. 315, Suppl. 1, S1–S8 (2021). https://doi.org/10.1134/S0081543821060079
A. A. Davydov and E. V. Vinnikov, “Optimal cyclic dynamic of distributed population under permanent and impulse harvesting,” Dynamic Control and Optimization: DCO 2021 (Springer, Cham, 2023), pp. 101–112. https://doi.org/10.1007/978-3-031-17558-9_5
D. V. Tunitsky, “On solvability of semilinear second-order elliptic equations on closed manifolds,” Izv. Math. 86 (5), 925–942 (2022). https://doi.org/10.1070/IM9261
D. V. Tunitsky, “On initial value problem for semilinear second order parabolic equations on spheres,” Proceedings of the 15th International Conference on Management of Large-Scale System Development (MLSD), Septem-ber 26–28, 2022, Moscow, Russia (IEEE Explore, 2022), pp. 1–4. https://doi.org/10.1109/MLSD55143.2022.9934193
L. I. Nicolaescu, Lectures on the Geometry of Manifolds (World Scientific, Hackensack, N.J., 2021).
D. V. Tunitsky, “On solvability of second-order semilinear elliptic equations on spheres,” in Proceedings of the 14th International Conference on Management of Large-Scale System Development (MLSD), Septem-ber 27–29, 2021, Moscow, Russia (IEEE Explore, 2021), pp. 1–4. https://doi.org/10.1109/MLSD52249.2021.9600203
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (Am. Math. Soc., Providence, RI, 1997).
J. L. Lions, Equations differentielles operationnelles et problemes aux limites (Springer-Verlag, Berlin, 1961).
R. S. Palais, Seminar on the Atiyah–Singer Index Theorem (Princeton Univ. Press, Princeton, N.J., 1965).
R. O. Wells, Differential Analysis on Complex Manifolds (Springer, New York, 2008).
B. O. Koopman, “The theory of search: III. The optimum distribution of search effort,” Oper. Res. 5 (5), 613–626 (1957).
V. V. Zhikov, “Mathematical problems of search theory,” in Proceedings of the Vladimir Polytechnic Institute (Vysshaya Shkola, Moscow, 1968), pp. 263–270 [in Russian].
G. M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, Hackensack, N.J., 2005).
Funding
This work was supported by the Russian Science Foundation, project no. 19-11-00223, and was performed at Lomonosov Moscow State University.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Vinnikov, E.V., Davydov, A.A. & Tunitsky, D.V. Existence of a Maximum of Time-Averaged Harvesting in the KPP Model on Sphere with Permanent and Impulse Harvesting. Dokl. Math. 108, 472–476 (2023). https://doi.org/10.1134/S1064562423701387
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562423701387