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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2024 Volume 30, Number 3, Pages 113–121 (Mi timm2108)

This article is cited in 1 paper

Existence of an optimal stationary solution in the KPP model under nonlocal competition

A. A. Davydovab, A. S. Platovc, D. V. Tunitskyd

a Lomonosov Moscow State University
b International Institute for Applied Systems Analysis, Laxenburg
c National University of Science and Technology «MISIS», Moscow
d V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow

Abstract: We consider a resource distributed on a compact closed connected manifold without edge, for example, on a two-dimensional sphere representing the Earth surface. The dynamics of the resource is governed by a model of the Fisher–Kolmogorov–Petrovsky–Piskunov type with coefficients in the reaction term depending on the total amount of the resource, which makes the model equation nonlocal. Under natural assumptions about the model parameters, it is shown that there is at most one nontrivial nonnegative stationary distribution of the resource. Moreover, in the case of constant distributed resource harvesting, there is a harvesting strategy under which such a distribution maximizes the time-averaged resource harvesting over the stationary states.

Keywords: KPP model, stationary solution, time-averaged harvesting, optimal strategy.

UDC: 517.97

MSC: 35K57, 49J20, 92D25, 91B76, 35A01

Received: 24.03.2024
Revised: 13.06.2024
Accepted: 17.06.2024

DOI: 10.21538/0134-4889-2024-30-3-113-121


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2024, 327, suppl. 1, S66–S73

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© Steklov Math. Inst. of RAS, 2025