Geodesic
Existence of Optimal Stationary States of Exploited Populations with Diffusion
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 135-142.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study population dynamics with diffusion described by a parabolic equation with a logistic reaction term in the presence of exploitation consisting in constant harvesting of a part of the population density. Under natural constraints on the parameters of the model, we prove that there exists a stable stationary state of the population that provides the maximum profit of exploitation in the natural form. 
@article{TM_2020_310_a8,
     author = {A. A. Davydov},
     title = {Existence of {Optimal} {Stationary} {States} of {Exploited} {Populations} with {Diffusion}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {135--142},
     publisher = {mathdoc},
     volume = {310},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2020_310_a8/}
}
TY  - JOUR
AU  - A. A. Davydov
TI  - Existence of Optimal Stationary States of Exploited Populations with Diffusion
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2020
SP  - 135
EP  - 142
VL  - 310
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2020_310_a8/
LA  - ru
ID  - TM_2020_310_a8
ER  - 
%0 Journal Article
%A A. A. Davydov
%T Existence of Optimal Stationary States of Exploited Populations with Diffusion
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2020
%P 135-142
%V 310
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2020_310_a8/
%G ru
%F TM_2020_310_a8
A. A. Davydov. Existence of Optimal Stationary States of Exploited Populations with Diffusion. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 135-142. http://geodesic.mathdoc.fr/item/TM_2020_310_a8/

[1] V. I. Arnold, “Optimization in Mean and Phase Transitions in Controlled Dynamical Systems”, Funct. Anal. Appl., 36:2 (2002), 83–92 | DOI | MR | Zbl

[2] S. M. Aseev and A. V. Kryazhimskii, “The Pontryagin maximum principle and optimal economic growth problems”, Proc. Steklov Inst. Math., 257 (2007), 1–255 | DOI | MR | Zbl

[3] S. M. Aseev and V. M. Veliov, “Another view of the maximum principle for infinite-horizon optimal control problems in economics”, Russ. Math. Surv., 74:6 (2019), 963–1011 | DOI | MR | Zbl

[4] Belyakov A.O., Davydov A.A., Veliov V.M., “Optimal cyclic exploitation of renewable resources”, J. Dyn. Control Syst., 21:3 (2015), 475–494 | DOI | MR | Zbl

[5] Berestycki H., Hamel F., Roques L., “Analysis of the periodically fragmented environment model. I: Species persistence”, J. Math. Biol., 51:1 (2005), 75–113 | DOI | MR | Zbl

[6] Colonius F., Kliemann W., “Infinite time optimal control and periodicity”, Appl. Math. Optimization, 20:1 (1989), 113–130 | DOI | MR | Zbl

[7] A. A. Davydov, “Generic profit singularities in Arnold's model of cyclic processes”, Proc. Steklov Inst. Math., 250 (2005), 70–84 | MR | Zbl

[8] A. A. Davydov and H. Mena Matos, “Generic phase transitions and profit singularities in Arnol'd's model”, Sb. Math., 198:1 (2007), 17–37 | DOI | MR | Zbl

[9] Euler L., “Recherches générales sur la mortalité et la multiplication du genre humain”, Mém. acad. sci. Berlin, 1767, 144–164

[10] A. F. Filippov, “On some problems in optimal control theory”, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., Astron., Fiz., Khim., 1959, no. 2, 25–32 | MR | Zbl

[11] A. F. Filippov, “Differential equations with many-valued discontinuous right-hand side”, Sov. Math., Dokl., 4 (1963), 941–945 | MR | Zbl

[12] Fisher R.A., “The wave of advance of advantageous genes”, Ann. Eugenics, 7:4 (1937), 355–369 | DOI | Zbl

[13] A. Kolmogoroff, I. Pretrovsky, and N. Piscounoff, “Étude de l'équation de la diffusion avec croissance de la quantite de matière et son application à un problème biologique”, Bull. Univ. État Moscou, Sér. Int., Sect. A: Math. Méc., 1:6 (1937), 1–26

[14] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, Nauka, Moscow, 1967 | MR | MR

[15] Am. Math. Soc., Providence, RI, 1968 | MR | MR

[16] Malthus T.R., An essay on the principle of population, as it affects the future improvement of society with remarks on the speculations of Mr. Godwin, M. Condorcet, and other writers, J. Johnson, London, 1798

[17] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1964 | MR | Zbl

[18] A. M. Tsirlin, Methods of Averaging Optimization and Their Applications, Nauka, Moscow, 1997 (in Russian) | MR | Zbl

[19] Verhulst P.-F., “Notice sur la loi que la population suit dans son accroissement”, Correspondance math. phys., 10 (1838), 113–121