Comparison theorem for systems of differential equations and its application to estimate the average time benefit from resource collection

A. A. Bazulkina ; L. I. Rodina

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 63 (2024), pp. 3-17.

Résumé

One of the versions of the comparison theorem for systems of ordinary differential equations is proved, the consequence of which is the property of monotonicity of solutions with respect to initial data. We consider the problem of estimating the average time profit from resource extraction for a structured population consisting of individual species $x_1,\ldots,x_n$, or divided into $n$ age groups. We assume that the dynamics of the population in the absence of exploitation is given by a system of differential equations $\dot x =f(x)$, and at times $\tau(k)=kd$, $d>0$, a certain share of the biological resource is extracted from the population $u(k)=(u_1(k),\ldots,u_n(k))\in [0,1]^n$, $k=1,2,\ldots.$ It is shown that using the comparison theorem it is possible to find estimates of the average time benefit in cases when analytical solutions for relevant systems are not known. The results obtained are illustrated for models of interaction between two species, such as symbiosis and competition. It is shown that for models of symbiosis, commensalism and neutralism, the greatest value of the average time profit is achieved with the simultaneous exploitation of two types of resources. For populations between which an interaction of the “competition” type is observed, cases in which it is advisable to extract only one type of resource are highlighted.

Détail

Keywords: comparison theorems, average time profit
Mots-clés : optimal exploitation

Bibliographie

[1] Chaplygin S.A., “Foundations of a new method of approximate integration of differential equations”, Collected works, v. 1, Gostekhizdat, Moscow, 1919, 348–368

[2] Vasil’eva A.B., Nefedov N.N., Comparison theorems. Chaplygin’s method of differential inequalities, Moscow State University, Moscow, 2007

[3] Chaplygin S.A., Approximate integration of a system of two differential equations, Gostekhizdat, Moscow, 1920, 402–419

[4] Babkin B.N., “Approximate integration of systems of ordinary differential equations of the first order by the method of S.A. Chaplygin”, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 18:5 (1954), 477–484 (in Russian) | MR | Zbl

[5] Khan Z.A., “On some fundamental integrodifferential inequalities”, Applied Mathematics, 5:19 (2014), 2968–2973 | DOI

[6] Takamura H., “Improved Kato’s lemma on ordinary differential inequality and its application to semilinear wave equations”, Nonlinear Analysis, 125 (2015), 227–240 | DOI | MR | Zbl

[7] Zhukovskiy E.S., Takhir Kh.M.T., “Comparison of solutions to boundary-value problems for linear functional-differential equations”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 28:3 (2018), 284–292 (in Russian) | DOI | MR | Zbl

[8] Arutyunov A.V., Zhukovskiy E.S., Zhukovskiy S.E., “Coincidence points principle for set-valued mappings in partially ordered spaces”, Topology and its Applications, 201 (2016), 330–343 | DOI | MR | Zbl

[9] Benarab S., “On Chaplygin’s theorem for an implicit differential equation of order $n$”, Russian Universities Reports. Mathematics, 26:135 (2021), 225–233 (in Russian) | DOI | Zbl

[10] Benarab S., Zhukovskiy E.S., “On the conditions of existence coincidence points for mappings in partially ordered spaces”, Tambov University Reports. Series: Natural and Technical Sciences, 23:121 (2018), 10–16 (in Russian) | DOI

[11] Zhukovskiy E.S., “On order covering maps in ordered spaces and Chaplygin-type inequalities”, St. Petersburg Mathematical Journal, 30:1 (2019), 73–94 | DOI | MR

[12] Kuzenkov O.A., Ryabova E.A., Mathematical modeling of processes of selection, Nizhny Novgorod State University, Nizhny Novgorod, 2007

[13] Matrosov V.M., “The comparison principle with a vector-valued Ljapunov function. I”, Differential Equations, 4:8 (1968), 1374–1386 (in Russian) | Zbl

[14] Ważewski T., “Systèmes des èquations et des inègalitès diffèrentieles ordinaires aux deuxièmes membres monotones et leurs applications”, Annales de la Sociètè Polonaise de Mathèmatique, 23 (1950), 112–166 | Zbl

[15] Rodina L.I., Woldeab M.S., “On the monotonicity of solutions of nonlinear systems with respect to the initial conditions”, Differential Equations, 59:8 (2023), 1025–1031 | DOI | DOI | Zbl

[16] Frisman Y.Y., Kulakov M.P., Revutskaya O.L., Zhdanova O.L., Neverova G.P., “The key approaches and review of current researches on dynamics of structured and interacting populations”, Computer Research and Modeling, 11:1 (2019) (in Russian) | DOI

[17] Kirillov A.N., Sazonov A.M., “A hybrid model of population dynamics with refuge–regime: regularization and self-organization”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 33:3 (2023), 467–482 (in Russian) | DOI

[18] Rodina L.I., “Optimization of average time profit for a probability model of the population subject to a craft”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 28:1 (2018) (in Russian) | DOI | Zbl

[19] Rodina L.I., “Properties of average time profit in stochastic models of harvesting a renewable resourse”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 28:2 (2018), 213–221 (in Russian) | DOI | Zbl

[20] Woldeab M.S., Rodina L.I., “About the methods of extracting a biological resource that provide the maximum average time benefit”, Russian Mathematics, 66:1 (2022), 8–18 | DOI | DOI | Zbl

[21] Riznichenko G.Yu., Lectures on mathematical models in biology. Part 1. Mathematical biology, biophysics, Regular and Chaotic Dynamics, Izhevsk, 2002

[22] Woldeab M.S., Rodina L.I., “About the methods of renewable resource extraction from the structured population”, Russian Universities Reports. Mathematics, 27:137 (2022), 16–26 (in Russian) | DOI

[23] Belyakov A.O., Davydov A.A., “Efficiency optimization for the cyclic use of a renewable resource”, Proceedings of the Steklov Institute of Mathematics, 299:1 (2017), 14–21 | DOI | DOI

[24] Aseev S.M., Veliov V.M., “Another view of the maximum principle for infinite-horizon optimal control problems in economics”, Russian Mathematical Surveys, 74:6 (2019), 963–1011 | DOI | DOI | Zbl

[25] Rodina L.I., Hammadi A.H., “Optimization problems for models of harvesting a renewable resourse”, Journal of Mathematical Sciences, 250:1 (2020), 113–122 | DOI | Zbl

[26] Rodina L.I., Chernikova A.V., “On infinite-horizon optimal exploitation of a renewable resource”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 29:1 (2023), 167–179 (in Russian) | DOI