Abstract
The paper presents a vector approach to control problems for systems of equations. The method is described in the case of Kolmogorov systems which arise frequently in the dynamics of populations. Three types of problems are discussed: problems with control of both per capita growth rates, problems with control parameters acting on the growth rates, and problems which combine the first two types. The controllability is obtained via a vector approach based on the Perov fixed point theorem and matrices which are convergent to zero. Four concrete illustrative examples are added
Authors
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Alexandru Hofman
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Keywords
Kolmogorov system; control problem, fixed point; matrix convergent to zero; differential equations and systems; Volterra-Fredholm integral equation; Lotka-Volterra system.
Paper coordinates
A. Hofman, R. Precup, Vector fixed point approach to control of Kolmogorov differential systems, Contemporary Mathematics, 2 (2024) no. 5, pp. 1311-1425. https://doi.org/10.37256/cm.5220242840
About this paper
Journal
Contemporary Mathematics
Publisher Name
Universal Wiser
Print ISSN
2705-1064
Online ISSN
2705-1056
google scholar link
[2] Leggett R, Williams L., A fixed point theorem with application to an infectious disease model. Journal of Mathematical Analysis and Applications. 1980; 76(1): 91-97. Available from: https://doi.org/10.1016/0022-247X(80)90062-1.
[3] Li X, Liu Z, Migórski S., Approximate controllability for second order nonlinear evolution hemivariational inequalities. Electronic Journal of Qualitative Theory of Differential Equations. 2015; 100: 1-16. Available from: https://doi.org/10.14232/ejqtde.2015.1.100.
[4] Mahmudov NI, Udhayakumar R, Vijayakumar V., On the approximate controllability of second-order evolution hemivariational inequalities. Results in Mathematics. 2020; 75: 160. Available from: https://doi.org/10.1007/s00025-020-01293-2.
[5] Dineshkumar C, Udhayakumar R, Vijayakumar V, Shukla A, Nisar KS., New discussion regarding approximate controllability for Sobolev-type fractional stochastic hemivariational inequalities of order r∈ (1, 2). Communications in Nonlinear Science and Numerical Simulation. 2023; 116: 106891. Available from: https://doi.org/10.1016/j.cnsns.2022.106891.
[6] Coron JM. Control and nonlinearity., Mathematical surveys and monographs, vol. 136. United States of America: American. Mathematical. Society. 2007. Available from: https://doi.org/10.1090/surv/136.
[7] Precup R., On some applications of the controllability principle for fixed point equations. Results in Applied Mathematics. 2022; 13: 100236. Available from: https://doi.org/10.1016/j.rinam.2021.100236.
[8] Haplea IŞ, Parajdi LG, Precup R., On the controllability of a system modeling cell dynamics related to leukemia. Symmetry. 2021; 13(10) : 1867. Available from: https://doi.org/10.3390/sym13101867.
[9] Hofman A, Precup R., On some control problems for Kolmogorov type systems. Mathematical Modelling and Control. 2022; 2(3): 90-99. Available from: https://doi.org/10.3934/mmc.2022011.
[10] Hofman A., An algorithm for solving a control problem for Kolmogorov systems. Studia Universitatis Babeş-Bolyai Mathematica. 2023; 68(2): 331-340. Available from: http://doi.org/10.24193/subbmath.2023.2.09.
[11] Precup R., The role of matrices that are convergent to zero in the study of semilinear operator systems. Mathematical and Computer Modelling. 2009; 49(3-4): 703-708. Available from: https://doi.org/10.1016/j.mcm.2008.04.006.
[12] Precup R., Methods in nonlinear integral equations. Dordrecht: Springer; 2002. Available from: https://doi.org/10.1007/978-94-015-9986-3.
[13] K. Sigmund, Kolmogorov and population dynamics. In: Charpentier É, Lesne A, Nikolski NK (eds.) Kolmogorov’s heritage in mathematics. Berlin: Springer; 2007. p.177-186. Available from: https://doi.org/10.1007/978-3-540-36351-4_9.
[14] Kolmogorov AN., Sulla teoria di Volterra della lotta per l’esistenza. Giornale dell Istituto Italiano degli Attuari. 1936; 7: 74-80.
[15] Murray JD., Mathematical Biology. I: An introduction. In: Murray JD (ed.) Interdisciplinary Applied Mathematics. Vol. 1. New York: Springer; 2011. Available from: https://doi.org/10.1007/b98868.
[16] Brauer F, Castillo-Chavez C., Mathematical models in population biology and epidemiology. 2th ed. Berlin: Springer; 2012. Available from: https://doi.org/10.1007/978-1-4614-1686-9.
[17] Allen LJS, An introduction to mathematical biology. London: Pearson Education; 2006.
[18] Lin Q., Allee effect increasing the final density of the species subject to the Allee effect in a Lotka-Volterra commensal symbiosis model. Advances in Difference Equations. 2018; 2018(196): 1-9. Available from: https://doi.org/10.1186/s13662-018-1646-3.
[19] He X, Zhu Z, Chen J, Chen F., Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect. Open Mathematics. 2022; 20(1): 646-665. Available from: https://doi.org/10.1515/math-2022-0055.
[20] Granas A, Dugundji J., Fixed Point Theory. Springer monographs in mathematics. New York: Springer; 2003. Available from: https://doi.org/10.1007/978-0-387-21593-8.
[21] Parajdi LG, Pặtrulescu F, Precup R, Haplea IŞ., Two numerical methods for solving a nonlinear system of integral equations of mixed Volterra-Fredholm type arising from a control problem related to leukemia. Journal of Applied Analysis & Computation. 2023; 13(4): 1797-1812. Available from: https://doi.org/10.11948/20220197.