Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

Permanence Analysis for Continuous and Discrete-time Generalized Lotka-Volterra Models with Delay and Switching of Parameters

Author(s):

Alexander Yur'evich Aleksandrov

Doctor of Physical and Mathematical Sciences, Professor, Head of the Department of Control of Medical and Biological Systems, St. Petersburg State University,

a.u.aleksandrov@spbu.ru

Abstract:

The paper is addressed to the permanence problem for a generalized Lotka-Volterra system modeling interaction of species in a biological community. The impact of a constant delay and switching of parameters on the dynamics of the system is taken into account. Our analysis is based on the Lyapunov direct method. An original construction of a Lyapunov--Krasovskii functional is proposed. The conditions for the existence of such a functional are formulated in terms of feasibility of special systems of linear algebraic inequalities. It is proved that, under these conditions, the investigated system is permanent for any constant positive delay and any admissible switching signal. In addition, a discrete-time counterpart of the considered model is studied for which the permanence analysis is fulfilled, as well. A comparison of the obtained permanence conditions with known ones is provided. It is shown that the constraints on the system parameters derived in this paper are less conservative.

Keywords

References:

  1. Hofbauer J., Sigmund K. Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge (1998)
  2. Pykh Yu. A. Ravnovesie i ustojchivost' v modelyah populyacionnoj dinamiki [Equilibrium and Stability in Models of Population Dynamics]. Nauka Publ., Moscow (1983). (In Russ. )
  3. Britton N. F. Essential Mathematical Biology, Springer, London, Berlin, Heidelberg (2003)
  4. Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191, Academic Press Inc., Boston (1993)
  5. Chen F. D. Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems. Appl. Math. Comput., 182 (1), 3-12 (2006)
  6. Chen F., Wu L., Li Z. Permanence and global attractivity of the discrete Gilpin-Ayala type population model. Computers and Mathematics with Applications, 53, 1214-1227 (2007)
  7. Lu Z., Wang W. Permanence and global attractivity for Lotka-Volterra difference systems. J. Math. Biol., 39, 269-282 (1999)
  8. Zhu C., Yin G. On hybrid competitive Lotka-Volterra ecosystems. Nonlinear Analysis. Theory, Methods & Applications, 71 (12), e1370-e1379 (2009)
  9. Buscarino A., Belhamel L., Bucolo M., Palumbo P., Manes C. Modeling a population of switches via chaotic dynamics. IFAC-PapersOnLine, 53 (2), 16791-16795 (2020)
  10. Xu R., Chaplain M. A. J., Davidson F. A. Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure. J. Math. Anal. Appl., 303, 602-621 (2005)
  11. Lenhart S. M., Travis C. C. Global stability of a biological model with time delay. Proc. Am. Math. Soc., 96 (1), 75-78 (1986)
  12. Ruan S. G. On nonlinear dynamics of predator-prey models with discrete delay. Math. Model. Nat. Phenom., 4 (2), 140-188 (2009)
  13. Lu G., Lu Z., Lian X. Delay effect on the permanence for Lotka-Volterra cooperative system. Nonlinear Analysis, 11, 2810-2816 (2010)
  14. Wu Zh., Huang H., Wang L. Stochastic delay population dynamics under regime switching: Permanence and asymptotic estimation. Abstract and Applied Analysis, 2013, Article ID 129072 (2013)
  15. Aleksandrov A. Yu., Aleksandrova E. B., Platonov A. V. Ultimate boundedness conditions for a hybrid model of population dynamics. Proc. 21st Mediterranean conference on Control and Automation, June 25-28, 2013. Platanias-Chania, Crite, Greece, 2013, pp. 622-627
  16. Aleksandrov A. Yu., Chen Y., Platonov A. V. Permanence and ultimate boundedness for discrete-time switched models of population dynamics. Nonlinear Dynamics and Systems Theory, 14 (1), 1-10 (2014)
  17. Li X., Yin G. Logistic models with regime switching: Permanence and ergodicity. J. Math. Anal. Appl., 441 (2), 593-611 (2016)
  18. Hu H., Wang K., Wu D. Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays. J. Math. Anal. Appl., 377, 145-160 (2011)
  19. Nakata Y., Muroya Y. Permanence for nonautonomous Lotka-Volterra cooperative systems with delays. Nonlinear Analysis, 11, 528-534 (2010)
  20. Aleksandrov A. Yu. Permanence conditions for models of population dynamics with switches and delay. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 16 (2), 88-99 (2020). (In Russ. )
  21. Pastravanu O. C., Matcovschi M. -H.. Max-type copositive Lyapunov functions for switching positive linear systems. Automatica, 50 (12), 3323-3327 (2014)
  22. Aleksandrov A. Y. Construction of the Lyapunov-Krasovskii functionals for some classes of positive delay systems. Siberian Math. J., 59 (5), 753-762 (2018)
  23. Aleksandrov A. Y., Mason O. On diagonal stability of positive systems with switches and delays. Automation and Remote Control, 79 (12), 2114-2127 (2018)
  24. Aleksandrov A. On the existence of diagonal Lyapunov-Krasovskii functionals for a class of nonlinear positive time-delay systems. Automatica, 160, Art. no. 111449. (2024)

Full text (pdf)