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Journal of Mathematical Biology

, Volume 78, Issue 4, pp 1225–1244 | Cite as

A discrete-time dynamical system and an evolution algebra of mosquito population

  • U. A. RozikovEmail author
  • M. V. Velasco
Article
  • 121 Downloads

Abstract

Recently, continuous-time dynamical systems of mosquito populations have been studied. In this paper, we consider a discrete-time dynamical system, generated by an evolution quadratic operator of a mosquito population, and show that this system has two fixed points, which become saddle points under some conditions on the parameters of the system. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results.

Keywords

Mathematical model Mosquito dispersal Discrete-time Fixed point Limit point 

Mathematics Subject Classification

92D25 (34C60 34D20 92D30 92D40 ) 

Notes

Acknowledgements

The work partially supported by Projects MTM2016-76327-C3-2-P and MTM2016- 79661-P of the Spanish Ministerio of Economía and Competitividad, and Research Group FQM 199 of the Junta de Andalucía (Spain), all of them include European Union FEDER support; grant 853/2017 Plan Propio University of Granada (Spain); Kazakhstan Ministry of Education and Science, grant 0828/GF4. We thank all (three) referees and Mark Lewis for their suggestions which were helpful to improve readability of the paper. We are very grateful to Farkhod Eshmatov for checking the English of this paper.

References

  1. Cabrera CY, Siles MM, Velasco MV (2016) Evolution algebras of arbitrary dimension and their decompositions. Linear Algebra Appl 495:122–162MathSciNetCrossRefzbMATHGoogle Scholar
  2. Casas JM, Ladra M, Omirov BA, Rozikov UA (2014) On evolution algebras. Algebra Colloq 21(2):331–342MathSciNetCrossRefzbMATHGoogle Scholar
  3. Devaney RL (2003) An introduction to chaotic dynamical system. Westview Press, BoulderzbMATHGoogle Scholar
  4. Galor O (2007) Discrete dynamical systems. Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. Ganikhodzhaev RN, Mukhamedov FM, Rozikov UA (2011) Quadratic stochastic operators and processes: results and open problems. Infin Dimens Anal Quantum Probab Relat Fields 14(2):279–335MathSciNetCrossRefzbMATHGoogle Scholar
  6. Lutambi AM, Penny MA, Smith T, Chitnis N (2013) Mathematical modelling of mosquito dispersal in a heterogeneous environment. Math Biosci 241(2):198–216MathSciNetCrossRefzbMATHGoogle Scholar
  7. Lyubich YI (1992) Mathematical structures in population genetics. Springer, BerlinCrossRefzbMATHGoogle Scholar
  8. Mayer CD (2000) Matrix analysis and applied linear algebra. SIAM, PhiladelphiaCrossRefGoogle Scholar
  9. Teschl G (2012) Ordinary differential equations and dynamical systems. American Mathematical Society, ProvidenceCrossRefzbMATHGoogle Scholar
  10. Tian JP (2008) Evolution algebras and their applications. Lecture notes in mathematics, vol 1921. Springer, BerlinGoogle Scholar
  11. Velasco MV. The Jacobson radical of an evolution algebra. J Spectr Theory (EMS) (http://www.ems-ph.org/journals/forthcoming.php?jrn=jst) (to appear)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsTashkentUzbekistan
  2. 2.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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