Skip to main content
SpringerLink
Log in

Numerical Optimization Identification of a Keller-Segel Model for Thermoregulation in Honey Bee Colonies in Winter

  • Conference paper
  • First Online:
Modelling and Development of Intelligent Systems (MDIS 2022)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1761))

Abstract

The present work is inspired by laboratory experiments providing measurements in a few places of the hive for a long period of time. Based on Keller-Segel model in form of coupled nonlinear parabolic equations for the local temperature T and the bee density \(\rho \ge 0\), using the real data, we investigate numerically the thermoregulation in honey bee colonies in winter. We propose a numerical approach, based on conjugate gradient method into two stages: first, we solve a semilinear parabolic inverse problem to recover the density \(\rho \) and the temperature T. On the second stage we solve the strongly nonlinear convection-diffusion equation to recover again the density \(\rho \). The numerical tests show the efficiency of the method at the calibration of thermoregulation model.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alifanov, O.M., Artioukhine, E.A., Rumyantsev, S.V.: Extreme Methods for Solving Ill-posed problems with Applications to Inverse Heat Transfer Problems. Begell House, New York (1995)

    MATH  Google Scholar 

  2. Atanasov, A.Z., Koleva, M.N., Vulkov, L.G.: Numerical analysis of thermoregulation in honey bee colonies in winter based on sign-changing chemotactic coefficient model. In: Springer Proceedings in Mathematics and Statistics, submitted, July 2022

    Google Scholar 

  3. Atanasov, A.Z., Georgiev, S.G., Vulkov, L.G.: Reconstruction analysis of honeybee colony collapse disorder modeling. Optim. Eng. 22(4), 2481–2503 (2021). https://doi.org/10.1007/s11081-021-09678-0

    Article  MathSciNet  MATH  Google Scholar 

  4. Bagheri, S., Mirzaie, M.: A mathematical model of honey bee colony dynamics to predict the effect of pollen on colony failure. PLoS ONE 14(11), e0225632 (2019)

    Article  Google Scholar 

  5. Bakhvalov, N.S.: Numerical Methods, p. 663. Mir Publishers, Moscow (1977). Translated from Russian to English

    Google Scholar 

  6. Bastaansen, R., Doelman, A., van Langevede, F., Rottschafer, V.: Modeling honey bee colonies in winter using a Keller-Segel model with a sign-changing chemotactic coefficient. SIAM J. Appl. Math. 80(20), 839–863 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller-Segel models of patern formulation in biological tissues. Math. Models Methods Appl. Sci. 25(09), 1663–1763 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calovi, M., Grozinger, C.M., Miller, D.A., Goslee, S.C.: Summer weather conditions influence winter survival of honey bees (Apis mellifera) in the northeastern United States. Sci. Rep. 11, 1553 (2021)

    Article  Google Scholar 

  9. Cao, K., Lesnic, D.: Reconstruction of the perfusion coefficient from temperature measurements using the conjugate gradient method. Int. J. Comput. Math. 95(4), 797–814 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chavent, G.: Nonlinear Least Squares for Inverse Problems: Theoretical Foundation and Step-by Guide for Applications. Springer, Cham (2009)

    MATH  Google Scholar 

  11. Chen, J., DeGrandi-Hoffman, G., Ratti, V., Kang, Y.: Review on mathematical modeling of honeybee population dynamics. Math. Biosci. Eng. 18(6), 9606–9650 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Esch, H.: Über die körpertemperaturen und den wärmehaushalt von Apis mellifica. Z. Vergleich. Physiol. 43, 305–335 (1960). https://doi.org/10.1007/BF00298066

    Article  Google Scholar 

  13. Fakhraie, M., Shidfar, A., Garshasbi, M.: A computational procedure for estimation of an unknown coefficient in an inverse boundary value problem. Appl. Math. Comput. 187, 1120–1125 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Guzman-Novoa, E., Eccles, C.Y., McGowan, J., Kelly, P.G., Correa- Bentez, A.: Varroa destructor is the main culprit for the death and reduced populations of overwintered honey bee (Apis mellifera) colonies in Ontario, Canada. Apidologie 41, 443–450 (2010)

    Article  Google Scholar 

  15. Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems, p. 144. Chapman and Hall/CRC, NewYork (1995)

    MATH  Google Scholar 

  16. Hasanov, A., Romanov, V.: Introduction to the Inverse Problems for Differential Equations. Springer, NewYork (2017). https://doi.org/10.1007/978-3-319-62797-7

    Book  MATH  Google Scholar 

  17. Heinrich, B.: Energetics of honeybee swarm thermoregulation. Science 212, 565–566 (1981)

    Article  Google Scholar 

  18. Kevan, P.G., Guzman, E., Skinner A., van Engelsdorp, D.: Colony collapse disorder in Canada: do we have a problem? HiveLights, pp. 14–16, May (2007). http://hdl.handle.net/10214/2418

  19. Krivorotko, O., Kabanikhin, S., Zhang, S., Kashtanova, V.: Global and local optimization in identification of parabolic systems. J. Inverse Ill-Posed Prob. 28(6), 899–913 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lemke, M., Lamprecht, A.: A model of heat production and thermoregulation in winter clusters of honey bee using differential heat conduction equations. J. Theor. Biol. 142(2), 261–273 (1990)

    Article  Google Scholar 

  21. Lesnic, D.: Inverse Problems with Applications in Science and Engineering. CRC Press, London (2020)

    MATH  Google Scholar 

  22. Ocko, S.A., Mahadevan, L.: Collective thermoregulation in bee clusters. J. R. Soc. Interface 11(91), 20131033 (2014)

    Article  Google Scholar 

  23. Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc, New York (2001)

    Book  MATH  Google Scholar 

  24. Peters, J.M., Peleg, O., Mahadevan, L.: Collective ventilation in honeybee nest. J. R. Soc. Interface 16(150), 20180561 (2019)

    Article  Google Scholar 

  25. Ratti, V., Kevan, P.G., Eberl, H.J.: A mathematical model of forager loss in honeybee colonies infested with varroa destructor and the acute bee paralysis virus. Bull. Math. Biol. 79(6), 1218–1253 (2017). https://doi.org/10.1007/s11538-017-0281-6

    Article  MathSciNet  MATH  Google Scholar 

  26. Stabentheiner, A., Kovac, H., Brodschneider, R.: Honeybee colony thermoregulation regulatory mechanisms and contribution of individuals in dependence on age, location and thermal stress. PLoS ONE 5(1), e8967 (2010)

    Article  Google Scholar 

  27. Tautz, J.: The Buzz about Bees: Biology of a Superorganism. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-78729-7

    Book  Google Scholar 

  28. Wang, B., Liu, J.: Recovery of thermal conductivity in two-dimensional media with nonlinear source by optimizations. Appl. Math. Lett. 60, 73–80 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Watmough, J., Camazine, S.: Self-organized thermoregulation of honeybee clusters. J. Theor. Biol. 176(3), 391–402 (1995)

    Article  Google Scholar 

Download references

Acknowledgment

This work is supported by the Bulgarian National Science Fund under the Project KP-06-PN 46-7 “Design and research of fundamental technologies and methods for precision apiculture”.

Author information

Authors and Affiliations

  1. University of Ruse, 8 Studentska str., 7017, Ruse, Bulgaria

    Atanas Z. Atanasov, Miglena N. Koleva & Lubin Vulkov

Authors
  1. Atanas Z. Atanasov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miglena N. Koleva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lubin Vulkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miglena N. Koleva .

Editor information

Editors and Affiliations

  1. Lucian Blaga University of Sibiu, Sibiu, Romania

    Dana Simian

  2. Lucian Blaga University of Sibiu, Sibiu, Romania

    Laura Florentina Stoica

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Atanasov, A.Z., Koleva, M.N., Vulkov, L. (2023). Numerical Optimization Identification of a Keller-Segel Model for Thermoregulation in Honey Bee Colonies in Winter. In: Simian, D., Stoica, L.F. (eds) Modelling and Development of Intelligent Systems. MDIS 2022. Communications in Computer and Information Science, vol 1761. Springer, Cham. https://doi.org/10.1007/978-3-031-27034-5_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-27034-5_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-27033-8

  • Online ISBN: 978-3-031-27034-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation