Skip to main content
SpringerLink
Log in

Hybrid Modelling of Individual Movement and Collective Behaviour

  • Chapter
  • First Online:
Dispersal, Individual Movement and Spatial Ecology

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2071))

Abstract

Mathematical models of dispersal in biological systems are often written in terms of partial differential equations (PDEs) which describe the time evolution of population-level variables (concentrations, densities). A more detailed modelling approach is given by individual-based (agent-based) models which describe the behaviour of each organism. In recent years, an intermediate modelling methodology—hybrid modelling—has been applied to a number of biological systems. These hybrid models couple an individual-based description of cells/animals with a PDE-model of their environment. In this chapter, we overview hybrid models in the literature with the focus on the mathematical challenges of this modelling approach. The detailed analysis is presented using the example of chemotaxis, where cells move according to extracellular chemicals that can be altered by the cells themselves. In this case, individual-based models of cells are coupled with PDEs for extracellular chemical signals. Travelling waves in these hybrid models are investigated. In particular, we show that in contrary to the PDEs, hybrid chemotaxis models only develop a transient travelling wave.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. J. Adler, Chemotaxis in bacteria. Science 153, 708–716 (1966)

    Google Scholar 

  2. J. Adler, Chemoreceptors in bacteria. Science 166, 1588–1597 (1969)

    Google Scholar 

  3. T. Alarcón, H. Byrne, P. Maini, A cellular automaton model for tumour growth in inhomogeneous environment. J. Theor. Biol. 225, 257–274 (2003)

    Article  Google Scholar 

  4. M. Alber, N. Chen, P. Lushnikov, S. Newman, Continuous macroscopic limit of a discrete stochastic model for interaction of living cells. Phys. Rev. Lett. 99, 168102 (2007)

    Article  Google Scholar 

  5. A. Anderson, A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol.: J. IMA 22, 163–186 (2005)

    Google Scholar 

  6. N. Barkai, S. Leibler, Bacterial chemotaxis - united we sense …Nature 393, 18–21 (1998)

    Google Scholar 

  7. H. Berg, How bacteria swim. Sci. Am. 233, 36–44 (1975)

    Article  Google Scholar 

  8. H. Berg, D. Brown, Chemotaxis in Esterichia coli analysed by three-dimensional tracking. Nature 239, 500–504 (1972)

    Article  Google Scholar 

  9. G. Bobashev, M. Goedecke, F. Yu, J. Epstein, A hybrid epidemic model: combining the advantages of agent-based and equation-based approaches. In Proceedings of the 2007 Winter Simulation Conference, ed. by S. Henderson (IEEE, New York, 2007), pp. 1532–1537

    Chapter  Google Scholar 

  10. J. Buhl, D. Sumpter, I. Couzin, J. Hale, E. Despland, E. Miller, S. Simpson, From disorder to order in marching locusts. Science 312, 1402–1406 (2006)

    Article  Google Scholar 

  11. P. Chavanis, Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations. Eur. Phys. J. B 62, 179–208 (2008)

    MATH  Google Scholar 

  12. I. Couzin, J. Krause, R. James, G. Ruxton, N. Franks, Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218(1), 1–11 (2002)

    Article  MathSciNet  Google Scholar 

  13. I. Couzin, J. Krause, N. Franks, S. Levin, Effective leadership and decision-making in animal groups on the move. Nature 433, 513–516 (2005)

    Article  Google Scholar 

  14. J. Dallon, H. Othmer, A discrete cell model with adaptive signalling for aggregation of dictyostelium discoideum. Phil. Trans. R. Soc. B: Biol. Sci. 352(1351), 391–417 (1997)

    Article  Google Scholar 

  15. R. Erban, From individual to collective behaviour in biological systems, Ph.D. thesis, University of Minnesota, 2005

    Google Scholar 

  16. R. Erban, S.J. Chapman, Reactive boundary conditions for stochastic simulations of reaction-diffusion processes. Phys. Biol. 4(1), 16–28 (2007)

    Article  Google Scholar 

  17. R. Erban, S.J. Chapman, Time scale of random sequential adsorption. Phys. Rev. E 75(4), 041116 (2007)

    Google Scholar 

  18. R. Erban, H. Othmer, From individual to collective behaviour in bacterial chemotaxis. SIAM J. Appl. Math. 65(2), 361–391 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Erban, H. Othmer, From signal transduction to spatial pattern formation in E. coli: a paradigm for multi-scale modeling in biology. Multiscale Model. Simul. 3(2), 362–394 (2005)

    Google Scholar 

  20. R. Erban, H. Othmer, Taxis equations for amoeboid cells. J. Math. Biol. 54(6), 847–885 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Erban, I. Kevrekidis, H. Othmer, An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal. Physica D 215(1), 1–24 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. M.Flegg, J. Chapman, R. Erban, Two Regime Method for optimizing stochastic reaction-diffusion simulations. J. R. Soc. Interface. 9, 859–868 (2012)

    Article  Google Scholar 

  23. B. Franz, Synchronisation Properties of an Agent-Based Animal Behaviour Model, Master’s thesis, University of Oxford, 2009

    Google Scholar 

  24. P. Gerlee, A. Anderson, An evolutionary hybrid cellular automaton model of solid tumour growth. J. Theor. Biol. 246, 583–603 (2007)

    Article  MathSciNet  Google Scholar 

  25. R. Grima, Multiscale modelling of biological pattern formation. Curr. Top. Dev. Biol. 81, 435–460 (2008)

    Article  Google Scholar 

  26. Z. Guo, P. Sloot, J.C. Tay, A hybrid agent-based approach for modeling microbiological systems. J. Theor. Biol. 255(2), 163–175 (2008)

    Article  MathSciNet  Google Scholar 

  27. D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic. Nature 407, 487–490 (2000)

    Article  Google Scholar 

  28. F. Hellweger, V. Bucci, A bunch of tiny individuals: individual-based modeling for microbes. Ecol. Model. 220, 8–22 (2009)

    Article  Google Scholar 

  29. A. Heppenstall, A. Evans, M. Birkin, A hybrid multi-agent/spatial interaction model system for petrol price setting. Trans. GIS 9, 35–51 (2005)

    Article  Google Scholar 

  30. A. Heppenstall, A. Evans, M. Birkin, Using hybrid agent-based systems to model spatially-influenced retail markets. J. Artif. Soc. Soc. Simul. 9, 2 (2006)

    Google Scholar 

  31. A. Heppenstall, A. Evans, M. Birkin, Genetic algorithm optimisation of an agent-based model for simulating a retail market. Environ. Plann. B: Plann. Des. 34, 1051–1070 (2007)

    Article  Google Scholar 

  32. D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences i. Jahresbericht der DMV 105(3), 103–165 (2003)

    MathSciNet  MATH  Google Scholar 

  33. N. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd edn. (North-Holland, Amsterdam, 2007)

    Google Scholar 

  34. G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations. Stud. Math. 143, 175–197 (2000)

    MathSciNet  MATH  Google Scholar 

  35. E. Keller, L. Segel, Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)

    Article  Google Scholar 

  36. E. Keller, L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)

    Article  Google Scholar 

  37. J. Landsberg, R. Waring, A generalised model of forest productivity using simplified concepts of radiation-use efficiency, carbon balance and partitioning. Forest Ecol. Manag. 95, 209–228 (1997)

    Article  Google Scholar 

  38. R. Lui, Z.A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models. J. Math. Biol. 61(5), 739–761 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Murray, Mathematical Biology (Springer, Berlin, 2002)

    MATH  Google Scholar 

  40. T. Newman, R. Grima, Many-body theory of chemotactic cell-cell interactions. Phys. Rev. E 70, 051916 (2004)

    Article  Google Scholar 

  41. P. O’Rourke, J. Brackbill, On particle-grid interpolation and calculating chemistry in particle-in-cell methods. J. Comput. Phys. 103, 37–52 (1993)

    Article  MathSciNet  Google Scholar 

  42. J. Osborne, A. Walter, S. Kershaw, G. Mirams, A. Fletcher, P. Pathmanathan, D. Gavaghan, O. Jensen, P. Maini, H. Byrne, A hybrid approach to multi-scale modelling of cancer. Phil. Trans. R. Soc. A: Math. Phys. Eng. Sci. 368, 5013–5028 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. H. Othmer, S. Dunbar, W. Alt, Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Patel, E. Gawlinski, S. Lemieux, R. Gatenby, A cellular automaton model of early tumor growth and invasion. J. Theor. Biol. 213, 315–331 (2001)

    Article  MathSciNet  Google Scholar 

  45. B. Ribba, T. Alarcón, K. Marron, P. Maini, Z. Agur, The use of hybrid cellular automaton models for improving cancer therapy, in ACRI 2004, ed. by P. Sloot, B. Chopard, A. Hoekstra. Lecture Notes in Computer Science, vol. 3305 (Springer, Berlin, 2004), pp. 444–453

    Google Scholar 

  46. F. Schweitzer, L. Schimansky-Geier, Clustering of “active” walkers in a two-component system. Physica A 206, 359–379 (1994)

    Article  Google Scholar 

  47. K. Smallbone, R. Gatenby, R. Gillies, P. Maini, D. Gavaghan, Metabolic changes during carcinogenesis: potential impact on invasiveness. J. Theor. Biol. 244, 703–713 (2007)

    Article  MathSciNet  Google Scholar 

  48. P. Spiro, J. Parkinson, H. Othmer, A model of excitation and adaptation in bacterial chemotaxis. Proc. Natl. Acad. Sci. USA 94, 7263–7268 (1997)

    Article  Google Scholar 

  49. A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61, 183–212 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. D. Sumpter, The principles of collective animal behaviour. Phil. Trans. R. Soc. B: Biol. Sci. 361, 5–22 (2006)

    Article  Google Scholar 

  51. D. Sumpter, Collective Animal Behavior (Princeton University Press, Princeton, 2010)

    MATH  Google Scholar 

  52. M. Wand, M. Jones, Kernel Smoothing (Chapman & Hall/CRC, London, 1994)

    Google Scholar 

  53. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (MIT, Cambridge, 1964)

    Google Scholar 

  54. M. Wooldridge, An Introduction to Multi-Agent Systems (Wiley, New York, 2002)

    Google Scholar 

  55. C. Xue, H. Othmer, Multiscale models of taxis-driven patterning in bacterial populations. SIAM J. Appl. Math. 70(1), 133–167 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  56. C. Xue, H.J. Hwang, K. Painter, R. Erban, Travelling waves in hyperbolic chemotaxis equations. Bull. Math. Biol. 1, 1–29 (2010)

    Google Scholar 

  57. C. Xue, E. Budrene, H. Othmer, Radial and spiral stream formation in Proteus mirabilis colonies. PLoS Comput. Biol. e1002332 (2011)

    Google Scholar 

Download references

Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement No. 239870. This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). RE would also like to thank Somerville College, University of Oxford, for a Fulford Junior Research Fellowship; Brasenose College, University of Oxford, for a Nicholas Kurti Junior Fellowship; the Royal Society for a University Research Fellowship; and the Leverhulme Trust for a Philip Leverhulme Prize.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Benjamin Franz & Radek Erban

Authors
  1. Benjamin Franz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radek Erban
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Radek Erban .

Editor information

Editors and Affiliations

  1. Mathematical & Statistical Sciences, University of Alberta, Central Academic Building 632, Edmonton, T6G 2G1, Alberta, Canada

    Mark A. Lewis

  2. Mathematical Institute, Centre for Mathematical Biology, University of Oxford, St. Giles Street 24-29, Oxford, OX1 3LB, United Kingdom

    Philip K. Maini

  3. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Sergei V. Petrovskii

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Franz, B., Erban, R. (2013). Hybrid Modelling of Individual Movement and Collective Behaviour. In: Lewis, M., Maini, P., Petrovskii, S. (eds) Dispersal, Individual Movement and Spatial Ecology. Lecture Notes in Mathematics(), vol 2071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35497-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation