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. © 2013 Springer-Verlag Berlin Heidelberg.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The research leading to these results has received funding from the EuropeanResearch Council under theEuropean Community’sSeventh Framework Programme(FP7/2007–2013)/ERCgrant agreementNo. 239870. This publication was based on work supported in partby 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 FulfordJunior Research Fellowship; Brasenose College, University of Oxford, for a Nicholas Kurti JuniorFellowship; the Royal Society for a University Research Fellowship; and the Leverhulme Trust fora Philip Leverhulme Prize.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.