Hybrid Modelling of Individual Movement and Collective Behaviour

Authored by Benjamin Franz, Radek Erban

Date Published: 2013

DOI: 10.1007/978-3-642-35497-7_5

Sponsors: European Union European Research Council (ERC)

Platforms: No platforms listed

Model Documentation: Other Narrative Flow charts Mathematical description

Model Code URLs: Model code not found

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.

Tags

Simulations systems Bacterial chemotaxis Animal groups Tumor-growth Populations Cellular-automaton model Escherichia-coli Equations Pattern-formation