Corrected mean-field models for spatially-dependent advection-diffusion-reaction phenomena
Simpson, Matthew & Baker, Ruth (2011) Corrected mean-field models for spatially-dependent advection-diffusion-reaction phenomena. Physical Review E, 83(5), 051922-1--051922-15.
In the exclusion-process literature, mean-field models are often derived by assuming that the occupancy status of lattice sites is independent. Although this assumption is questionable, it is the foundation of many mean-field models. In this work we develop methods to relax the independence assumption for a range of discrete exclusion process-based mechanisms motivated by applications from cell biology. Previous investigations that focussed on relaxing the independence assumption have been limited to studying initially-uniform populations and ignored any spatial variations. By ignoring spatial variations these previous studies were greatly simplified due to translational invariance of the lattice. These previous corrected mean-field models could not be applied to many important problems in cell biology such as invasion waves of cells that are characterised by moving fronts. Here we propose generalised methods that relax the independence assumption for spatially inhomogeneous problems, leading to corrected mean-field descriptions of a range of exclusion process-based models that incorporate
(i) unbiased motility, (ii) biased motility, and (iii) unbiased motility with agent birth and death processes.
The corrected mean-field models derived here are applicable to spatially variable processes including invasion wave type problems. We show that there can be large deviations between simulation data and traditional mean-field models based on invoking the independence assumption. Furthermore, we show that the corrected mean-field models give an improved match to the simulation data in all cases considered.
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|Item Type:||Journal Article|
|Keywords:||cellular automata, exclusion process, mean-field, correlation|
|Subjects:||Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200)|
|Divisions:||Past > QUT Faculties & Divisions > Faculty of Science and Technology|
Current > Institutes > Institute of Health and Biomedical Innovation
Past > Schools > Mathematical Sciences
|Copyright Owner:||Copyright 2011 American Physical Society|
|Deposited On:||11 Apr 2011 15:17|
|Last Modified:||05 Jan 2012 10:06|
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