Journal
KINETIC AND RELATED MODELS
Volume 8, Issue 3, Pages 413-441Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/krm.2015.8.413
Keywords
Self-organised aggregations; kinetic models; macroscopic models; non-local interactions; asymptotic preserving methods
Categories
Funding
- Engineering and Physical Sciences Research Council (UK) [EP/K008404/1, EP/K033689/1, EP/H023348/1]
- Royal Society through a Wolfson Research Merit Award
- [MTM2011-27739-C04-02]
- Engineering and Physical Sciences Research Council [1360938, EP/K033689/1, EP/K008404/1] Funding Source: researchfish
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The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient epsilon is varied from epsilon = 1 (for 1D transport models) to epsilon = 0 (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit epsilon -> 0, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams.
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