2014/07/08 by José A. Carrillo, Raluca Eftimie, Carrillo, José A. +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #35A99 #35Q92 #Analysis of PDEs (math.AP) #Evolution and Genetic Dynamics #FOS: Biological sciences #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE)
paper · pdf · doi:10.48550/arxiv.1407.2099
openalex publication_date 2014/07/08 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
The last two decades have seen a surge in kinetic and macroscopic models\nderived to investigate the multi-scale aspects of self-organised biological\naggregations. Because the individual-level details incorporated into the\nkinetic models (e.g., individual speeds and turning rates) make them somewhat\ndifficult to investigate, one is interested in transforming these models into\nsimpler macroscopic models, by using various scaling techniques that are\nimposed by the biological assumptions of the models. Here, we consider three\nscaling approaches (parabolic, hydrodynamic and grazing collision limits) that\ncan be used to reduce a class of non-local 1D and 2D models for biological\naggregations to simpler models existent in the literature. Next, we investigate\nhow some of the spatio-temporal patterns exhibited by the original kinetic\nmodels are preserved via these scalings. To this end, we focus on the parabolic\nscaling for non-local 1D models and apply asymptotic preserving numerical\nmethods, which allow us to analyse changes in the patterns as the scaling\ncoefficient \ε is varied from \ε=1 (for 1D transport models) to\n\ε=0 (for 1D parabolic models). We show that some patterns (describing\nstationary aggregations) are preserved in the limit \ε\→ 0, while\nother patterns (describing moving aggregations) are lost in this limit. To\nunderstand the loss of these patterns, we construct bifurcation diagrams.\n