arXiv2014
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. Here, we consider three scaling approaches (parabolic, hydrodynamic 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. Next, 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 $ε$ is varied from $ε=1$ (for 1D transport models) to $ε=0$ (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit $ε\to 0$, while other patterns (describing moving aggregations) are lost in this limit. To understand the loss of these patterns, we construct bifurcation diagrams.