Non-local kinetic and macroscopic models for self-organised animal aggregations
Non-local kinetic and macroscopic models for self-organised animal aggregations
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自组织动物聚集的非局部动力学和宏观模型
DOI：
10.3934/krm.2015.8.413
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
F. Hoffmann
中科院分区：
文献类型：
--
作者：
J. Carrillo;R. Eftimie;F. Hoffmann
在过去的二十年里，动力学和宏观模型的激增，用于研究自组织生物聚集的多尺度方面。因为结合到动力学模型中的个人层面的细节（例如，个体速度和转弯速率）使得它们在某种程度上难以研究，人们感兴趣的是通过使用由模型的生物学假设强加的各种缩放技术将这些模型转换成更简单的宏观模型。在这里，我们考虑三个缩放方法（抛物线，流体动力学和放牧碰撞限制），可用于减少一类非本地的1D和2D模型的生物聚集在文献中存在的更简单的模型。接下来，我们将研究如何通过这些缩放保留原始动力学模型所表现出的一些时空模式。为此，我们专注于非局部一维模型的抛物线标度，并应用渐近保持数值方法，这使我们能够分析的变化模式的标度系数$\$是从$\=1$（一维运输模型）到$\=0$（一维抛物模型）。我们发现，一些模式（描述固定的聚合）被保存在限制$\xA 1\0$，而其他模式（描述移动的聚合）在此限制中丢失。为了理解这些模式的损失，我们构建了分叉图。
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 $\epsilon$ 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 $\epsilon\to 0$, while other patterns (describing moving aggregations) are lost in this limit. To understand the loss of these patterns, we construct bifurcation diagrams.
DOI：
10.1137/130932272
发表时间：
2014-11
期刊：
SIAM J. Appl. Dyn. Syst.
影响因子：
--
作者：
P. Buono;R. Eftimie
通讯作者：
P. Buono;R. Eftimie