An investigation of a nonlocal hyperbolic model for self-organization of biological groups
An investigation of a nonlocal hyperbolic model for self-organization of biological groups
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DOI：
10.1007/s00285-009-0311-6
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发表时间：
2010-10-01
影响因子：
1.9
通讯作者：
Eftimie, Raluca
中科院分区：
文献类型：
--
作者：
Fetecau, Razvan C.;Eftimie, Raluca
在这篇文章中，我们引入并研究了一种新的非局部双曲线模型，用于动物聚集的形成和移动。我们假设个体之间的非局部吸引、排斥和排列相互作用可以影响群体成员的速度和转向率。我们使用解析和数值技术来研究这些非局域相互作用对模型所展示的图案的长期行为的影响。证明了非线性双曲型方程组局部解的存在唯一性，并证明了该系统不存在激波解(梯度爆破)。根据吸引和排斥的相对大小，我们证明了该模型的解在时间上是全局存在的，或者可能呈现有限时间的振幅爆破。我们数值地说明了模型中的色散聚集、有限大小群和爆破模式所显示的各种模式，后者对应于可能坍塌到一点的聚集。从有限大小到爆炸模式的转变受社会相互作用的大小和随机翻转速度的控制。这些类型的模式的存在和没有冲击是关于速度和翻转速度函数的形式的生物学相关假设的结果，以及描述社会相互作用的核。
In this article, we introduce and study a new nonlocal hyperbolic model for the formation and movement of animal aggregations. We assume that the nonlocal attractive, repulsive, and alignment interactions between individuals can influence both the speed and the turning rates of group members. We use analytical and numerical techniques to investigate the effect of these nonlocal interactions on the long-time behavior of the patterns exhibited by the model. We establish the local existence and uniqueness and show that the nonlinear hyperbolic system does not develop shock solutions (gradient blow-up). Depending on the relative magnitudes of attraction and repulsion, we show that the solutions of the model either exist globally in time or may exhibit finite-lime amplitude blow-up. We illustrate numerically the various patterns displayed by the model dispersive aggregations, finite-size groups and blow-up patterns, the latter corresponding to aggregations which may collapse to a point. The transition from finite-size to blow-up patterns is governed by the magnitude of the social interactions and the random turning rates The presence of these types of patterns and the absence of shocks are consequences of the biologically relevant assumptions regarding the form of the speed and the turning rate functions, as well as of the kernels describing the social interactions.