Ring patterns and their bifurcations in a nonlocal model of biological swarms
Ring patterns and their bifurcations in a nonlocal model of biological swarms
复制标题
生物群非局部模型中的环模式及其分叉
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
J. V. Brecht
中科院分区：
文献类型：
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作者：
A. Bertozzi;T. Kolokolnikov;Hui Sun;D. Uminsky;J. V. Brecht
本文研究了二维生物群集运动聚集模型的斑图形成问题。群体由粒子表示，动力学由具有局部排斥长程吸引结构的非局部相互作用势的梯度流驱动。我们回顾和扩大这类问题的最新发展，以及目前的新成果。在以前的工作中，我们利用一个共同的维度一制定的连续梯度ow来表征一般的相互作用内核的环解决方案的稳定性。在制度的长波不稳定性，我们表明，由此产生的基态是一个低模式的分叉远离环，并使用弱非线性分析提供的条件时，这种分叉是一个干草叉。在制度的短波不稳定性，我们表明，环破裂成完全二维基态的大粒子限制。我们分析了环的稳定性对粒子数的依赖性，并提供了复杂的多环分岔行为的例子，随着粒子数的增加。我们也能够提供一个解决方案的\设计师潜力的问题在2D。最后，我们描述了二阶动力学群集模型中旋转环的稳定性。
In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient ow of a non-local interaction potential which has a local repulsion long range attraction structure. We review and expand upon recent developments of this class of problems as well as present new results. As in previous work, we leverage a co-dimension one formulation of the continuum gradient ow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence of the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the \designer potential" problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model.
影响因子：
2.5
作者：
Balague, D.;Carrillo, J. A.;Raoul, G.
通讯作者：
Raoul, G.