Mathematical models for self-propelled motions are often utilized for understanding the mechanism of collective motions observed in biological systems. Indeed, several patterns of collective motions of camphor disks have been reported in experimental systems. In this paper, we show the existence of asymmetrically rotating solutions of a two-camphor model and give necessary conditions for their existence and non-existence. The main theorem insists that the function describing the surface tension should have a concave part so that asymmetric motions of two camphor disks appear. Our result provides a clue for the dependence between the surfactant concentration and the surface tension in the mathematical model, which is difficult to be measured in experiments.

中文翻译：

---

自走运动数学模型非对称旋转解的存在与不存在

自推进运动的数学模型通常用于理解在生物系统中观察到的集体运动的机制。事实上，在实验系统中已经报道了樟脑盘的几种集体运动模式。在本文中，我们证明了双樟脑模型的非对称旋转解的存在，并给出了它们存在和不存在的必要条件。主定理坚持描述表面张力的函数应该有一个凹面，从而出现两个樟脑盘的不对称运动。我们的结果为数学模型中表面活性剂浓度与表面张力之间的依赖性提供了线索，这在实验中是难以测量的。