Existence and non-existence of asymmetrically rotating solutions to a mathematical model of self-propelled motion

Abstract

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.

On a numerical bifurcation analysis of a particle reaction-diffusion model for a motion of two self-propelled disks

“…In order to theoretically understand the mechanism of self-propelled material motion, much computer-aided and formal analysis using mathematical models have been investigated [32][33][34][35][36]. Furthermore, as rigorous mathematical analysis of the mathematical model ( 1), the global existence of solutions in the whole space and the existence of unique traveling wave solutions [37], rotational motion solutions for one-dimensional periodic boundary conditions in space have been shown for the piecewise constant function supply term [38]. In addition, the existence of weak solutions is shown for the delta function supply term [39].…”