SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2783-2802, January 2020.
We study the effect of noise on dynamics of a single spike for the classical Gierer--Meinhardt model on a finite interval. When spatio-temporal noise is introduced in the equation for the activator, we derive a stochastic ODE that describes the motion of a single spike on a slow time scale. The steady state is described by a density distribution for spike positions, obtained via the corresponding Fokker--Planck PDE. For sufficiently small noise level, the spike performs random fluctuations near the center of the domain. As noise level is increased, the spike can deviate from the domain center but remains effectively “trapped” within a certain subinterval that includes the center. For even larger noise levels, the spike starts to undergo large excursions that eventually collide with the domain boundary and temporarily trap the spike there. By reformulating this problem in terms of mean first passage time, we derive the expected time for the spike to collide with the boundary. Several new open problems are also presented.

中文翻译：

---

存在噪声时的尖峰动力学

SIAM应用动力系统杂志，第19卷，第4期，第2783-2802页，2020年1月。
对于经典Gierer-Meinhardt模型在有限间隔内，我们研究了噪声对单个尖峰动力学的影响。当在激活器的方程式中引入时空噪声时，我们得到了一个随机ODE，它描述了单个尖峰在慢时间尺度上的运动。稳态是通过相应的Fokker-Planck PDE获得的尖峰位置的密度分布来描述的。对于足够小的噪声水平，尖峰在域中心附近执行随机波动。随着噪声水平的提高，尖峰可能会偏离磁畴中心，但仍有效地“困在”包含该中心的某个子间隔内。对于更大的噪声水平，尖峰开始经历较大的偏移，最终与域边界发生碰撞并暂时将尖峰捕获在此处。通过根据平均首次通过时间重新构造此问题，我们得出了峰值与边界碰撞的预期时间。还提出了几个新的开放问题。