论文标题
混合反应传输模型的多尖峰溶液
Multi-spike solutions of a hybrid reaction-transport model
论文作者
论文摘要
经典模式形成反应扩散系统的数值模拟表明它们通常在强烈的非线性方向上运行，最终的稳态由局部尖峰的空间重复模式组成。在激活器抑制剂系统中，例如两个组件Gierer-Meinhardt（GM）型号，可以考虑单个极限$ d_a \ ll d_h $，其中$ d_a $和$ d_h $是激活剂和抑制剂的扩散性。然后，渐近分析可用于分析多尖峰溶液的存在和线性稳定性。在本文中，我们分析了混合反应传输模型中的多峰值溶液，该模型由缓慢扩散的激活剂和主动运输的抑制剂组成，该抑制剂以右运动和左旋速度状态之间的速率$α$切换。最近引入了这类模型，以说明{\ em C. elegrans}幼虫发育过程中突触点的形成和体内稳态调节。我们利用了一个事实，即可以将混合模型映射到快速开关限制$α\ rightarrow \ infty $中的经典GM模型，这使我们能够确定多尖峰解决方案的存在。关于多峰值溶液的线性化导致非本地特征值问题，该问题用于得出有限$α$的多尖峰解决方案的稳定性条件
Numerical simulations of classical pattern forming reaction-diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady-state consisting of a spatially repeating pattern of localized spikes. In activator-inhibitor systems such as the two-component Gierer-Meinhardt (GM) model, one can consider the singular limit $D_a\ll D_h$, where $D_a$ and $D_h$ are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyze the existence and linear stability of multi-spike solutions. In this paper, we analyze multi-spike solutions in a hybrid reaction-transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate $α$ between right-moving and left-moving velocity states. This class of model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in {\em C. elegans}. We exploit the fact that that the hybrid model can be mapped onto the classical GM model in the fast switching limit $α\rightarrow \infty$, which allows us to establish the existence of multi-spike solutions. Linearization about the multi-spike solution leads to a non-local eigenvalue problem that is used to derive stability conditions for the multi-spike solution for finite $α$
