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 ≪ D_h, where D_a and D_h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse 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. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans. We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α → ∞ with a graphical construction for finite α.

经典模式形成反应扩散系统的仿真表明，它们通常在强烈的非线性状态下运行，最终的稳态由局部尖峰的空间重复模式组成。在活化剂-抑制剂系统，例如双组分Gierer-迈进（GM）模型，可以考虑单数限制d_一个 «  d _ħ，其中d_一和d _ħ分别是激活剂和抑制剂的扩散率。然后可以使用渐近分析来分析多尖峰解的存在性和线性稳定性。在本文中，我们在混合反应-运输模型中分析了多峰解决方案，该模型由缓慢扩散的活化剂和主动转移的抑制剂组成，后者以速率α在右移和左移速度状态之间切换。最近引入了这样的模型，以解决秀丽隐杆线虫幼虫发育过程中突触点的形成和稳态调节。我们利用以下事实：可以在快速切换极限α中将混合模型映射到经典GM模型上 →∞，它建立了多尖峰解的存在。关于多钉解决方案的线性化产生了一个非局部特征值问题，该问题用于通过将α  →∞的分析结果与有限α的图形构造相结合来研究多钉解决方案的稳定性。