An analysis is undertaken of the formation and stability of localised patterns in a 1D Schanckenberg model, with source terms in both the activator and inhibitor fields. The aim is to illustrate the connection between semi-strong asymptotic analysis and the theory of localised pattern formation within a pinning region created by a subcritical Turing bifurcation. A two-parameter bifurcation diagram of homogeneous, periodic and localised patterns is obtained numerically. A natural asymptotic scaling for semi-strong interaction theory is found where an activator source term \[a = O(\varepsilon )\] and the inhibitor source \[b = O({\varepsilon ^2})\], with ε2 being the diffusion ratio. The theory predicts a fold of spike solutions leading to onset of localised patterns upon increase of b from zero. Non-local eigenvalue arguments show that both branches emanating from the fold are unstable, with the higher intensity branch becoming stable through a Hopf bifurcation as b increases beyond the \[O(\varepsilon )\] regime. All analytical results are found to agree with numerics. In particular, the asymptotic expression for the fold is found to be accurate beyond its region of validity, and its extension into the pinning region is found to form the low b boundary of the so-called homoclinic snaking region. Further numerical results point to both sub and supercritical Hopf bifurcation and novel spikeinsertion dynamics.

中文翻译：

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半强相互作用机制中新型 Schnakenberg 模型的尖峰和局部模式

对 1D Schanckenberg 模型中局部模式的形成和稳定性进行了分析，在激活剂和抑制剂领域都有源项。目的是说明半强渐近分析与亚临界图灵分岔产生的钉扎区域内局部模式形成理论之间的联系。数值得到了均匀、周期性和局部模式的两参数分岔图。发现半强相互作用理论的自然渐近标度，其中激活源项\[a = O(\varepsilon )\]和抑制剂来源\[b = O({\varepsilon ^2})\]， 和ε2为扩散比。该理论预测峰值溶液的折叠导致局部模式在增加时出现b从零开始。非局部特征值参数表明，从折叠发出的两个分支都是不稳定的，较高强度的分支通过 Hopf 分岔变得稳定b增加超过\[O(\伐普西隆)\]政权。发现所有分析结果都与数字一致。特别是，折叠的渐近表达式被发现在其有效区域之外是准确的，并且发现其扩展到钉扎区域形成了低b所谓的同宿蛇行区域的边界。进一步的数值结果指向亚临界和超临界 Hopf 分岔和新的尖峰插入动力学。