Reaction–diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., 2017), i.e. forward–backward–forward diffusion. Numerical simulations suggest these RDEs support shock-fronted travelling waves when the reaction term includes an Allee effect. In this work we formalise these preliminary numerical observations by analysing the shock-fronted travelling waves through embedding the RDE into a larger class of higher order partial differential equations (PDEs). Subsequently, we use geometric singular perturbation theory to study this larger class of equations and prove the existence of these shock-fronted travelling waves. Most notable, we show that different embeddings yield shock-fronted travelling waves with different properties.

反应扩散方程（RDE）通常作为基于晶格的离散模型的连续极限。最近，有人提出了一种离散模型，该模型允许运动，增殖和死亡的速度取决于是否分离出这些药物，并且这种方法给出了各种RDE，其中扩散项是凸的并且可以变为负数（Johnston等，2017）。 ），即向前-向后-向前扩散。数值模拟表明，当反应项包括Allee效应时，这些RDE支持冲击波行进。在这项工作中，我们通过将RDE嵌入较大类的高阶偏微分方程（PDE）中来分析激波传播波，从而形式化了这些初步的数值观测。随后，我们使用几何奇异摄动理论来研究这一更大的方程组，并证明这些冲击波前行波的存在。最值得注意的是，我们证明了不同的嵌入会产生具有不同特性的震荡行波。