Plant autotoxicity has proved to play an essential role in the behaviour of local vegetation. We analyse a reaction–diffusion-ODE model describing the interactions between vegetation, water, and autotoxicity. The presence of autotoxicity is seen to induce movement and deformation of spot patterns in some parameter regimes, a phenomenon which does not occur in classical biomass–water models. We aim to analytically quantify this novel feature, by studying travelling wave solutions in one spatial dimension. We use geometric singular perturbation theory to prove the existence of symmetric, stationary and non-symmetric, travelling pulse solutions, by constructing appropriate homoclinic orbits in the associated 5-dimensional dynamical system. In the singularly perturbed context, we perform an extensive scaling analysis of the dynamical system, identifying multiple asymptotic scaling regimes where (travelling) pulses may or may not be constructed. We show that, while the analytically constructed stationary pulse corresponds to its numerical counterpart, there is a discrepancy between the numerically observed travelling pulse and its analytical counterpart. Our findings indicate how the inclusion of an additional ODE may significantly influence the properties of classical biomass–water models of Klausmeier/Gray–Scott type.

植物自毒已被证明在当地植被的行为中起着至关重要的作用。我们分析了描述植被、水和自毒作用之间相互作用的反应扩散 ODE 模型。在某些参数范围内，自体毒性的存在会引起斑点模式的运动和变形，这种现象在经典的生物质-水模型中不会发生。我们的目标是通过研究一个空间维度的行波解来分析量化这一新特征。我们通过在相关的 5 维动力系统中构建适当的同宿轨道，使用几何奇异摄动理论来证明对称、平稳和非对称的行进脉冲解的存在。在奇异扰动的背景下，我们对动力系统进行了广泛的缩放分析，识别多个渐近缩放制度，其中（旅行）脉冲可能会或可能不会被构造。我们表明，虽然分析构造的静止脉冲与其数值对应物相对应，但数值观察到的行进脉冲与其解析对应物之间存在差异。我们的研究结果表明，包含额外的 ODE 可能如何显着影响 Klausmeier/Gray-Scott 类型的经典生物质 - 水模型的特性。