We explore the dynamics of invasive weeds by partial differential equation (PDE) modelling and applying dynamical system and phase portrait techniques. We begin by applying the method of characteristics to a preexisting PDE model of the spreading of T. fluminensis , an invasive weed which has been responsible for native forest depletion. We explore the system both at particular points in space and over all of space, in one dimension, as a function of time. Our model suggests that an increase in the rate of spread of the weed through space will increase the efficacy of control measures taken at the weed’s spatial boundary. We then propose new competition models based on the previous model and explore the existence of travelling wave solutions. These models represent both the cases with (i) a competing native plant species which spreads through the forest and (ii) a non-mobile, established native plant species. In the former case, the model suggests that an increased mass-action coefficient between the competing species is sufficient and necessary for the transition of the forest into a state of coexistence. In the latter case, the result is not as strong: a sufficiently large rate of competition between the species excludes the possibility of native plant extinction and hence suggests that forest depletion will not occur, but does not imply coexistence. We perform some numerical simulations to support our analytic results. In all cases, we give a discussion on the physical and biological interpretations of our results. We conclude with some suggestions for future work and with a discussion of the advantages and disadvantages of the methods.

中文翻译：

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竞争性入侵杂草动力学的数学建模

我们通过偏微分方程 (PDE) 建模和应用动力系统和相图技术探索侵入性杂草的动态。我们首先将特征方法应用于 T.fluminensis 传播的预先存在的 PDE 模型，T.fluminensis 是一种导致原生森林枯竭的入侵杂草。我们在空间中的特定点和整个空间，在一维中，作为时间的函数来探索系统。我们的模型表明，增加杂草在空间中的传播速度将提高在杂草空间边界采取的控制措施的有效性。然后，我们在先前模型的基础上提出新的竞争模型，并探索行波解的存在。这些模型代表了以下两种情况：（i）在森林中传播的竞争性本地植物物种和（ii）非移动的、已建立的本地植物物种。在前一种情况下，模型表明竞争物种之间增加的质量作用系数对于森林过渡到共存状态是充分和必要的。在后一种情况下，结果不那么强烈：物种之间足够大的竞争率排除了本地植物灭绝的可能性，因此表明不会发生森林枯竭，但并不意味着共存。我们进行了一些数值模拟来支持我们的分析结果。在所有情况下，我们都会对结果的物理和生物学解释进行讨论。