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.